Planet Musings

February 20, 2017

David Hoggstellar twins and stellar age indicators

In the stars group meeting at CCA, Keith Hawkins (Columbia) blew us away with examples of stellar twins, identified with HARPS spectra. They were chosen to have identical derived spectroscopic parameters in three or four labels, but were amazingly identical at signal-to-noise of hundreds. He then showed us some he found in the APOGEE data, using very blunt tools to identify twins. This led to a long discussion of what we could do with twins, and things we expect to find in the data, especially regarding failures of spectroscopic twins to be identical in other respects, and failures of twins identified through means other than spectroscopic to be identical spectroscopically. Lots to do!

This was followed by Ruth Angus (Columbia) walking us through all the age-dating methods we have found for stars. The crowd was pretty unimpressed with many of our age indicators! But they agreed that we should take a self-calibration approach to assemble them and cross-calibrate them. It also interestingly connects to the twins discussion that preceded. Angus and I followed the meeting with a more detailed discussion about our plans, in part so that she can present them in a talk in her near future.

BackreactionFake news wasn’t hard to predict – But what’s next?

In 2008, I wrote a blogpost which began with a dark vision – a presidential election led astray by fake news. I’m not much of a prophet, but it wasn’t hard to predict. Journalism, for too long, attempted the impossible: Make people pay for news they don’t want to hear. It worked, because news providers, by and large, shared an ethical code. Journalists aspired to tell the truth; their

John PreskillIt’s CHAOS!

My brother and I played the video game Sonic the Hedgehog on a Sega Dreamcast. The hero has spiky electric-blue fur and can run at the speed of sound.1 One of us, then the other, would battle monsters. Monster number one oozes onto a dark city street as an aquamarine puddle. The puddle spreads, then surges upward to form limbs and claws.2 The limbs splatter when Sonic attacks: Aqua globs rain onto the street.

chaos-vs-sonic

The monster’s master, Dr. Eggman, has ginger mustachios and a body redolent of his name. He scoffs as the heroes congratulate themselves.

“Fools!” he cries, the pauses in his speech heightening the drama. “[That monster is] CHAOS…the GOD…of DE-STRUC-TION!” His cackle could put a Disney villain to shame.

Dr. Eggman’s outburst comes to mind when anyone asks what topic I’m working on.

“Chaos! And the flow of time, quantum theory, and the loss of information.”

eggman

Alexei Kitaev, a Caltech physicist, hooked me on chaos. I TAed his spring-2016 course. The registrar calls the course Ph 219c: Quantum Computation. I call the course Topics that Interest Alexei Kitaev.

“What do you plan to cover?” I asked at the end of winter term.

Topological quantum computation, Alexei replied. How you simulate Hamiltonians with quantum circuits. Or maybe…well, he was thinking of discussing black holes, information, and chaos.

If I’d had a tail, it would have wagged.

“What would you say about black holes?” I asked.

untitled-2

Sonic’s best friend, Tails the fox.

I fwumped down on the couch in Alexei’s office, and Alexei walked to his whiteboard. Scientists first noticed chaos in classical systems. Consider a double pendulum—a pendulum that hangs from the bottom of a pendulum that hangs from, say, a clock face. Imagine pulling the bottom pendulum far to one side, then releasing. The double pendulum will swing, bend, and loop-the-loop like a trapeze artist. Imagine freezing the trapeze artist after an amount t of time.

What if you pulled another double pendulum a hair’s breadth less far? You could let the pendulum swing, wait for a time t, and freeze this pendulum. This pendulum would probably lie far from its brother. This pendulum would probably have been moving with a different speed than its brother, in a different direction, just before the freeze. The double pendulum’s motion changes loads if the initial conditions change slightly. This sensitivity to initial conditions characterizes classical chaos.

A mathematical object F(t) reflects quantum systems’ sensitivities to initial conditions. [Experts: F(t) can evolve as an exponential governed by a Lyapunov-type exponent: \sim 1 - ({\rm const.})e^{\lambda_{\rm L} t}.] F(t) encodes a hypothetical process that snakes back and forth through time. This snaking earned F(t) the name “the out-of-time-ordered correlator” (OTOC). The snaking prevents experimentalists from measuring quantum systems’ OTOCs easily. But experimentalists are trying, because F(t) reveals how quantum information spreads via entanglement. Such entanglement distinguishes black holes, cold atoms, and specially prepared light from everyday, classical systems.

Alexei illustrated, on his whiteboard, the sensitivity to initial conditions.

“In case you’re taking votes about what to cover this spring,” I said, “I vote for chaos.”

We covered chaos. A guest attended one lecture: Beni Yoshida, a former IQIM postdoc. Beni and colleagues had devised quantum error-correcting codes for black holes.3 Beni’s foray into black-hole physics had led him to F(t). He’d written an OTOC paper that Alexei presented about. Beni presented about a follow-up paper. If I’d had another tail, it would have wagged.

tails-2

Sonic’s friend has two tails.

Alexei’s course ended. My research shifted to many-body localization (MBL), a quantum phenomenon that stymies the spread of information. OTOC talk burbled beyond my office door.

At the end of the summer, IQIM postdoc Yichen Huang posted on Facebook, “In the past week, five papers (one of which is ours) appeared . . . studying out-of-time-ordered correlators in many-body localized systems.”

I looked down at the MBL calculation I was performing. I looked at my computer screen. I set down my pencil.

“Fine.”

I marched to John Preskill’s office.

boss

The bosses. Of different sorts, of course.

The OTOC kept flaring on my radar, I reported. Maybe the time had come for me to try contributing to the discussion. What might I contribute? What would be interesting?

We kicked around ideas.

“Well,” John ventured, “you’re interested in fluctuation relations, right?”

Something clicked like the “power” button on a video-game console.

Fluctuation relations are equations derived in nonequilibrium statistical mechanics. They describe systems driven far from equilibrium, like a DNA strand whose ends you’ve yanked apart. Experimentalists use fluctuation theorems to infer a difficult-to-measure quantity, a difference \Delta F between free energies. Fluctuation relations imply the Second Law of Thermodynamics. The Second Law relates to the flow of time and the loss of information.

Time…loss of information…Fluctuation relations smelled like the OTOC. The two had to join together.

on-button

I spent the next four days sitting, writing, obsessed. I’d read a paper, three years earlier, that casts a fluctuation relation in terms of a correlator. I unearthed the paper and redid the proof. Could I deform the proof until the paper’s correlator became the out-of-time-ordered correlator?

Apparently. I presented my argument to my research group. John encouraged me to clarify a point: I’d defined a mathematical object A, a probability amplitude. Did A have physical significance? Could anyone measure it? I consulted measurement experts. One identified A as a quasiprobability, a quantum generalization of a probability, used to model light in quantum optics. With the experts’ assistance, I devised two schemes for measuring the quasiprobability.

The result is a fluctuation-like relation that contains the OTOC. The OTOC, the theorem reveals, is a combination of quasiprobabilities. Experimentalists can measure quasiprobabilities with weak measurements, gentle probings that barely disturb the probed system. The theorem suggests two experimental protocols for inferring the difficult-to-measure OTOC, just as fluctuation relations suggest protocols for inferring the difficult-to-measure \Delta F. Just as fluctuation relations cast \Delta F in terms of a characteristic function of a probability distribution, this relation casts F(t) in terms of a characteristic function of a (summed) quasiprobability distribution. Quasiprobabilities reflect entanglement, as the OTOC does.

pra-image

Collaborators and I are extending this work theoretically and experimentally. How does the quasiprobability look? How does it behave? What mathematical properties does it have? The OTOC is motivating questions not only about our quasiprobability, but also about quasiprobability and weak measurements. We’re pushing toward measuring the OTOC quasiprobability with superconducting qubits or cold atoms.

Chaos has evolved from an enemy to a curiosity, from a god of destruction to an inspiration. I no longer play the electric-blue hedgehog. But I remain electrified.

 

1I hadn’t started studying physics, ok?

2Don’t ask me how the liquid’s surface tension rises enough to maintain the limbs’ shapes.

3Black holes obey quantum mechanics. Quantum systems can solve certain problems more quickly than ordinary (classical) computers. Computers make mistakes. We fix mistakes using error-correcting codes. The codes required by quantum computers differ from the codes required by ordinary computers. Systems that contain black holes, we can regard as performing quantum computations. Black-hole systems’ mistakes admit of correction via the code constructed by Beni & co. 


February 19, 2017

David Hoggabundance dimensionality, optimized photometric estimators

Kathryn Johnston (Columbia) organized a Local-Group meeting of locals, or a local group of Local Group researchers. There were various discussions of things going on in the neighborhood. Natalie Price-Jones (Toronto) started up a lot of discussion with her work on the dimensionality of chemical-abundance space, working purely with the APOGEE spectral data. That is, they are inferring the dimensionality without explicitly measuring chemical abundances or interpreting the spectra at all. Much of the questioning centered on how they know that the diversity they see is purely or primarily chemical rather than, say, instrumental or stellar nuisances.

At lunch time there were amusing things said at the Columbia Astro Dept Pizza Lunch. One was a very nice presentation by Benjamin Pope (Oxford) about how to do precise photometry of saturated stars in the Kepler data. He has developed a method that fully scoops me in one of my unfinished projects: The OWL, in which the pixel weights used in his soft-aperture aperture photometry are found through the optimization of a (very clever, in Pope's case) convex objective function. After the Lunch, we discussed a huge space of generalizations, some in the direction of more complex (but still convex) objectives, and others in the direction of train-and-test to ameliorate over-fitting.

John BaezAzimuth Backup Project (Part 4)

The Azimuth Climate Data Backup Project is going well! Our Kickstarter campaign ended on January 31st and the money has recently reached us. Our original goal was $5000. We got $20,427 of donations, and after Kickstarter took its cut we received $18,590.96.

Next time I’ll tell you what our project has actually been doing. This time I just want to give a huge “thank you!” to all 627 people who contributed money on Kickstarter!

I sent out thank you notes to everyone, updating them on our progress and asking if they wanted their names listed. The blanks in the following list represent people who either didn’t reply, didn’t want their names listed, or backed out and decided not to give money. I’ll list people in chronological order: first contributors first.

Only 12 people backed out; the vast majority of blanks on this list are people who haven’t replied to my email. I noticed some interesting but obvious patterns. For example, people who contributed later are less likely to have answered my email yet—I’ll update this list later. People who contributed more money were more likely to answer my email.

The magnitude of contributions ranged from $2000 to $1. A few people offered to help in other ways. The response was international—this was really heartwarming! People from the US were more likely than others to ask not to be listed.

But instead of continuing to list statistical patterns, let me just thank everyone who contributed.

thank-you-message2_edited-1

Daniel Estrada
Ahmed Amer
Saeed Masroor
Jodi Kaplan
John Wehrle
Bob Calder
Andrea Borgia
L Gardner

Uche Eke
Keith Warner
Dean Kalahan
James Benson
Dianne Hackborn

Walter Hahn
Thomas Savarino
Noah Friedman
Eric Willisson
Jeffrey Gilmore
John Bennett
Glenn McDavid

Brian Turner

Peter Bagaric

Martin Dahl Nielsen
Broc Stenman

Gabriel Scherer
Roice Nelson
Felipe Pait
Kenneth Hertz

Luis Bruno


Andrew Lottmann
Alex Morse

Mads Bach Villadsen
Noam Zeilberger

Buffy Lyon

Josh Wilcox

Danny Borg

Krishna Bhogaonker
Harald Tveit Alvestrand


Tarek A. Hijaz, MD
Jouni Pohjola
Chavdar Petkov
Markus Jöbstl
Bjørn Borud


Sarah G

William Straub

Frank Harper
Carsten Führmann
Rick Angel
Drew Armstrong

Jesimpson

Valeria de Paiva
Ron Prater
David Tanzer

Rafael Laguna
Miguel Esteves dos Santos 
Sophie Dennison-Gibby




Randy Drexler
Peter Haggstrom


Jerzy Michał Pawlak
Santini Basra
Jenny Meyer


John Iskra

Bruce Jones
Māris Ozols
Everett Rubel



Mike D
Manik Uppal
Todd Trimble

Federer Fanatic

Forrest Samuel, Harmos Consulting








Annie Wynn
Norman and Marcia Dresner



Daniel Mattingly
James W. Crosby








Jennifer Booth
Greg Randolph





Dave and Karen Deeter

Sarah Truebe










Jeffrey Salfen
Birian Abelson

Logan McDonald

Brian Truebe
Jon Leland






Sarah Lim







James Turnbull




John Huerta
Katie Mandel Bruce
Bethany Summer






Anna Gladstone



Naom Hart
Aaron Riley

Giampiero Campa

Julie A. Sylvia


Pace Willisson









Bangskij










Peter Herschberg

Alaistair Farrugia


Conor Hennessy




Stephanie Mohr




Torinthiel


Lincoln Muri 
Anet Ferwerda 


Hanna





Michelle Lee Guiney

Ben Doherty
Trace Hagemann







Ryan Mannion


Penni and Terry O'Hearn



Brian Bassham
Caitlin Murphy
John Verran






Susan


Alexander Hawson
Fabrizio Mafessoni
Anita Phagan
Nicolas Acuña
Niklas Brunberg

Adam Luptak
V. Lazaro Zamora






Branford Werner
Niklas Starck Westerberg
Luca Zenti and Marta Veneziano 


Ilja Preuß
Christopher Flint

George Read 
Courtney Leigh

Katharina Spoerri


Daniel Risse



Hanna
Charles-Etienne Jamme
rhackman41



Jeff Leggett

RKBookman


Aaron Paul
Mike Metzler


Patrick Leiser

Melinda

Ryan Vaughn
Kent Crispin

Michael Teague

Ben



Fabian Bach
Steven Canning


Betsy McCall

John Rees

Mary Peters

Shane Claridge
Thomas Negovan
Tom Grace
Justin Jones


Jason Mitchell




Josh Weber
Rebecca Lynne Hanginger
Kirby


Dawn Conniff


Michael T. Astolfi



Kristeva

Erik
Keith Uber

Elaine Mazerolle
Matthieu Walraet

Linda Penfold




Lujia Liu



Keith



Samar Tareem


Henrik Almén
Michael Deakin 


Erin Bassett
James Crook



Junior Eluhu
Dan Laufer
Carl
Robert Solovay






Silica Magazine







Leonard Saers
Alfredo Arroyo García



Larry Yu













John Behemonth


Eric Humphrey








Øystein Risan Borgersen
David Anderson Bell III











Ole-Morten Duesend







Adam North and Gabrielle Falquero

Robert Biegler 


Qu Wenhao






Steffen Dittmar




Shanna Germain






Adam Blinkinsop







John WS Marvin (Dread Unicorn Games)


Bill Carter
Darth Chronis 



Lawrence Stewart

Gareth Hodges

Colin Backhurst
Christopher Metzger

Rachel Gumper


Mariah Thompson

Falk Alexander Glade
Johnathan Salter




Maggie Unkefer
Shawna Maryanovich






Wilhelm Fitzpatrick
Dylan “ExoByte” Mayo
Lynda Lee




Scott Carpenter



Charles D, Payet
Vince Rostkowski


Tim Brown
Raven Daegmorgan
Zak Brueckner


Christian Page

Adi Shavit


Steven Greenberg
Chuck Lunney



Adriel Bustamente

Natasha Anicich



Bram De Bie
Edward L






Gray Detrick
Robert


Sarah Russell

Sam Leavin

Abilash Pulicken

Isabel Olondriz
James Pierce
James Morrison


April Daniels



José Tremblay Champagne


Chris Edmonds

Hans & Maria Cummings
Bart Gasiewiski


Andy Chamard



Andrew Jackson

Christopher Wright



ichimonji10


Alan Stern
Alison W


Dag Henrik Bråtane





Martin Nilsson


William Schrade


David HoggJWST opportunity

Benjaming Pope (Oxford) arrived in New York today for a few days of visit, to discuss projects of mutual interest, with the hope of starting collaborations that will continue in his (upcoming) postdoc years. One thing we discussed was the JWST Early Release Science proposal call. The idea is to ask for observations that would be immediately scientifically valuable, but also create good archival opportunities for other researchers, and also help the JWST community figure out what are the best ways to make best use of the spacecraft in its (necessarily) limited lifetime. I am kicking around four ideas, one of which is about photometric redshifts, one of which is about precise time-domain photometry, one of which is about exoplanet transit spectroscopy, and one of which is about crowded-field photometry. The challenge we face is: Although there is tons of time to write a proposal, letters of intent are required in just a few weeks!

Tommaso DorigoAnomaly! Now Available As E-Book

Today I would like to mention that my book "Anomaly! Collider Physics and the Quest for New Phenomena at Fermilab" is now available for purchase as E-Book at its World Scientific site.

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February 18, 2017

Tommaso DorigoTwo Physics Blogs You Should Not Miss

I would like to use this space to advertise a couple of blogs you might be interesting to know about. Many of you who erratically read this blog may probably have already bumped into those sites, but I figured that as the readership of a site varies continuously, there is always the need to do some periodic evangelization.

read more

n-Category Café Distributive Laws

Guest post by Liang Ze Wong

The Kan Extension Seminar II continues and this week, we discuss Jon Beck’s “Distributive Laws”, which was published in 1969 in the proceedings of the Seminar on Triples and Categorical Homology Theory, LNM vol 80. In the previous Kan seminar post, Evangelia described the relationship between Lawvere theories and finitary monads, along with two ways of combining them (the sum and tensor) that are very natural for Lawvere theories but less so for monads. Distributive laws give us a way of composing monads to get another monad, and are more natural from the monad point of view.

Beck’s paper starts by defining and characterizing distributive laws. He then describes the category of algebras of the composite monad. Just as monads can be factored into adjunctions, he next shows how distributive laws between monads can be “factored” into a “distributive square” of adjunctions. Finally, he ends off with a series of examples.

Before we dive into the paper, I would like to thank Emily Riehl, Alexander Campbell and Brendan Fong for allowing me to be a part of this seminar, and the other participants for their wonderful virtual company. I would also like to thank my advisor James Zhang and his group for their insightful and encouraging comments as I was preparing for this seminar.

First, some differences between this post and Beck’s paper:

  • I’ll use the standard, modern convention for composition: the composite XFYGZ\mathbf{X} \overset{F}{\to} \mathbf{Y} \overset{G}{\to} \mathbf{Z} will be denoted GFGF. This would be written FGFG in Beck’s paper.

  • I’ll use the terms “monad” and “monadic” instead of “triple” and “tripleable”.

  • I’ll rely quite a bit on string diagrams instead of commutative diagrams. These are to be read from right to left and top to bottom. You can learn about string diagrams through these videos or this paper (warning: they read string diagrams in different directions than this post!).

  • All constructions involving the category of SS-algebras, X S\mathbf{X}^S, will be done in an “object-free” manner involving only the universal property of X S\mathbf{X}^S.

The last two points have the advantage of making the resulting theory applicable to 22-categories or bicategories other than Cat\mathbf{Cat}, by replacing categories/ functors/ natural transformations with 0/1/2-cells.

Since string diagrams play a key role in this post, here’s a short example illustrating their use. Suppose we have functors F:XYF: \mathbf{X} \to \mathbf{Y} and U:YXU: \mathbf{Y} \to \mathbf{X} such that FUF \dashv U. Let η:1 XUF\eta: 1_{\mathbf{X}} \Rightarrow{UF} be the unit and ε:FU1 Y\varepsilon: FU \Rightarrow 1_{\mathbf{Y}} the counit of the adjunction. Then the composite FFηFUFεFFF \overset{F \eta}{\Rightarrow} FUF \overset{\varepsilon F}{\Rightarrow} F can be drawn thus:

Sample string diagram

Most diagrams in this post will not be as meticulously labelled as the above. Unlabelled white regions will always stand for a fixed category X\mathbf{X}. If FUF \dashv U, I’ll use the same colored string to denote them both, since they can be distinguished from their context: above, FF goes from a white to red region, whereas UU goes from red to white (remember to read from right to left!). The composite monad UFUF (not shown above) would also be a string of the same color, going from a white region to a white region.

Motivating examples

Example 1: Let SS be the free monoid monad and TT be the free abelian group monad over Set\mathbf{Set}. Then the elementary school fact that multiplication distributes over addition means we have a function STXTSXSTX \to TSX for XX a set, sending (a+b)(c+d)(a+b)(c+d), say, to ac+ad+bc+bdac+ad+bc+bd. Further, the composition of TT with SS is the free ring monad, TSTS.

Example 2: Let AA and BB be monoids in a braided monoidal category (𝒱,,1)(\mathcal{V}, \otimes, 1). Then ABA \otimes B is also a monoid, with multiplication:

ABABAtw BABAABBμ Aμ BAB, A \otimes B \otimes A \otimes B \xrightarrow{A \otimes tw_{BA} \otimes B} A \otimes A \otimes B \otimes B \xrightarrow{\mu_A \otimes \mu_B} A \otimes B,

where tw BA:BAABtw_{BA}: B \otimes A \to A \otimes B is provided by the braiding in 𝒱\mathcal{V}.

In example 1, there is also a monoidal category in the background: the category (End(Set),,Id)\left(\mathbf{End}(\mathbf{Set}), \circ, \text{Id}\right) of endofunctors on Set\mathbf{Set}. But this category is not braided – which is why we need distributive laws!

Distributive laws, composite and lifted monads

Let (S,η S,μ S)(S,\eta^S, \mu^S) and (T,η T,μ T)(T,\eta^T,\mu^T) be monads on a category X\mathbf{X}. I’ll use Scarlet and Teal strings to denote SS and TT, resp., and white regions will stand for X\mathbf{X}.

A distributive law of SS over TT is a natural transformation :STTS\ell:ST \Rightarrow TS, denoted

Definition of distributive law

satisfying the following equalities:

Axioms for a distributive law

A distributive law looks somewhat like a braiding in a braided monoidal category. In fact, it is a local pre-braiding: “local” in the sense of being defined only for SS over TT, and “pre” because it is not necessarily invertible.

As the above examples suggest, a distributive law allows us to define a multiplication m:TSTSTSm:TSTS \Rightarrow TS:

Multiplication for the composite monad

It is easy to check visually that this makes TSTS a monad, with unit η Tη S\eta^T \eta^S. For instance, the proof that mm is associative looks like this:

Associativity for the composite monad

Not only is TSTS a monad, we also have monad maps Tη S:TTST \eta^S: T \Rightarrow TS and η TS:STS\eta^T S: S \Rightarrow TS:

Monad maps to the composite monad

Asserting that Tη ST \eta^S is a monad morphism is the same as asserting these two equalities:

Check that we have a monad map

Similar diagrams hold for η TS\eta^T S. Finally, the multiplication mm also satisfies a middle unitary law:

Middle unitary law

To get back the distributive law, we can simply plug the appropriate units at both ends of mm:

Recovering the distributive law

This last procedure (plugging units at the ends) can be applied to any m:TSTSTSm':TSTS \Rightarrow TS. It turns out that if mm' happens to satisfy all the previous properties as well, then we also get a distributive law. Further, the (distributive law \to multiplication) and (multiplication \to distributive law) constructions are mutually inverse:

Theorem   The following are equivalent: (1) Distributive laws :STTS\ell:ST \Rightarrow TS; (2) multiplications m:TSTSTSm:TSTS \Rightarrow TS such that (TS,η Tη S,m)(TS, \eta^T \eta^S, m) is a monad, Tη ST\eta^S and η TS\eta^T S are monad maps, and the middle unitary law holds.

In addition to making TSTS a monad, distributive laws also let us lift TT to the category of SS-algebras, X S\mathbf{X}^S. Before defining what we mean by “lift”, let’s recall the universal property of X S\mathbf{X}^S: Let Y\mathbf{Y} be another category; then there is an isomorphism of categories between Funct(Y,X S)\mathbf{Funct}(\mathbf{Y}, \mathbf{X}^S) – the category of functors G˜:YX S\tilde{G}: \mathbf{Y} \to \mathbf{X}^S and natural transformations between them, and SS-Alg(Y)\mathbf{Alg}(\mathbf{Y}) – the category of functors G:YXG: \mathbf{Y} \to \mathbf{X} equipped with an SS-action σ:SGG\sigma: SG \Rightarrow G and natural transformations that commute with the SS-action.

Universal property of S-algebras

Given G˜:YX S\tilde{G}: \mathbf{Y} \to \mathbf{X}^S, we get a functor YX\mathbf{Y} \to \mathbf{X} by composing with U SU^S. This composite U SG˜U^S \tilde{G} has an SS-action given by the canonical action on U SU^S. The universal property says that every such functor G:YXG: \mathbf{Y} \to \mathbf{X} with an SS-action is of the form U SG˜U^S \tilde{G}. Similar statements hold for natural transformations. We will call G˜\tilde{G} and ϕ˜\tilde{\phi} lifts of GG and ϕ\phi, resp.

A monad lift of TT to X S\mathbf{X}^S is a monad (T˜,η˜ T,μ˜ T)(\tilde{T}, \tilde{\eta}^T,\tilde{\mu}^T) on X S\mathbf{X}^S such that

U ST˜=TU S,U Sη˜ T=η TU S,U Sμ˜ T=μ TU S. U^S \tilde{T} = T U^S, \qquad U^S \tilde{\eta}^T = \eta^T U^S, \qquad U^S \tilde{\mu}^T = \mu^T U^S.

We may express U ST˜=TU SU^S \tilde{T} = T U^S via the following equivalent commutative diagrams:

Commutative diagrams for lifts

The diagram on the right makes it clear that T˜\tilde{T} being a monad lift of TT is equivalent to T˜,η˜ T,μ˜ T\tilde{T}, \tilde{\eta}^T, \tilde{\mu}^T being lifts of TU S,η TU S,μ TU STU^S, \eta^T U^S,\mu^T U^S, resp. Thus, to get a monad lift of TT, it suffices to produce an SS-action on TU STU^S and check that it is compatible with η TU S\eta^T U^S and μ TU S\mu^T U^S. We may simply combine the distributive law with the canonical SS-action on U SU^S to obtain the desired action on TU STU^S:

S-action on TU^S

(Recall that the unlabelled white region is X\mathbf{X}. In subsequent diagrams, we will leave the red region unlabelled as well, and this will always be X S\mathbf{X}^S. Similarly, teal regions will denote X T\mathbf{X}^T.)

Conversely, suppose we have a monad lift T˜\tilde{T} of TT. Then the equality U ST˜=TU SU^S \tilde{T} = T U^S can be expressed by saying that we have an invertible natural transformation χ:U ST˜TU S\chi: U^S \tilde{T} \Rightarrow TU^S. Using χ\chi and the unit and counit of the adjunction F SU SF^S \dashv U^S that gives rise to SS, we obtain a distributive law of SS over TT:

Getting a distributive law from a lift

The key steps in the proof that these constructions are mutually inverse are contained in the following two equalities:

Constructions are mutually inverse

The first shows that the resulting distributive law in the (distributive law \to monad lift \to distributive law) construction is the same as the original distributive law we started with. The second shows that in the (monad lift T˜\tilde{T} \to distributive law \to another lift T˜\tilde{T}') construction, the SS-action on U ST˜U^S \tilde{T}' (LHS of the equation) is the same as the original SS-action on U ST˜U^S \tilde{T} (RHS), hence T˜=T˜\tilde{T} = \tilde{T}' (by virtue of being lifts, T˜\tilde{T} and T˜\tilde{T}' can only differ in their induced SS-actions on U ST˜=U ST˜=TU SU^S \tilde{T} = U^S \tilde{T}' = TU^S). We thus have another characterization of distributive laws:

Theorem   The following are equivalent: (1) Distributive laws :STTS\ell:ST \Rightarrow TS; (3) monad lifts of TT to X S\mathbf{X}^S.

In fact, the converse construction did not rely on the universal property of X S\mathbf{X}^S, and hence applies to any adjunction giving rise to SS (with a suitable definition of a monad lift of TT in this situation). In particular, it applies to the Kleisli adjunction F SU SF_S \dashv U_S. Since the Kleisli category X S\mathbf{X}_S is equivalent to the subcategory of free SS-algebras (in the classical sense) in X S\mathbf{X}^S, this means that to get a distributive law of SS over TT, it suffices to lift TT to a monad over just the free SS-algebras! (Thanks to Jonathan Beardsley for pointing this out!) The resulting distributive law may be used to get another lift of TT, but we should not expect this to be the same as the original lift unless the original lift was “monadic” to begin with, in the sense of being a lift to X S\mathbf{X}^S.

There are two further characterizations of distributive laws that are not mentioned in Beck’s paper, but whose equivalences follow easily. Eugenia Cheng in Distrbutive laws for Lawvere theories states that distributive laws of SS over TT are also equivalent to extensions of SS to a monad S˜\tilde{S} on the Kleisli category X T\mathbf{X}_T. This follows by duality from the above theorem, since X T=(X op) T\mathbf{X}_T = (\mathbf{X}^{op})^T. Finally, Ross Street’s The formal theory of monads (which was also covered in a previous Kan Extension Seminar post) says that distributive laws in a 22-category K\mathbf{K} are precisely monads in Mnd(K)\mathbf{Mnd}(\mathbf{K}). It is a fun and easy exercise to draw string diagrams for objects of Mnd(Mnd(K))\mathbf{Mnd}(\mathbf{Mnd}(\mathbf{K})); it becomes visually obvious that these are the same as distributive laws.

Algebras for the composite monad

After characterizing distributive laws, Beck characterizes the algebras for the composite monad TSTS.

Just as a morphism of rings RRR \to R' induces a “restriction of scalars” functor RR'-ModR\mathbf{Mod} \to R-Mod\mathbf{Mod}, the monad maps Tη S:TTST \eta^S: T \Rightarrow TS and η TS:STS\eta^T S: S \Rightarrow TS induce functors U^ TS:X TSX T\hat{U}^{TS}: \mathbf{X}^{TS} \to \mathbf{X}^T and U˜ TS:X TSX S\tilde{U}^{TS}: \mathbf{X}^{TS} \to \mathbf{X}^S.

Equivalently, we have SS- and TT-actions on U TSU^{TS}, which we call σ:SU TSU TS\sigma: SU^{TS} \Rightarrow U^{TS} and τ:TU TSU TS\tau: T U^{TS} \Rightarrow U^{TS}. Let ε:TSU TSU TS\varepsilon: TS\, U^{TS} \Rightarrow U^{TS} be the canonical TSTS-action on U TSU^{TS}. The middle unitary law then implies that ε=Tστ\varepsilon = T\sigma \cdot \tau:

Actions of T, S and TS

Further, σ\sigma is distributes over τ\tau in the following sense:

S-action distributes over T-action

The properties of these actions allow us to characterize TSTS-algebras:

Theorem   The category of algebras for TSTS coincides with that of T˜\tilde{T}:

X TS(X S) T˜ \mathbf{X}^{TS} \cong (\mathbf{X}^S)^{\tilde{T}}

To prove this, Beck constructs Φ:(X S) T˜X TS\Phi: (\mathbf{X}^{S})^{\tilde{T}} \to \mathbf{X}^{TS} and its inverse Φ 1:X TS(X S) T˜\Phi^{-1}: \mathbf{X}^{TS} \to (\mathbf{X}^S)^{\tilde{T}} . These constructions are best summarized in the following diagram of lifts:

Diagram of required lifts

On the left half of the diagram, we see that to get Φ 1\Phi^{-1}, we must first produce a functor U˜ TS:X TSX S\tilde{U}^{TS}: \mathbf{X}^{TS} \to \mathbf{X}^S with a T˜\tilde{T}-action. We already have U˜ TS\tilde{U}^{TS} as a lift of U TSU^{TS}, given by the SS-action σ\sigma. We also have the TT-action τ\tau on U TSU^{TS}, which σ\sigma distributes over. This is precisely what is required to get a lift of τ\tau to a T˜\tilde{T}-action τ˜\tilde{\tau} on U˜ TS\tilde{U}^{TS}, which gives us Φ 1\Phi^{-1}.

On the right half of the diagram, to get Φ\Phi we need to produce a functor (X S) T˜X(\mathbf{X}^{S})^{\tilde{T}} \to \mathbf{X} with a TSTS-action. The obvious functor is U SU T˜U^S U^{\tilde{T}}, and we get an action by using the canonical actions of SS on U SU^S and T˜\tilde{T} on U T˜U^{\tilde{T}}:

Lifts for Phi

All that’s left to prove the theorem is to check that Φ\Phi and Φ 1\Phi^{-1} are inverses. In a similar fashion, we can prove the dual statement (again found in Cheng’s paper but not Beck’s):

Theorem   The Kleisli category of TSTS coincides with that of S˜\tilde{S}:

X TS(X T) S˜ \mathbf{X}_{TS} \cong (\mathbf{X}_T)_{\tilde{S}}

Distributivity for adjoints

From now on, we identify X TS\mathbf{X}^{TS} with (X S) T˜(\mathbf{X}^S)^{\tilde{T}}. Under this identification, it turns out that U˜ TSU T˜\tilde{U}^{TS} \cong U^{\tilde{T}}, and we obtain what Beck calls a distributive adjoint situation comprising 3 pairs of adjunctions:

Distributive adjoint situation

For this to qualify as a distributive adjoint situation, we also require that both composites from X TS\mathbf{X}^{TS} to X\mathbf{X} are naturally isomorphic, and both composites from X S\mathbf{X}^S to X T\mathbf{X}^T are naturally isomorphic. This can be expressed in the following diagram by requiring both blue circles to be mutually inverse, and both red circles to be mutually inverse:

Distributive adjoints in string

(Recall that colored regions are categories of algebras for the corresponding monads, and the cap and cup are the unit and counit of F SU SF^S \dashv U^S.)

This diagram is very similar to the diagram for getting a distributive law out of a lift T˜\tilde{T}, and it is easy to believe that any such distributive adjoint situation (with 3 pairs of adjoints - not necessarily monadic - and the corresponding natural isomorphisms) leads to a distributive law.

Finally, suppose the “restriction of scalars” functor U^ TS\hat{U}^{TS} has an adjoint. This adjoint behaves like an “extension of scalars” functor, and Beck fittingly calls it () SF T(\,) \otimes_S F^T at the start of his paper. I’ll use F^ TS\hat{F}^{TS} instead, to highlight its relationship with U^ TS\hat{U}^{TS}.

In such a situation, we get an adjoint square consisting of 4 pairs of adjunctions. By drawing these 4 adjoints in the following manner, it becomes clear which natural transformations we require in order to get a distributive law:

Distributive adjoints again

(Recall that S=U SF SS = U^S F^S and T=U TF TT = U^T F^T, so this is a “thickened” version of what a distributive law looks like.)

It turns out that given the natural transformation uu between the composite right adjoints U SU T˜U^S U^{\tilde{T}} and U TU^ TSU^T \hat{U}^{TS}, we can get the natural transformation ff as the mate of uu between the corresponding composite left adjoints F^ TSF T\hat{F}^{TS} F^T and F T˜F SF^{\tilde{T}} F^S. Note that ff is invertible if and only if uu is. We may use uu or ff, along with the units and counits of the relevant adjunctions, to construct ee:

Getting e from u or f

But ee is in the wrong direction, so we have to further require that ee is invertible, to get e 1e^{-1}. We get ee' from u 1u^{-1} or f 1f^{-1} in a similar manner. Since ee' will turn out to already be in the right direction, we will not require it to be invertible. Finally, given any 4 pairs of adjoints that look like the above, along with natural transformations u,f,e,eu,f,e,e' satisfying the above properties, we will get a distributive law!

What next?

Beck ends his paper with some examples, two of which I’ve already mentioned at the start of this post. During our discussion, there were some remarks on these and other examples, which I hope will be posted in the comments below. Instead of repeating those examples, I’d like to end by pointing to some related works:

  • Since we’ve been talking about Lawvere theories, we can ask what distributive laws look like for Lawvere theories. Cheng’s Distributive laws for Lawvere theories, which I’ve already referred to a few times, does exactly that. But first, she comes up with 4 settings in which to define Lawvere theories! She also has a very readable introduction to the correspondence between Lawvere theories and finitary monads.

  • As Beck mentions in his paper, we can similarly define distributive laws between comonads, as well as mixed distributive laws between a monad and a comonad. Just as we can define bimonoids/bialgebras, and thus Hopf monoids/algebras, in a braided monoidal category, such distributive laws allow us to define bimonads and Hopf monads. There are in fact two distinct notions of Hopf monads: the first is described in this paper by Alain Bruguières and Alexis Virelizier (with a follow-up paper coauthored with Steve Lack, and a diagrammatic approach with amazing surface diagrams by Simon Willerton); the second is this paper by Bachuki Mesablishvili and Robert Wisbauer. The difference between these two approaches is described in the Mesablishvili-Wisbauer paper, but both involve mixed distributive laws. Gabriella Böhm also recently gave a talk entitled The Unifying Notion of Hopf Monad, in which she shows how the many generalizations of Hopf algebras are just instances of Hopf monads (in the first sense) in an appropriate monoidal bicategory!

  • We also saw that distributive laws are monads in a category of monads. Instead of thinking of distributive laws as merely a means of composing monads, we can study distributive laws as objects in their own right, just as monoids in a category of monoids (a.k.a. abelian monoids) are studied in their own right! The story for monoids terminates at this step: monoids in abelian monoids are just abelian monoids. But for distributive laws, we can keep going! See Cheng’s paper on Iterated Distributive Laws, where she shows the connection between iterated distributive laws and nn-categories. In addition to requiring distributive laws between each pair of monads involved, it is also necessary to have a Yang-Baxter equation between every three monads:

Yang-Baxter condition

  • Finally, there seems to be a strange connection between distributive laws and factorization systems (e.g. here, here and even in “Distributive Laws for Lawvere theories” mentioned above). I can’t say more because I don’t know much about factorization systems, but hopefully someone else can say something illuminating about this!

February 17, 2017

BackreactionBlack Hole Information - Still Lost

[Illustration of black hole.Image: NASA] According to Google, Stephen Hawking is the most famous physicist alive, and his most famous work is the black hole information paradox. If you know one thing about physics, therefore, that’s what you should know. Before Hawking, black holes weren’t paradoxical. Yes, if you throw a book into a black hole you can’t read it anymore. That’s because what

February 16, 2017

Jordan EllenbergBraid monodromy and the dual curve

Nick Salter gave a great seminar here about this paper; hmm, maybe I should blog about that paper, which is really interesting, but I wanted to make a smaller point here.  Let C be a smooth curve in P^2 of degree n. The lines in P^2 are parametrized by the dual P^2; let U be the open subscheme of the dual P^2 parametrizing those lines which are not tangent to C; in other words, U is the complement of the dual curve C*.  For each point u of U, write L_u for the corresponding line in P^2.

This gives you a fibration X -> U where the fiber over a point u in U is L_u – (L_u intersect C).  Since L_u isn’t tangent to C, this fiber is a line with n distinct points removed.  So the fibration gives you an (outer) action of pi_1(U) on the fundamental group of the fiber preserving the puncture classes; in other words, we have a homomorphism

\pi_1(U) \rightarrow B_n

where B_n is the n-strand braid group.

When you restrict to a line L* in U (i.e. a pencil of lines through a point in the original P^2) you get a map from a free group to B_n; this is the braid monodromy of the curve C, as defined by Moishezon.  But somehow it feels more canonical to consider the whole representation of pi_1(U).  Here’s one place I see it:  Proposition 2.4 of this survey by Libgober shows that if C is a rational nodal curve, then pi_1(U) maps isomorphically to B_n.  (OK, C isn’t smooth, so I’d have to be slightly more careful about what I mean by U.)

 


BackreactionA new theory SMASHes problems

Most of my school nightmares are history exams. But I also have physics nightmares, mostly about not being able to recall Newton’s laws. Really, I didn’t like physics in school. The way we were taught the subject, it was mostly dead people’s ideas. On the rare occasion our teacher spoke about contemporary research, I took a mental note every time I heard “nobody knows.” Unsolved problems were

February 15, 2017

John PreskillTen finalists selected for film festival “Quantum Shorts”

“Crazy enough”, “visually very exciting”, “compelling from the start”, “beautiful cinematography”: this is what members of the Quantum Shorts festival shortlisting panel had to say about films selected for screening. As a member of the panel, and as someone who has experienced the power of visual storytelling firsthand (Anyone Can Quantum, Quantum Is Calling), I was excited to see filmmakers and students from around the world try their hand at interpreting the weirdness of the quantum realm in fresh ways.

The ten shortlisted films were chosen from a total of 203 submissions received during the festival’s 2016 call for entries. Some of the finalists are dramatic, some funny, some abstract. Some are live-action film, some animation. Each is under five minutes long. Find the titles and synopses of the shortlisted films below.

Screenings of the films start February 23 with confirmed events in Waterloo (23 February) and Vancouver (23 February), Canada; Singapore (25-28 February); Glasgow, UK (17 March); and Brisbane, Australia (24 March).

More details can be found at shorts.quantumlah.org, where viewers can also watch the films online
and vote for their favorite to help decide a ‘People’s Choice’ prize. The website also hosts interviews with the filmmakers.

The Quantum Shorts festival is run by the Centre for Quantum Technologies at the National University of Singapore with a constellation of prestigious partners including Scientific American magazine and the journal Nature. The festival’s media partners, scientific partners and screening partners span five countries. The Institute for Quantum Information and Matter at Caltech is a proud sponsor.

For making the shortlist, the filmmakers receive a $250 award, a one-year digital subscription to Scientific American and certificates.

The festival’s top prize of US $1500 and runner-up prize of US $1000 will now be decided by a panel of eminent judges. The additional People’s Choice prize of $500 will be decided by public vote on the shortlist, with voting open on the festival website until March 26th. Prizes will be announced by the end of March.

Quantum Shorts 2016: FINALISTS

ampersand2 approachingreality3 bolero2 novae3 theguardian1 therealthing1 togetherparalleluniverse3 tomsbreakfast1 triangulation whitecap3

Ampersand

What unites everything on Earth? That we are all ultimately composed of something that is both matter & wave

Submitted by Erin Shea, United States

Approaching Reality

Dancing cats, a watchful observer and a strange co-existence. It’s all you need to understand the essence of quantum mechanics

Submitted by Simone De Liberato, United Kingdom

Bolero

The coin is held fast, but is it heads or tails? As long as the fist remains closed, you are a winner – and a loser

Submitted by Ivan D’Antonio, Italy

Novae

What happens when a massive star reaches the end of its life? Something that goes way beyond the spectacular, according to this cosmic poem about the infinite beauty of a black hole’s birth

Submitted by Thomas Vanz, France

The Guardian

A quantum love triangle, where uncertainty is the only winner

Submitted by Chetan Kotabage, India

The Real Thing

Picking up a beverage shouldn’t be this hard. And it definitely shouldn’t take you through the multiverse…

Submitted by Adam Welch, United States

Together – Parallel Universe

It’s a tale as old as time: boy meets girl, girl is not as interested as boy hoped. So boy builds spaceship and travels through multi-dimensional reality to find the one universe where they can be together

Submitted by Michael Robertson, South Africa

Tom’s Breakfast

This is one of those days when Tom’s morning routine doesn’t go to plan – far from it, in fact. The only question is, can he be philosophical about it?

Submitted by Ben Garfield, United Kingdom

Triangulation

Only imagination can show us the hidden world inside of fundamental particles

Submitted by Vladimir Vlasenko, Ukraine

Whitecap

Dr. David Long has discovered how to turn matter into waveforms. So why shouldn’t he experiment with his own existence?

Submitted by Bernard Ong, United States


qs_trailer_final

Jordan EllenbergMark Metcalf

Have you ever heard of this guy?  I hadn’t.  Or thought I hadn’t.  But: he was Niedermeyer in Animal House

and the dad in Twisted Sister’s “We’re Not Gonna Take It” video

and the Master, the Big Bad of Buffy the Vampire Slayer season 1.

That’s a hell of a career! Plus: he lived in suburban Milwaukee until three years ago! And he used to go out with Glenn Close and Carrie Fisher! OK. Now I’ve heard of Mark Metcalf and so have you.


February 14, 2017

n-Category Café Functional Equations II: Shannon Entropy

In the second instalment of the functional equations course that I’m teaching, I introduced Shannon entropy. I also showed that up to a constant factor, it’s uniquely characterized by a functional equation that it satisfies: the chain rule.

Notes for the course so far are here. For a quick summary of today’s session, read on.

You can read the full story in the notes, but here I’ll state the main result as concisely as I can.

For n0n \geq 0, let Δ n\Delta_n denote the set of probability distributions p=(p 1,,p n)\mathbf{p} = (p_1, \ldots, p_n) on {1,,n}\{1, \ldots, n\}. The Shannon entropy of pΔ n\mathbf{p} \in \Delta_n is

H(p)= i:p i>0p ilogp i. H(\mathbf{p}) = - \sum_{i \colon p_i \gt 0} p_i \log p_i.

Now, given

wΔ n,p 1Δ k 1,,p nΔ k n, \mathbf{w} \in \Delta_n, \,\, \mathbf{p}^1 \in \Delta_{k_1}, \ldots, \mathbf{p}^n \in \Delta_{k_n},

we obtain a composite distribution

w(p 1,,p n)=(w 1p 1 1,,w 1p k 1 1,,w np 1 n,,w np k n n). \mathbf{w} \circ (\mathbf{p}^1, \ldots, \mathbf{p}^n) = (w_1 p^1_1, \ldots, w_1 p^1_{k_1}, \, \ldots, \, w_n p^n_1, \ldots, w_n p^n_{k_n}).

The chain rule for HH states that

H(w(p 1,,p n))=H(w)+ i=1 nw iH(p i). H(\mathbf{w} \circ (\mathbf{p}^1, \ldots, \mathbf{p}^n)) = H(\mathbf{w}) + \sum_{i = 1}^n w_i H(\mathbf{p}^i).

So, (H:Δ n +) n1(H: \Delta_n \to \mathbb{R}^+)_{n \geq 1} is a sequence of continuous functions satisfying the chain rule. Clearly, the same is true of any nonnegative scalar multiple of HH.

Theorem (Faddeev, 1956)   The only sequences of continuous functions (Δ n +) n1(\Delta_n \to \mathbb{R}^+)_{n \geq 1} satisfying the chain rule are the scalar multiples of entropy.

One interesting aspect of the proof is where the difficulty lies. Let I:Δ n +I: \Delta_n \to \mathbb{R}^+ be continuous functions satisfying the chain rule; we have to show that II is proportional to HH. All the effort and ingenuity goes into showing that II is proportional to HH when restricted to the uniform distributions. In other words, the hard part is to show that there exists a constant cc such that

I(1/n,,1/n)=cH(1/n,,1/n) I(1/n, \ldots, 1/n) = c H(1/n, \ldots, 1/n)

for all n1n \geq 1. But once that’s done, showing that I(p)=cH(p)I(\mathbf{p}) = c H(\mathbf{p}) is a pushover. The notes show you how!

Mark Chu-CarrollDoes well-ordering contradict Cantor?

The other day, I received an email that actually excited me! It’s a question related to Cantor’s diagonalization, but there’s absolutely nothing cranky about it! It’s something interesting and subtle. So without further ado:

Cantor’s diagonalization says that you can’t put the reals into 1 to 1 correspondence with the integers. The well-ordering theorem seems to suggest that you can pick a least number from every set including the reals, so why can’t you just keep picking least elements to put them into 1 to 1 correspondence with the reals. I understand why Cantor says you can’t. I just don’t see what is wrong with the other arguments (other than it must be wrong somehow). Apologies for not being able to state the argument in formal maths, I’m around 20 years out of practice for formal maths.

As we’ve seen in too many discussions of Cantor’s diagonalization, it’s a proof that shows that it is impossible to create a one-to-one correspondence between the natural numbers and the real numbers.

The Well-ordering says something that seems innoccuous at first, but which, looked at in depth, really does appear to contradict Cantor’s diagonalization.

A set S is well-ordered if there exists a total ordering <= on the set, with the additional property that for any subset T \subseteq S, T has a smallest element.

The well-ordering theorem says that every non-empty set can be well-ordered. Since the set of real numbers is a set, that means that there exists a well-ordering relation over the real numbers.

The problem with that is that it appears that that tells you a way of producing an enumeration of the reals! It says that the set of all real numbers has a least element: Bingo, there’s the first element of the enumeration! Now you take the set of real numbers excluding that one, and it has a least element under the well-ordering relation: there’s the second element. And so on. Under the well-ordering theorem, then, every set has a least element; and every element has a unique successor! Isn’t that defining an enumeration of the reals?

The solution to this isn’t particularly satisfying on an intuitive level.

The well-ordering theorem is, mathematically, equivalent to the axiom of choice. And like the axiom of choice, it produces some very ugly results. It can be used to create “existence” proofs of things that, in a practical sense, don’t exist in a usable form. It proves that something exists, but it doesn’t prove that you can ever produce it or even identify it if it’s handed to you.

So there is an enumeration of the real numbers under the well ordering theorem. Only the less-than relation used to define the well-ordering is not the standard real-number less than operation. (It obviously can’t be, because under well-ordering, every set has a least element, and standard real-number less-than doesn’t have a least element.) In fact, for any ordering relation \le_x that you can define, describe, or compute, \le_x is not the well-ordering relation for the reals.

Under the well-ordering theorem, the real numbers have a well-ordering relation, only you can’t ever know what it is. You can’t define any element of it; even if someone handed it to you, you couldn’t tell that you had it.

It’s very much like the Banach-Tarski paradox: we can say that there’s a way of doing it, only we can’t actually do it in practice. In the B-T paradox, we can say that there is a way of cutting a sphere into these strange pieces – but we can’t describe anything about the cut, other than saying that it exists. The well-ordering of the reals is the same kind of construct.

How does this get around Cantor? It weasels its way out of Cantor by the fact that while the well-ordering exists, it doesn’t exist in a form that can be used to produce an enumeration. You can’t get any kind of handle on the well-ordering relation. You can’t produce an enumeration from something that you can’t create or identify – just like you can’t ever produce any of the pieces of the Banach-Tarski cut of a sphere. It exists, but you can’t use it to actually produce an enumeration. So the set of real numbers remains non-enumerable even though it’s well-ordered.

If that feels like a cheat, well… That’s why a lot of people don’t like the axiom of choice. It produces cheatish existence proofs. Connecting back to something I’ve been trying to write about, that’s a big part of the reason why intuitionistic type theory exists: it’s a way of constructing math without stuff like this. In an intuitionistic type theory (like the Martin-Lof theory that I’ve been writing about), it doesn’t exist if you can’t construct it.

February 13, 2017

n-Category Café Functional Equations I: Cauchy's Equation

This semester, I’m teaching a seminar course on functional equations. Why? Among other reasons:

  1. Because I’m interested in measures of biological diversity. Dozens (or even hundreds?) of diversity measures have been proposed, but it would be a big step forward to have theorems of the form: “If you want your measure to have this property, this property, and this property, then it must be that measure. No other will do.”

  2. Because teaching a course on functional equations will force me to learn about functional equations.

  3. Because it touches on lots of mathematically interesting topics, such as entropy of various kinds and the theory of large deviations.

Today was a warm-up, focusing on Cauchy’s functional equation: which functions f:f: \mathbb{R} \to \mathbb{R} satisfy

f(x+y)=f(x)+f(y)x,y? f(x + y) = f(x) + f(y) \,\,\,\, \forall x, y \in \mathbb{R}?

(I wrote about this equation before when I discovered that one of the main references is in Esperanto.) Later classes will look at entropy, means, norms, diversity measures, and a newish probabilistic method for solving functional equations.

Read on for today’s notes and an outline of the whole course.

I don’t want to commit to TeXing up notes every week, as any such commitment would suck joy out of something I’m really doing for intellectual fulfilment (also known as “fun”). However, I seem to have done it this week. Here they are. For those who came to the class, the parts in black ink are pretty much exactly what I wrote on the board.

Here’s the overall plan. We’ll take it at whatever pace feels natural, so the section numbers below don’t correspond to weeks. The later sections are pretty tentative — plans might change!

  1. Warm-up   Which functions ff satisfy f(x+y)=f(x)+f(y)f(x + y) = f(x) + f(y)? Which functions of two variables can be separated as a product of functions of one variable?

  2. Shannon entropy   Basic ideas. Characterizations of entropy by Shannon, Faddeev, Rényi, etc. Relative entropy.

  3. Deformed entropies   Rényi and “Tsallis” entropies. Characterizations of them. Relative Rényi entropy.

  4. Probabilistic methods   Cramér’s large deviation theorem. Characterization of pp-norms and power means.

  5. Diversity of a single community   Background and introduction. Properties of diversity measures. Value. Towards a uniqueness theorem.

  6. Diversity of a metacommunity   Background: diversity within and between subcommunities; beta-diversity in ecology. Link back to relative entropy. Properties.

Doug NatelsonLosing a colleague and friend - updated

Blogging is taking a back seat right now.  I'm only posting because I know some Rice connections and alumni read here and may not have heard about this.  Here is a longer article, though I don't know how long it will be publicly accessible.

Update:  This editorial was unexpected (at least by me) and much appreciated.  There is also a memorial statement here.

Update 2:  The Houston Chronicle editorial is now behind a pay-wall.  I suspect they won't mind me reproducing it here:

"If I have seen further it is by standing on the shoulders of giants."

Isaac Newton was not the first to express this sentiment, though he was perhaps the most brilliant. But even a man of his stature knew that he only peered further into the secrets of our universe because of the historic figures who preceded him.

Those giants still walk among us today. They work at the universities, hospitals and research laboratories that dot our city. They explore the uncharted territory of human knowledge, their footsteps laying down paths that lead future generations.

Dr. Marjorie Corcoran was one of those giants. The Rice University professor had spent her career uncovering the unknown - the subatomic levels where Newton's physics fall apart. She was killed after being struck by a Metro light rail train last week.

Corcoran's job was to ask the big questions about the fundamental building blocks and forces of the universe. Why does matter have mass? Why does physics act the way it does?
She worked to understand reality and unveil eternity. To the layperson, her research was a secular contemplation of the divine.

Our city spent years of work and millions of dollars preparing for the super-human athletic feats witnessed at the Super Bowl. But advertisers didn't exactly line up to sponsor Corcoran - and for good reason. Anyone can marvel in a miraculous catch. It is harder to grasp the wonder of a subatomic world, the calculations that bring order to the universe, the research that hopes to explain reality itself.

Only looking backward can we fully grasp the incredible feats done by physicists like Corcoran.
"A lot of people don't have a very long timeline. They're thinking what's going to happen to them in the next hour or the next day, maybe the next week," Andrea Albert, one of Corcoran's former students, told the editorial board. "No, we're laying the foundation so that your grandkids are going to have an awesome, cool technology. I don't know what it is yet. But it is going to be awesome."

Houston is already home to some of the unexpected breakthroughs of particle physics. Accelerators once created to smash atoms now treat cancer patients with proton therapy.

All physics is purely academic - until it isn't. From the radio to the atom bomb, modern civilization is built on the works of giants.

But the tools that we once used to craft the future are being left to rust.

Federal research funding has fallen from its global heights. Immigrants who help power our labs face newfound barriers. Our nation shouldn't forget that Albert Einstein and Edward Teller were refugees.
"How are we going to foster the research mission of the university?" Rice University President David Leebron posed to the editorial board last year. "I think as we see that squeeze, you look at the Democratic platform or the Republican platform or the policies out of Austin, I worry about the level of commitment."

In a competitive field, Corcoran went out of her way to help new researchers. In a field dominated by men, she stood as a model for young women. And in a nation focused on quarterly earnings, her work was dedicated to the next generation.

Marjorie Corcoran was a giant. The world stands taller because of her.

n-Category Café M-theory from the Superpoint

You may have been following the ‘Division algebra and supersymmetry’ story, the last instalment of which appeared a while ago under the title M-theory, Octonions and Tricategories. John (Baez) was telling us of some work by his former student John Huerta which relates these entities. The post ends with a declaration which does not suffer from comparison to Prospero’s in The Tempest

But this rough magic

I here abjure. And when I have required

Some heavenly music – which even now I do –

To work mine end upon their senses that

This airy charm is for, I’ll break my staff,

Bury it certain fathoms in the earth,

And deeper than did ever plummet sound

I’ll drown my book.

Well, maybe not quite so poetic:

And with the completion of this series, I can now relax and forget all about these ideas, confident that at this point, the minds of a younger generation will do much better things with them than I could.

Anyway, you may be interested to know that the younger generation has pressed on. John Huerta teamed up with Urs Schreiber to write M-theory from the Superpoint (updated versions here), which looks to grow out of a mere superpoint Lorentzian spacetimes, D-branes and M-branes by the simple device of successive invariant higher central extensions.

It’s like a magical Whitehead tower where you can’t see how they put the rabbit in.

Jordan EllenbergDifference sets missing a Hamming sphere

I tend to think of the Croot-Lev-Pach method (as used, for instance, in the cap set problem) as having to do with n-tensors, where n is bigger than 2.  But you can actually also use it in the case of 2-tensors, i.e. matrices, to say things that (as far as I can see) are not totally trivial.

Write m_d for the number of squarefree monomials in x_1, .. x_n of degree at most d; that is,

m_d = 1 + {n \choose 1} + {n \choose 2} + \ldots + {n \choose d}.

Claim:  Let P be a polynomial of degree d in F_2[x_1, .. x_n] such that P(0) = 1.  Write S for the set of nonzero vectors x such that P(x) = 1.  Let A be a subset of F_2^n such that no two elements of A have difference lying in S.  Then |A| < 2m_{d/2}.

Proof:  Write M for the A x A matrix whose (a,b) entry is P(a-b).  By the Croot-Lev-Pach lemma, this matrix has rank at most 2m_{d/2}.  By hypothesis on A, M is the identity matrix, so its rank is |A|.

Remark: I could have said “sum” instead of “difference” since we’re in F_2 but for larger finite fields you really want difference.

The most standard context in which you look for large subsets of F_2^n with restricted difference sets is that of error correcting codes, where you ask that no two distinct elements of A have difference with Hamming weight (that is, number of 1 entries) at most k.

It would be cool if the Croot-Lev-Pach lemma gave great new bounds on error-correcting codes, but I don’t think it’s to be.  You would need to find a polynomial P which vanishes on all nonzero vectors of weight larger than k, but which doesn’t vanish at 0. Moreover, you already know that the balls of size k/2 around the points of A are disjoint, which gives you the “volume bound”

|A| < 2^n / m_{k/2}.

I think that’ll be hard to beat.

If you just take a random polynomial P, the support of P will take up about half of F_2^n; so it’s not very surprising that a set whose difference misses that support has to be small!

Here’s something fun you can do, though.  Let s_i be the i-th symmetric function on x_1, … x_n.  Then

s_i(x) = {wt(x) \choose i}

where wt(x) denotes Hamming weight.  Recall also that the binomial coefficient

{k \choose 2^a}

is odd precisely when the a’th binary digit of k is 1.

Thus,

(1-s_1(x))(1-s_2(x))(1-s_4(x))\ldots(1-s_{2^{b-1}}(x))

is a polynomial of degree 2^b-1 which vanishes on x unless the last b digits of wt(x) are 0; that is, it vanishes unless wt(x) is a multiple of 2^b.  Thus we get:

Fact:  Let A be a subset of F_2^n such that the difference of two nonzero elements in A never has weight a multiple of 2^b.  Then

|A| \leq 2m_{2^{b-1} - 1}.

Note that this is pretty close to sharp!  Because if we take A to be the set of vectors of  weight at most 2^{b-1} – 1, then A clearly has the desired property, and already that’s half as big as the upper bound above.  (What’s more, you can throw in all the vectors of weight 2^{b-1} whose first coordinate is 1; no two of these sum to something of weight 2^b.  The Erdös-Ko-Rado theorem says you can do no better with those weight 2^{b-1} vectors.)

Is there an easier way to prove this?

When b=1, this just says that a set with no differences of even Hamming weight has size at most 2; that’s clear, because two vectors whose Hamming weight has the same parity differ by a vector of even weight.  Even for b=2 this isn’t totally obvious to me.  The result says that a subset of F_2^n with no differences of weight divisible by 4 has size at most 2+2n.  On the other hand, you can get 1+2n by taking 0, all weight-1 vectors, and all weight-2 vectors with first coordinate 1.  So what’s the real answer, is it 1+2n or 2+2n?

Write H(n,k) for the size of the largest subset of F_2^n having no two vectors differing by a vector of Hamming weight exactly k.  Then if 2^b is the largest power of 2 less than n, we have shown above that

m_{2^{b-1} - 1 } \leq H(n,2^b) \leq 2m_{2^{b-1} - 1}.

On the other hand, if k is odd, then H(n,k) = 2^{n-1}; we can just take A to be the set of all even-weight vectors!  So perhaps H(n,k) actually depends on k in some modestly interesting 2-adic way.

The sharpness argument above can be used to show that H(4m,2m) is as least

2(1 + 4m + {4m \choose 2} + \ldots + {4m \choose m-1} + {4m-1 \choose m-1}). (*)

I was talking to Nigel Boston about this — he did some computations which make it looks like H(4m,2m) is exactly equal to (*) for m=1,2,3.  Could that be true for general m?

(You could also ask about sets with no difference of weight a multiple of k; not sure which is the more interesting question…)

Update:  Gil Kalai points out to me that much of this is very close to and indeed in some parts a special case of the Frankl-Wilson theorem…  I will investigate further and report back!


February 12, 2017

Tommaso DorigoThe Six-Month Cycle Of The Experimental Physicist

Every year, at about this time, the level of activity of physicists working in experimental collaborations at high-energy colliders and elsewhere increases dramatically. We are approaching the time of "winter conferences", so called in order to distinguish them from "summer conferences". 
During winter conferences, which take place between mid-February and the end of March in La Thuile, Lake Louise, and other fashionable places close to ski resorts, experimentalists gather to show off their latest results. The same ritual repeats during the summer in a few more varied locations around the world. 

read more

Doug NatelsonWhat is a time crystal?

Recall a (conventional, real-space) crystal involves a physical system with a large number of constituents spontaneously arranging itself in a way that "breaks" the symmetry of the surrounding space.  By periodically arranging themselves, the atoms in an ordinary crystal "pick out" particular length scales (like the spatial period of the lattice) and particular directions.

Back in 2012, Frank Wilczek proposed the idea of time crystals, here and here, for classical and quantum versions, respectively.  The original idea in a time crystal is that a system with many dynamical degrees of freedom, can in its ground state spontaneously break the smooth time translation symmetry that we are familiar with.  Just as a conventional spatial crystal would have a certain pattern of, e.g., density that repeats periodically in space, a time crystal would spontaneously repeat its motion periodically in time.  For example, imagine a system that, somehow while in its ground state, rotates at a constant rate (as described in this viewpoint article).  In quantum mechanics involving charged particles, it's actually easier to think about this in some ways.  [As I wrote about back in the ancient past, the Aharonov-Bohm phase implies that you can have electrons producing persistent current loops in the ground state in metals.]

The "ground state" part of this was not without controversy.   There were proofs that this kind of spontaneous periodic groundstate motion is impossible in classical systems.  There were proofs that this is also a challenge in quantum systems.  [Regarding persistent currents, this gets into a definitional argument about what is a true time crystal.]

Now people have turned to the idea that one can have (with proper formulation of the definitions) time crystals in driven systems.  Perhaps it is not surprising that driving a system periodically can result in periodic response at integer multiples of the driving period, but there is more to it than that.  Achieving some kind of steady-state with spontaneous time periodicity and a lack of runaway heating due to many-body interacting physics is pretty restrictive.  A good write-up of this is here.  A theoretical proposal for how to do this is here, and the experiments that claim to demonstrate this successfully are here and here.   This is another example of how physicists are increasingly interested in understanding and classifying the responses of quantum systems driven out of equilibrium (see here and here).

BackreactionAway Note

I'm traveling next week and will be offline for some days. Blogging may be insubstantial, if existent, and comments may be stuck in the queue longer than usual. But I'm sure you'll survive without me ;) And since you haven't seen the girls for a while, here is a recent photo. They'll be starting school this year in the fall and are very excited about it.

Jordan EllenbergCurry apple chicken

I didn’t have time to do a real shop and had nothing for Friday night dinner so I bought some chicken and a bag of apples and made something that came out surprisingly well; I hereby record it.

Ingredients:

2-3 lb boneless chicken breasts

5 apples, cubed

some scallions

some vegetable oil, whatever kind, doesn’t matter, I used olive

1 tbsp ground coriander

1 tbsp ground cumin

1/2 tsp turmeric

1 tsp salt

however much minced garlic you’re into

1/2-1 tsp garam masala

some crushed tomatoes but you could use actual tomatoes if it weren’t the middle of winter

Recipe:

Get oil hot.  Throw apples and scallions in.  Stir and cook 5 mins until apples soft.  Clear off some pan space and put coriander, cumin, turmeric, salt in the oil, let it cook 30 sec – 1 min, then throw in all the chicken, which by the way you cut into chunks, saute it all up until it’s cooked through.  Put the minced garlic in and let that cook for a minute.  Then put in however much tomato you need to combine with everything else in the pan and make a sauce.  (Probably less than you think, you don’t want soup.)  Turn heat down to warm and mix in garam masala.  You could just eat it like this or you could have been making some kind of starch in parallel.  I made quinoa.  CJ liked this, AB did not.

I took the spice proportions from a Madhur Jaffrey recipe but this is in no way meant as actual Indian food, obviously.  I guess I was just thinking about how when I was a kid you would totally get a “curry chicken salad” which was shredded chicken with curry powder, mayonnaise, and chunked up apple, and I sort of wanted a hot mayonnaiseless version of that.  Also, when I was in grad school learning to cook from Usenet with David Carlton, we used to make a salad with broiled chicken and curry mayonnaise and grapes.  I think it was this.  Does that sound right, David?   Yes, that recipe calls for 2 cups of mayonnaise.  It was a different time.  I feel like we would make this and then put it on top of like 2 pounds of rotini and have food for days.

 

 


February 11, 2017

David Hoggnucleosynthesis and stellar ages

Benoit Coté (Victoria & MSU) came to NYU for the day. He gave a great talk about nucleosynthetic models for the origin of the elements. He is building a full pipeline from raw nuclear physics through to cosmological simulations of structure formation, to get it all right. There were many interesting aspects to his talk and our discussions afterwards. One was about the i-process, intermediate between r and s. Another was about how r-process elements (like Eu) put very strong constraints on the rate at which stars form within their gas. Another was about how we have to combine nucleosynthetic chemistry observations with other kinds of observations (of, say, the PDMF, and neutron-star binaries, and so on) to really get a reliable and true picture of the nucleosynthetic story.

Late in the afternoon, I met with Ruth Angus (Columbia) to further discuss our project on cross-calibrating (or really, self-calibrating) all stellar age indicators. We wrote down some probability expressions, designed a rough design for the code, and discussed how we might structure a Gibbs sampler for this model, which is inherently hierarchical. We also drew a cool chalk-board graphical model (in this tweet), which has overlapping plates, which I am not sure is permitted in PGMs?

David Hoggmaking our own Gaia pipeline

My writing today was in the introduction to the paper Lauren Anderson (Flatiron) and I are writing about the color-magnitude diagram and statistical shrinkage in the Gaia TGAS—2MASS overlap. My view is that the idea behind the project is the same as the fundamental idea behind the Gaia Mission: The astrometry data (the parallaxes) give distances to the nearby stars; these are used to calibrate spectrophotometric models, which deliver distances for the (far more numerous) distant stars. Our goal is to show that this can be done without any involvement of stellar physics or physical models of stellar structure, evolution, or photospheres.

February 10, 2017

Terence TaoOpen thread for mathematicians on the immigration executive order

The self-chosen remit of my blog is “Updates on my research and expository papers, discussion of open problems, and other maths-related topics”.  Of the 774 posts on this blog, I estimate that about 99% of the posts indeed relate to mathematics, mathematicians, or the administration of this mathematical blog, and only about 1% are not related to mathematics or the community of mathematicians in any significant fashion.

This is not one of the 1%.

Mathematical research is clearly an international activity.  But actually a stronger claim is true: mathematical research is a transnational activity, in that the specific nationality of individual members of a research team or research community are (or should be) of no appreciable significance for the purpose of advancing mathematics.  For instance, even during the height of the Cold War, there was no movement in (say) the United States to boycott Soviet mathematicians or theorems, or to only use results from Western literature (though the latter did sometimes happen by default, due to the limited avenues of information exchange between East and West, and former did occasionally occur for political reasons, most notably with the Soviet Union preventing Gregory Margulis from traveling to receive his Fields Medal in 1978 EDIT: and also Sergei Novikov in 1970).    The national origin of even the most fundamental components of mathematics, whether it be the geometry (γεωμετρία) of the ancient Greeks, the algebra (الجبر) of the Islamic world, or the Hindu-Arabic numerals 0,1,\dots,9, are primarily of historical interest, and have only a negligible impact on the worldwide adoption of these mathematical tools. While it is true that individual mathematicians or research teams sometimes compete with each other to be the first to solve some desired problem, and that a citizen could take pride in the mathematical achievements of researchers from their country, one did not see any significant state-sponsored “space races” in which it was deemed in the national interest that a particular result ought to be proven by “our” mathematicians and not “theirs”.   Mathematical research ability is highly non-fungible, and the value added by foreign students and faculty to a mathematics department cannot be completely replaced by an equivalent amount of domestic students and faculty, no matter how large and well educated the country (though a state can certainly work at the margins to encourage and support more domestic mathematicians).  It is no coincidence that all of the top mathematics department worldwide actively recruit the best mathematicians regardless of national origin, and often retain immigration counsel to assist with situations in which these mathematicians come from a country that is currently politically disfavoured by their own.

Of course, mathematicians cannot ignore the political realities of the modern international order altogether.  Anyone who has organised an international conference or program knows that there will inevitably be visa issues to resolve because the host country makes it particularly difficult for certain nationals to attend the event.  I myself, like many other academics working long-term in the United States, have certainly experienced my own share of immigration bureaucracy, starting with various glitches in the renewal or application of my J-1 and O-1 visas, then to the lengthy vetting process for acquiring permanent residency (or “green card”) status, and finally to becoming naturalised as a US citizen (retaining dual citizenship with Australia).  Nevertheless, while the process could be slow and frustrating, there was at least an order to it.  The rules of the game were complicated, but were known in advance, and did not abruptly change in the middle of playing it (save in truly exceptional situations, such as the days after the September 11 terrorist attacks).  One just had to study the relevant visa regulations (or hire an immigration lawyer to do so), fill out the paperwork and submit to the relevant background checks, and remain in good standing until the application was approved in order to study, work, or participate in a mathematical activity held in another country.  On rare occasion, some senior university administrator may have had to contact a high-ranking government official to approve some particularly complicated application, but for the most part one could work through normal channels in order to ensure for instance that the majority of participants of a conference could actually be physically present at that conference, or that an excellent mathematician hired by unanimous consent by a mathematics department could in fact legally work in that department.

With the recent and highly publicised executive order on immigration, many of these fundamental assumptions have been seriously damaged, if not destroyed altogether.  Even if the order was withdrawn immediately, there is no longer an assurance, even for nationals not initially impacted by that order, that some similar abrupt and major change in the rules for entry to the United States could not occur, for instance for a visitor who has already gone through the lengthy visa application process and background checks, secured the appropriate visa, and is already in flight to the country.  This is already affecting upcoming or ongoing mathematical conferences or programs in the US, with many international speakers (including those from countries not directly affected by the order) now cancelling their visit, either in protest or in concern about their ability to freely enter and leave the country.  Even some conferences outside the US are affected, as some mathematicians currently in the US with a valid visa or even permanent residency are uncertain if they could ever return back to their place of work if they left the country to attend a meeting.  In the slightly longer term, it is likely that the ability of elite US institutions to attract the best students and faculty will be seriously impacted.  Again, the losses would be strongest regarding candidates that were nationals of the countries affected by the current executive order, but I fear that many other mathematicians from other countries would now be much more concerned about entering and living in the US than they would have previously.

It is still possible for this sort of long-term damage to the mathematical community (both within the US and abroad) to be reversed or at least contained, but at present there is a real risk of the damage becoming permanent.  To prevent this, it seems insufficient for me for the current order to be rescinded, as desirable as that would be; some further legislative or judicial action would be needed to begin restoring enough trust in the stability of the US immigration and visa system that the international travel that is so necessary to modern mathematical research becomes “just” a bureaucratic headache again.

Of course, the impact of this executive order is far, far broader than just its effect on mathematicians and mathematical research.  But there are countless other venues on the internet and elsewhere to discuss these other aspects (or politics in general).  (For instance, discussion of the qualifications, or lack thereof, of the current US president can be carried out at this previous post.) I would therefore like to open this post to readers to discuss the effects or potential effects of this order on the mathematical community; I particularly encourage mathematicians who have been personally affected by this order to share their experiences.  As per the rules of the blog, I request that “the discussions are kept constructive, polite, and at least tangentially relevant to the topic at hand”.

Some relevant links (please feel free to suggest more, either through comments or by email):


Filed under: math.HO, non-technical, opinion

n-Category Café The Heilbronn Institute and the University of Bristol

The Heilbronn Institute is the mathematical brand of the UK intelligence and spying agency GCHQ (Government Communications Headquarters). GCHQ is one of the country’s largest employers of mathematicians. And the Heilbronn Institute is now claiming to be the largest funder of “pure mathematics” in the country, largely through its many research fellowships at Bristol (where it’s based) and London.

In 2013, Edward Snowden leaked a massive archive of documents that shone a light on the hidden activities of GCHQ and its close partner, the US National Security Agency (NSA), including whole-population surveillance and deliberate stifling of peaceful activism. Much of this was carried out without the permission — or even knowledge — of the politicians who supposedly oversee them.

All this should obviously concern any mathematician with a soul, as I’ve argued. These are our major employers and funders. But you might wonder about the close-up picture. How do spy agencies such as GCHQ and the NSA work their way into academic culture? What do they do to ensure a continuing supply of mathematicians to employ, despite the suspicion with which most of us view them?

Alon Aviram of the Bristol Cable has just published an article on this, describing specific connections between GCHQ/Heilbronn and the University of Bristol — and, more broadly, academic mathematicians and computer scientists:

Alon Aviram, Bristol University working with the surveillance state. The Bristol Cable, 7 February 2017.

It includes some quotes from me and from legendary computer-security scientist Ross Anderson, as well as some nuggets from a long leaked Heilbronn “problem book” that’s interesting in its own right.

Terence TaoA bound on partitioning clusters

Daniel Kane and I have just uploaded to the arXiv our paper “A bound on partitioning clusters“, submitted to the Electronic Journal of Combinatorics. In this short and elementary paper, we consider a question that arose from biomathematical applications: given a finite family {X} of sets (or “clusters”), how many ways can there be of partitioning a set {A \in X} in this family as the disjoint union {A = A_1 \uplus A_2} of two other sets {A_1, A_2} in this family? That is to say, what is the best upper bound one can place on the quantity

\displaystyle | \{ (A,A_1,A_2) \in X^3: A = A_1 \uplus A_2 \}|

in terms of the cardinality {|X|} of {X}? A trivial upper bound would be {|X|^2}, since this is the number of possible pairs {(A_1,A_2)}, and {A_1,A_2} clearly determine {A}. In our paper, we establish the improved bound

\displaystyle | \{ (A,A_1,A_2) \in X^3: A = A_1 \uplus A_2 \}| \leq |X|^{3/p}

where {p} is the somewhat strange exponent

\displaystyle p := \log_3 \frac{27}{4} = 1.73814\dots, \ \ \ \ \ (1)

 

so that {3/p = 1.72598\dots}. Furthermore, this exponent is best possible!

Actually, the latter claim is quite easy to show: one takes {X} to be all the subsets of {\{1,\dots,n\}} of cardinality either {n/3} or {2n/3}, for {n} a multiple of {3}, and the claim follows readily from Stirling’s formula. So it is perhaps the former claim that is more interesting (since many combinatorial proof techniques, such as those based on inequalities such as the Cauchy-Schwarz inequality, tend to produce exponents that are rational or at least algebraic). We follow the common, though unintuitive, trick of generalising a problem to make it simpler. Firstly, one generalises the bound to the “trilinear” bound

\displaystyle | \{ (A_1,A_2,A_3) \in X_1 \times X_2 \times X_3: A_3 = A_1 \uplus A_2 \}|

\displaystyle \leq |X_1|^{1/p} |X_2|^{1/p} |X_3|^{1/p}

for arbitrary finite collections {X_1,X_2,X_3} of sets. One can place all the sets in {X_1,X_2,X_3} inside a single finite set such as {\{1,\dots,n\}}, and then by replacing every set {A_3} in {X_3} by its complement in {\{1,\dots,n\}}, one can phrase the inequality in the equivalent form

\displaystyle | \{ (A_1,A_2,A_3) \in X_1 \times X_2 \times X_3: \{1,\dots,n\} =A_1 \uplus A_2 \uplus A_3 \}|

\displaystyle \leq |X_1|^{1/p} |X_2|^{1/p} |X_3|^{1/p}

for arbitrary collections {X_1,X_2,X_3} of subsets of {\{1,\dots,n\}}. We generalise further by turning sets into functions, replacing the estimate with the slightly stronger convolution estimate

\displaystyle f_1 * f_2 * f_3 (1,\dots,1) \leq \|f_1\|_{\ell^p(\{0,1\}^n)} \|f_2\|_{\ell^p(\{0,1\}^n)} \|f_3\|_{\ell^p(\{0,1\}^n)}

for arbitrary functions {f_1,f_2,f_3} on the Hamming cube {\{0,1\}^n}, where the convolution is on the integer lattice {\bf Z}^n rather than on the finite field vector space {\bf F}_2^n. The advantage of working in this general setting is that it becomes very easy to apply induction on the dimension {n}; indeed, to prove this estimate for arbitrary {n} it suffices to do so for {n=1}. This reduces matters to establishing the elementary inequality

\displaystyle (ab(1-c))^{1/p} + (bc(1-a))^{1/p} + (ca(1-b))^{1/p} \leq 1

for all {0 \leq a,b,c \leq 1}, which can be done by a combination of undergraduate multivariable calculus and a little bit of numerical computation. (The left-hand side turns out to have local maxima at {(1,1,0), (1,0,1), (0,1,1), (2/3,2/3,2/3)}, with the latter being the cause of the numerology (1).)

The same sort of argument also gives an energy bound

\displaystyle E(A,A) \leq |A|^{\log_2 6}

for any subset {A \subset \{0,1\}^n} of the Hamming cube, where

\displaystyle E(A,A) := |\{(a_1,a_2,a_3,a_4) \in A^4: a_1+a_2 = a_3 + a_4 \}|

is the additive energy of {A}. The example {A = \{0,1\}^n} shows that the exponent {\log_2 6} cannot be improved.


Filed under: math.CO, paper

February 09, 2017

BackreactionNew Data from the Early Universe Does Not Rule Out Holography

[img src: entdeckungen.net] It’s string theorists’ most celebrated insight: The world is a hologram. Like everything else string theorists have come up with, it’s an untested hypothesis. But now, it’s been put to test with a new analysis that compares a holographic early universe with its non-holographic counterpart. Tl;dr: Results are inconclusive. When string theorists say we live in a

Jordan EllenbergVacuum cleaner on sale

I was explaining the “regular price” scam to CJ the other day. A store sells a vacuum cleaner for $79.95. One day, they put up a sign saying “SALE! Regular price, $109.95; now MARKED DOWN to $79.95.” The point is to create an imaginary past that never was, a past where vacuum cleaners cost $109.95, a difficult past from which the store has generously granted you respite.
This is what Trump’s team is doing. They’re trying to create an imaginary past in which the last 5 years of life in America was characterized by ubiquitous street crime, unchecked terrorism, and mass unemployment. So that life in America in 2017 and 2018 will seem comparatively placid, safe, and prosperous. Look how much I saved you on this goddamn vacuum cleaner. You’re welcome.


February 07, 2017

Noncommutative GeometryConnes 70

I am happy to report that to  celebrate Alain Connes' 70th birthday, 3 conferences on noncommutative geometry and its interactions with different fields are planned to take place in Shanghai, China. Students, postdocs, young faculty and all those interested in the subject are encouraged to participate. Please check the  Conference webpage for more details.

John PreskillHow do you hear electronic oscillations with light

For decades, understanding the origin of high temperature superconductivity has been regarded as the Holy Grail by physicists in the condensed matter community. The importance of high temperature superconductivity resides not only in its technological promises, but also in the dazzling number of exotic phases and elementary excitations it puts on display for physicists. These myriad phases and excitations give physicists new dimensions and building bricks for understanding and exploiting the world of collective phenomena. The pseudogap, charge-density-wave, nematic and spin liquid phases, for examples, are a few exotica that are found in cuprate high temperature superconductors. Understanding these phases is important for understanding the mechanism behind high temperature superconductivity, but they are also interesting in and of themselves.

The charge-density-wave (CDW) phase in the cuprates – a spontaneous emergence of a periodic modulation of charge density in real space – has particularly garnered a lot of attention. It emerges upon the destruction of the parent antiferromagnetic Mott insulating phase with doping and it appears to directly compete with superconductivity. Whether or not these features are generic, or maybe even necessary, for high temperature superconductivty is an important question. Unfortunately, currently there exists no other comparable high temperature superconducting materials family that enables such questions to be answered.

iqim

Recently, the iridates have emerged as a possible analog to the cuprates. The single layer variant Sr2IrO4, for example, exhibits signatures of both a pseudogap phase and a high temperature superconducting phase. However, with an increasing parallel being drawn between the iridates and the cuprates in terms of their electronic phases, CDW has so far eluded detection in any iridate, calling into question the validity of this comparison. Rather than studying the single layer variant, we decided to look at the bilayer iridate Sr3Ir2O7 in which a clear Mott insulator to metal transition has been reported with doping.

While CDW has been observed in many materials, what made it elusive in cuprates for many years is its spatially short-ranged (it extends only a few lattice spacings long) and often temporally short-ranged (it blinks in and out of existence quickly) nature. To get a good view of this order, experimentalists had to literally pin it down using external influences like magnetic fields or chemical dopants to suppress the temporal fluctuations and then use very sensitive diffraction or scanning tunneling based probes to observe them.

But rather than looking in real space for signatures of the CDW order, an alternative approach is to look for them in the time domain. Works by the Gedik group at MIT and the Orenstein group at U.C. Berkeley have shown that one can use ultrafast time-resolved optical reflectivity to “listen” for the tone of a CDW to infer its presence in the cuprates. In these experiments, one impulsively excites a coherent mode of the CDW using a femtosecond laser pulse, much like one would excite the vibrational mode of a tuning fork by impulsively banging it. One then stroboscopically looks for these CDW oscillations via temporally periodic modulations in its optical reflectivity, much like one would listen for the tone produced by the tuning fork. If you manage to hear the tone of the CDW, then you have established its existence!

We applied a similar approach to Sr3Ir2O7 and its doped versions [hear our experiment]. To our delight, the ringing of a CDW mode sounded immediately upon doping across its Mott insulator to metal transition, implying that the electronic liquid born from the doped Mott insulator is unstable to CDW formation, very similar to the case in cuprates. Also like the case of cuprates, this charge-density-wave is of a special nature: it is either very short-ranged, or temporally fluctuating. Whether or not there is a superconducting phase that competes with the CDW in Sr3Ir2O7 remains to be seen. If so, the phenomenology of the cuprates may really be quite generic. If not, the interesting question of why not is worth pursuing. And who knows, maybe the fact that we have a system that can be controllably tuned between the antiferromagnetic order and the CDW order may find use in technology some day.


February 06, 2017

John BaezSaving Climate Data (Part 5)

march-for-science-earth-day

There’s a lot going on! Here’s a news roundup. I will separately talk about what the Azimuth Climate Data Backup Project is doing.

I’ll start with the bad news, and then go on to some good news.

Tweaking the EPA website

Scientists are keeping track of how Trump administration is changing the Environmental Protection Agency website, with before-and-after photos, and analysis:

• Brian Kahn, Behold the “tweaks” Trump has made to the EPA website (so far), National Resources Defense Council blog, 3 February 2017.

There’s more about “adaptation” to climate change, and less about how it’s caused by carbon emissions.

All of this would be nothing compared to the new bill to eliminate the EPA, or Myron Ebell’s plan to fire most of the people working there:

• Joe Davidson, Trump transition leader’s goal is two-thirds cut in EPA employees, Washington Post, 30 January 2017.

If you want to keep track of this battle, I recommend getting a 30-day free subscription to this online magazine:

InsideEPA.com.

Taking animal welfare data offline

The Trump team is taking animal-welfare data offline. The US Department of Agriculture will no longer make lab inspection results and violations publicly available, citing privacy concerns:

• Sara Reardon, US government takes animal-welfare data offline, Nature Breaking News, 3 Feburary 2017.

Restricting access to geospatial data

A new bill would prevent the US government from providing access to geospatial data if it helps people understand housing discrimination. It goes like this:

Notwithstanding any other provision of law, no Federal funds may be used to design, build, maintain, utilize, or provide access to a Federal database of geospatial information on community racial disparities or disparities in access to affordable housing._

For more on this bill, and the important ways in which such data has been used, see:

• Abraham Gutman, Scott Burris, and the Temple University Center for Public Health Law Research, Where will data take the Trump administration on housing?, Philly.com, 1 February 2017.

The EDGI fights back

The Environmental Data and Governance Initiative or EDGI is working to archive public environmental data. They’re helping coordinate data rescue events. You can attend one and have fun eating pizza with cool people while saving data:

• 3 February 2017, Portland
• 4 February 2017, New York City
• 10-11 February 2017, Austin Texas
• 11 February 2017, U. C. Berkeley, California
• 18 February 2017, MIT, Cambridge Massachusetts
• 18 February 2017, Haverford Connecticut
• 18-19 February 2017, Washington DC
• 26 February 2017, Twin Cities, Minnesota

Or, work with EDGI to organize one your own data rescue event! They provide some online tools to help download data.

I know there will also be another event at UCLA, so the above list is not complete, and it will probably change and grow over time. Keep up-to-date at their site:

Environmental Data and Governance Initiative.

Scientists fight back

The pushback is so big it’s hard to list it all! For now I’ll just quote some of this article:

• Tabitha Powledge, The gag reflex: Trump info shutdowns at US science agencies, especially EPA, 27 January 2017.

THE PUSHBACK FROM SCIENCE HAS BEGUN

Predictably, counter-tweets claiming to come from rebellious employees at the EPA, the Forest Service, the USDA, and NASA sprang up immediately. At The Verge, Rich McCormick says there’s reason to believe these claims may be genuine, although none has yet been verified. A lovely head on this post: “On the internet, nobody knows if you’re a National Park.”

At Hit&Run, Ronald Bailey provides handles for several of these alt tweet streams, which he calls “the revolt of the permanent government.” (That’s a compliment.)

Bailey argues, “with exception perhaps of some minor amount of national security intelligence, there is no good reason that any information, data, studies, and reports that federal agencies produce should be kept from the public and press. In any case, I will be following the Alt_Bureaucracy feeds for a while.”

NeuroDojo Zen Faulkes posted on how to demand that scientific societies show some backbone. “Ask yourself: “Have my professional societies done anything more political than say, ‘Please don’t cut funding?’” Will they fight?,” he asked.

Scientists associated with the group_ 500 Women Scientists _donned lab coats and marched in DC as part of the Women’s March on Washington the day after Trump’s Inauguration, Robinson Meyer reported at the Atlantic. A wildlife ecologist from North Carolina told Meyer, “I just can’t believe we’re having to yell, ‘Science is real.’”

Taking a cue from how the Women’s March did its social media organizing, other scientists who want to set up a Washington march of their own have put together a closed Facebook group that claims more than 600,000 members, Kate Sheridan writes at STAT.

The #ScienceMarch Twitter feed says a date for the march will be posted in a few days. [The march will be on 22 April 2017.] The group also plans to release tools to help people interested in local marches coordinate their efforts and avoid duplication.

At The Atlantic, Ed Yong describes the political action committee 314Action. (314=the first three digits of pi.)

Among other political activities, it is holding a webinar on Pi Day—March 14—to explain to scientists how to run for office. Yong calls 314Action the science version of Emily’s List, which helps pro-choice candidates run for office. 314Action says it is ready to connect potential candidate scientists with mentors—and donors.

Other groups may be willing to step in when government agencies wimp out. A few days before the Inauguration, the Centers for Disease Control and Prevention abruptly and with no explanation cancelled a 3-day meeting on the health effects of climate change scheduled for February. Scientists told Ars Technica’s Beth Mole that CDC has a history of running away from politicized issues.

One of the conference organizers from the American Public Health Association was quoted as saying nobody told the organizers to cancel.

I believe it. Just one more example of the chilling effect on global warming. In politics, once the Dear Leader’s wishes are known, some hirelings will rush to gratify them without being asked.

The APHA guy said they simply wanted to head off a potential last-minute cancellation. Yeah, I guess an anticipatory pre-cancellation would do that.

But then—Al Gore to the rescue! He is joining with a number of health groups—including the American Public Health Association—to hold a one-day meeting on the topic Feb 16 at the Carter Center in Atlanta, CDC’s home base. Vox’s Julia Belluz reports that it is not clear whether CDC officials will be part of the Gore rescue event.

The Sierra Club fights back

The Sierra Club, of which I’m a proud member, is using the Freedom of Information Act or FOIA to battle or at least slow the deletion of government databases. They wisely started even before Trump took power:

• Jennifer A Dlouhy, Fearing Trump data purge, environmentalists push to get records, BloombergMarkets, 13 January 2017.

Here’s how the strategy works:

U.S. government scientists frantically copying climate data they fear will disappear under the Trump administration may get extra time to safeguard the information, courtesy of a novel legal bid by the Sierra Club.

The environmental group is turning to open records requests to protect the resources and keep them from being deleted or made inaccessible, beginning with information housed at the Environmental Protection Agency and the Department of Energy. On Thursday [January 9th], the organization filed Freedom of Information Act requests asking those agencies to turn over a slew of records, including data on greenhouse gas emissions, traditional air pollution and power plants.

The rationale is simple: Federal laws and regulations generally block government agencies from destroying files that are being considered for release. Even if the Sierra Club’s FOIA requests are later rejected, the record-seeking alone could prevent files from being zapped quickly. And if the records are released, they could be stored independently on non-government computer servers, accessible even if other versions go offline.


Mark Chu-CarrollUnderstanding Global Warming Scale Issues

Aside from the endless stream of Cantor cranks, the next biggest category of emails I get is from climate “skeptics”. They all ask pretty much the same question. For example, here’s one I received today:

My personal analysis, and natural sceptisism tells me, that there are something fundamentally wrong with the entire warming theory when it comes to the CO2.

If a gas in the atmosphere increase from 0.03 to 0.04… that just cant be a significant parameter, can it?

I generally ignore it, because… let’s face it, the majority of people who ask this question aren’t looking for a real answer. But this one was much more polite and reasonable than most, so I decided to answer it. And once I went to the trouble of writing a response, I figured that I might as well turn it into a post as well.

The current figures – you can find them in a variety of places from wikipedia to the US NOAA – are that the atmosphere CO2 has changed from around 280 parts per million in 1850 to 400 parts per million today.

Why can’t that be a significant parameter?

There’s a couple of things to understand to grasp global warming: how much energy carbon dioxide can trap in the atmosphere, and hom much carbon dioxide there actually is in the atmosphere. Put those two facts together, and you realize that we’re talking about a massive quantity of carbon dioxide trapping a massive amount of energy.

The problem is scale. Humans notoriously have a really hard time wrapping our heads around scale. When numbers get big enough, we aren’t able to really grasp them intuitively and understand what they mean. The difference between two numbers like 300 and 400ppm is tiny, we can’t really grasp how in could be significant, because we aren’t good at taking that small difference, and realizing just how ridiculously large it actually is.

If you actually look at the math behind the greenhouse effect, you find that some gasses are very effective at trapping heat. The earth is only habitable because of the carbon dioxide in the atmosphere – without it, earth would be too cold for life. Small amounts of it provide enough heat-trapping effect to move us from a frozen rock to the world we have. Increasing the quantity of it increases the amount of heat it can trap.

Let’s think about what the difference between 280 and 400 parts per million actually means at the scale of earth’s atmosphere. You hear a number like 400ppm – that’s 4 one-hundreds of one percent – that seems like nothing, right? How could that have such a massive effect?!

But like so many other mathematical things, you need to put that number into the appropriate scale. The earths atmosphere masses roughly 5 times 10^21 grams. 400ppm of that scales to 2 times 10^18 grams of carbon dioxide. That’s 2 billion trillion kilograms of CO2. Compared to 100 years ago, that’s about 800 million trillion kilograms of carbon dioxide added to the atmosphere over the last hundred years. That’s a really, really massive quantity of carbon dioxide! scaled to the number of particles, that’s something around 10^40th (plus or minus a couple of powers of ten – at this scale, who cares?) additional molecules of carbon dioxide in the atmosphere. It’s a very small percentage, but it’s a huge quantity.

When you talk about trapping heat, you also have to remember that there’s scaling issues there, too. We’re not talking about adding 100 degrees to the earths temperature. It’s a massive increase in the quantity of energy in the atmosphere, but because the atmosphere is so large, it doesn’t look like much: just a couple of degrees. That can be very deceptive – 5 degrees celsius isn’t a huge temperature difference. But if you think of the quantity of extra energy that’s being absorbed by the atmosphere to produce that difference, it’s pretty damned huge. It doesn’t necessarily look like all that much when you see it stated at 2 degrees celsius – but if you think of it terms of the quantity of additional energy being trapped by the atmosphere, it’s very significant.

Calculating just how much energy a molecule of CO2 can absorb is a lot trickier than calculating the mass-change of the quantity of CO2 in the atmosphere. It’s a complicated phenomenon which involves a lot of different factors – how much infrared is absorbed by an atom, how quickly that energy gets distributed into the other molecules that it interacts with… I’m not going to go into detail on that. There’s a ton of places, like here, where you can look up a detailed explanation. But when you consider the scale issues, it should be clear that there’s a pretty damned massive increase in the capacity to absorb energy in a small percentage-wise increase in the quantity of CO2.

February 04, 2017

Tommaso DorigoLHCb Finds Suppressed Lambda_B Decay

The so-called Lambda_b baryon is a well-studied particle nowadays, with several experiments having measured its main production properties and decay modes in the course of the past two decades. It is a particle made of quarks: three of them, like the proton and the neutron. Being electrically neutral, it is easily likened to the neutron, which has a quark composition "udd". In the space of quark configurations, the Lambda_b is in fact obtained by exchanging a down-type quark of the neutron with a bottom quark, getting the "udb" combination.

read more

Scott AaronsonFirst they came for the Iranians

Action Item: If you’re an American academic, please sign the petition against the Immigration Executive Order. (There are already more than eighteen thousand signatories, including Nobel Laureates, Fields Medalists, you name it, but it could use more!)

I don’t expect this petition to have the slightest effect on the regime, but at least we should demonstrate to the world and to history that American academia didn’t take this silently.


I’m sure there were weeks, in February or March 1933, when the educated, liberal Germans commiserated with each other over the latest outrages of their new Chancellor, but consoled themselves that at least none of it was going to affect them personally.

This time, it’s taken just five days, since the hostile takeover of the US by its worst elements, for edicts from above to have actually hurt my life and (much more directly) the lives of my students, friends, and colleagues.

Today, we learned that Trump is suspending the issuance of US visas to people from seven majority-Islamic countries, including Iran (but strangely not Saudi Arabia, the cradle of Wahhabist terrorism—not that that would be morally justified either).  This suspension might last just 30 days, but might also continue indefinitely—particularly if, as seems likely, the Iranian government thumbs its nose at whatever Trump demands that it do to get the suspension rescinded.

So the upshot is that, until further notice, science departments at American universities can no longer recruit PhD students from Iran—a country that, along with China, India, and a few others, has long been the source of some of our best talent.  This will directly affect this year’s recruiting season, which is just now getting underway.  (If Canada and Australia have any brains, they’ll snatch these students, and make the loss America’s.)

But what about the thousands of Iranian students who are already here?  So far, no one’s rounding them up and deporting them.  But their futures have suddenly been thrown into jeopardy.

Right now, I have an Iranian PhD student who came to MIT on a student visa in 2013.  He started working with me two years ago, on the power of a rudimentary quantum computing model inspired by (1+1)-dimensional integrable quantum field theory.  You can read our paper about it, with Adam Bouland and Greg Kuperberg, here.  It so happens that this week, my student is visiting us in Austin and staying at our home.  He’s spent the whole day pacing around, terrified about his future.  His original plan, to do a postdoc in the US after he finishes his PhD, now seems impossible (since it would require a visa renewal).

Look: in the 11-year history of this blog, there have been only a few occasions when I felt so strongly about something that I stood my ground, even in the face of widespread attacks from people who I otherwise respected.  One, of course, was when I spoke out for shy nerdy males, and for a vision of feminism broad enough to recognize their suffering as a problem.  A second was when I was more blunt about D-Wave, and about its and its supporters’ quantum speedup claims, than some of my colleagues were comfortable with.  But the remaining occasions almost all involved my defending the values of the United States, Israel, Zionism, or “the West,” or condemning Islamic fundamentalism, radical leftism, or the worldviews of such individuals as Noam Chomsky or my “good friend” Mahmoud Ahmadinejad.

Which is simply to say: I don’t think anyone on earth can accuse me of secret sympathies for the Iranian government.

But when it comes to student visas, I can’t see that my feelings about the mullahs have anything to do with the matter.  We’re talking about people who happen to have been born in Iran, who came to the US to do math and science.  Would we rather have these young scientists here, filled with gratitude for the opportunities we’ve given them, or back in Iran filled with justified anger over our having expelled them?

To the Trump regime, I make one request: if you ever decide that it’s the policy of the US government to deport my PhD students, then deport me first.  I’m practically begging you: come to my house, arrest me, revoke my citizenship, and tear up the awards I’ve accepted at the White House and the State Department.  I’d consider that to be the greatest honor of my career.

And to those who cheered Trump’s campaign in the comments of this blog: go ahead, let me hear you defend this.


Update (Jan. 27, 2017): To everyone who’s praised the “courage” that it took me to say this, thank you so much—but to be perfectly honest, it takes orders of magnitude less courage to say this, than to say something that any of your friends or colleagues might actually disagree with! The support has been totally overwhelming, and has reaffirmed my sense that the United States is now effectively two countries, an open and a closed one, locked in a cold Civil War.

Some people have expressed surprise that I’d come out so strongly for Iranian students and researchers, “given that they don’t always agree with my politics,” or given my unapologetic support for the founding principles (if not always the actions) of the United States and of Israel. For my part, I’m surprised that they’re surprised! So let me say something that might be clarifying.

I care about the happiness, freedom, and welfare of all the men and women who are actually working to understand the universe and build the technologies of the future, and of all the bright young people who want to join these quests, whatever their backgrounds and wherever they might be found—whether it’s in Iran or Israel, in India or China or right here in the US.  The system of science is far from perfect, and we often discuss ways to improve it on this blog.  But I have not the slightest interest in tearing down what we have now, or destroying the world’s current pool of scientific talent in some cleansing fire, in order to pursue someone’s mental model of what the scientific community used to look like in Periclean Athens—or for that matter, their fantasy of what it would look like in a post-gender post-racial communist utopia.  I’m interested in the actual human beings doing actual science who I actually meet, or hope to meet.

Understand that, and a large fraction of all the political views that I’ve ever expressed on this blog, even ones that might seem to be in tension with each other, fall out as immediate corollaries.

(Related to that, some readers might be interested in a further explanation of my views about Zionism. See also my thoughts about liberal democracy, in response to numerous comments here by Curtis Yarvin a.k.a. Mencius Moldbug a.k.a. “Boldmug.”)


Update (Jan. 29) Here’s a moving statement from my student Saeed himself, which he asked me to share here.

This is not of my best interest to talk about politics. Not because I am scared but because I know little politics. I am emotionally affected like many other fellow human beings on this planet. But I am still in the US and hopefully I can pursue my degree at MIT. But many other talented friends of mine can’t. Simply because they came back to their hometowns to visit their parents. On this matter, I must say that like many of my friends in Iran I did not have a chance to see my parents in four years, my basic human right, just because I am from a particular nationality; something that I didn’t have any decision on, and that I decided to study in my favorite school, something that I decided when I was 15. When, like many other talented friends of mine, I was teaching myself mathematics and physics hoping to make big impacts in positive ways in the future. And I must say I am proud of my nationality – home is home wherever it is. I came to America to do science in the first place. I still don’t have any other intention, I am a free man, I can do science even in desert, if I have to. If you read history you’ll see scientists even from old ages have always been traveling.

As I said I know little about many things, so I just phrase my own standpoint. You should also talk to the ones who are really affected. A good friend of mine, Ahmad, who studies Mechanical engineering in UC Berkeley, came back to visit his parents in August. He is one of the most talented students I have ever seen in my life. He has been waiting for his student visa since then and now he is ultimately depressed because he cannot finish his degree. The very least the academic society can do is to help students like Ahmad finish their degrees even if it is from abroad. I can’t emphasize enough I know little about many things. But, from a business standpoint, this is a terrible deal for America. Just think about it. All international students in this country have been getting free education untill 22, in the American point of reference, and now they are using their knowledge to build technology in the USA. Just do a simple calculation and see how much money this would amount to. In any case my fellow international students should rethink this deal, and don’t take it unless at the least they are treated with respect. Having said all of this I must say I love the people of America, I have had many great friends here, great advisors specially Scott Aaronson and Aram Harrow, with whom I have been talking about life, religion, freedom and my favorite topic the foundations of the universe. I am grateful for the education I received at MIT and I think I have something I didn’t have before. I don’t even hate Mr Trump. I think he would feel different if we have a cup of coffee sometime.


Update (Jan. 31): See also this post by Terry Tao.


Update (Feb. 2): If you haven’t been checking the comments on this post, come have a look if you’d like to watch me and others doing our best to defend the foundations of Enlightenment and liberal democracy against a regiment of monarchists and neoreactionaries, including the notorious Mencius Moldbug, as well as a guy named Jim who explicitly advocates abolishing democracy and appointing Trump as “God-Emperor” with his sons to succeed him. (Incidentally, which son? Is Ivanka out of contention?)

I find these people to be simply articulating, more clearly and logically than most, the worldview that put Trump into office and where it inevitably leads. And any of us who are horrified by it had better get over our incredulity, fast, and pick up the case for modernity and Enlightenment where Spinoza and Paine and Mill and all the others left it off—because that’s what’s actually at stake here, and if we don’t understand that then we’ll continue to be blindsided.

February 02, 2017

Chad OrzelPhysics Blogging Round-Up: January

It’s a new month now, so it’s time to share links to what I wrote for Forbes last month:

Small College Astronomers Predict Big Stellar Explosion: I mostly leave astronomy stories to others, but I heard about this from a friend at Calvin College, and it’s a story that hits a lot of my pet issues, so I wrote it up.

For Scientists, Recognition Is A Weird And Contingent Thing: Vera Rubin tops the list of great women in science who died in 2016, but AMO physics lost two great women but less famous women as well. I spent a while thinking about why they had such different levels of status.

Squeezing Through A Loophole In The Laws Of Physics To Cool A Drum: A detailed look at a new experiment from NIST in Boulder, using squeezed light to reach temperatures below the standard quantum limit.

How Much Scientific Research Is Wasted?: There was a claim floating around that 85% of biomedical research is a waste of time. On closer inspection, it turned out to be underwhelming, but the general question of what counts as waste in science is interesting.

Do Dark Matter And Dark Energy Affect Ordinary Atoms?: A post in which I almost violate Betteridge’s Law of Headlines, because the answer is “Yes, but not so you’d notice.”

As always, I’m basically happy with these, though I was running a fever while I wrote the last one, so it might not be the clearest thing I’ve ever done. I’m also very pleased to have gotten nice email from two of the people mentioned in these posts thanking me for the stories. I’m always nervous when I recognize the name of a scientist in my inbox, for fear that they’re writing to complain that I misunderstood or misrepresented them, so a “Nice job!” message is wonderful.

This month also showed the typical lack of correlation between effort and traffic– the post I worked hardest on was the one about Vera Rubin, Debbie Jin, and Katharine Gebbie, but it got the fewest views of the lot, even with the last couple being overshadowed by momentous political events. The astronomy one was dashed off really quickly, and did the best, because there’s a boundless audience for astronomy.

And, so, that was January. On to the next thing.

John BaezInformation Geometry (Part 16)

This week I’m giving a talk on biology and information:

• John Baez, Biology as information dynamics, talk for Biological Complexity: Can it be Quantified?, a workshop at the Beyond Center, 2 February 2017.

While preparing this talk, I discovered a cool fact. I doubt it’s new, but I haven’t exactly seen it elsewhere. I came up with it while trying to give a precise and general statement of ‘Fisher’s fundamental theorem of natural selection’. I won’t start by explaining that theorem, since my version looks rather different than Fisher’s, and I came up with mine precisely because I had trouble understanding his. I’ll say a bit more about this at the end.

Here’s my version:

The square of the rate at which a population learns information is the variance of its fitness.

This is a nice advertisement for the virtues of diversity: more variance means faster learning. But it requires some explanation!

The setup

Let’s start by assuming we have n different kinds of self-replicating entities with populations P_1, \dots, P_n. As usual, these could be all sorts of things:

• molecules of different chemicals
• organisms belonging to different species
• genes of different alleles
• restaurants belonging to different chains
• people with different beliefs
• game-players with different strategies
• etc.

I’ll call them replicators of different species.

Let’s suppose each population P_i is a function of time that grows at a rate equal to this population times its ‘fitness’. I explained the resulting equation back in Part 9, but it’s pretty simple:

\displaystyle{ \frac{d}{d t} P_i(t) = f_i(P_1(t), \dots, P_n(t)) \, P_i(t)   }

Here f_i is a completely arbitrary smooth function of all the populations! We call it the fitness of the ith species.

This equation is important, so we want a short way to write it. I’ll often write f_i(P_1(t), \dots, P_n(t)) simply as f_i, and P_i(t) simply as P_i. With these abbreviations, which any red-blooded physicist would take for granted, our equation becomes simply this:

\displaystyle{ \frac{dP_i}{d t}  = f_i \, P_i   }

Next, let p_i(t) be the probability that a randomly chosen organism is of the ith species:

\displaystyle{ p_i(t) = \frac{P_i(t)}{\sum_j P_j(t)} }

Starting from our equation describing how the populations evolve, we can figure out how these probabilities evolve. The answer is called the replicator equation:

\displaystyle{ \frac{d}{d t} p_i(t)  = ( f_i - \langle f \rangle ) \, p_i(t) }

Here \langle f \rangle is the average fitness of all the replicators, or mean fitness:

\displaystyle{ \langle f \rangle = \sum_j f_j(P_1(t), \dots, P_n(t)) \, p_j(t)  }

In what follows I’ll abbreviate the replicator equation as follows:

\displaystyle{ \frac{dp_i}{d t}  = ( f_i - \langle f \rangle ) \, p_i }

The result

Okay, now let’s figure out how fast the probability distribution

p(t) = (p_1(t), \dots, p_n(t))

changes with time. For this we need to choose a way to measure the length of the vector

\displaystyle{  \frac{dp}{dt} = (\frac{d}{dt} p_1(t), \dots, \frac{d}{dt} p_n(t)) }

And here information geometry comes to the rescue! We can use the Fisher information metric, which is a Riemannian metric on the space of probability distributions.

I’ve talked about the Fisher information metric in many ways in this series. The most important fact is that as a probability distribution p(t) changes with time, its speed

\displaystyle{  \left\| \frac{dp}{dt} \right\|}

as measured using the Fisher information metric can be seen as the rate at which information is learned. I’ll explain that later. Right now I just want a simple formula for the Fisher information metric. Suppose v and w are two tangent vectors to the point p in the space of probability distributions. Then the Fisher information metric is given as follows:

\displaystyle{ \langle v, w \rangle = \sum_i \frac{1}{p_i} \, v_i w_i }

Using this we can calculate the speed at which p(t) moves when it obeys the replicator equation. Actually the square of the speed is simpler:

\begin{array}{ccl}  \displaystyle{ \left\| \frac{dp}{dt}  \right\|^2 } &=& \displaystyle{ \sum_i \frac{1}{p_i} \left( \frac{dp_i}{dt} \right)^2 } \\ \\  &=& \displaystyle{ \sum_i \frac{1}{p_i} \left( ( f_i - \langle f \rangle ) \, p_i \right)^2 } \\ \\  &=& \displaystyle{ \sum_i  ( f_i - \langle f \rangle )^2 p_i }   \end{array}

The answer has a nice meaning, too! It’s just the variance of the fitness: that is, the square of its standard deviation.

So, if you’re willing to buy my claim that the speed \|dp/dt\| is the rate at which our population learns new information, then we’ve seen that the square of the rate at which a population learns information is the variance of its fitness!

Fisher’s fundamental theorem

Now, how is this related to Fisher’s fundamental theorem of natural selection? First of all, what is Fisher’s fundamental theorem? Here’s what Wikipedia says about it:

It uses some mathematical notation but is not a theorem in the mathematical sense.

It states:

“The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.”

Or in more modern terminology:

“The rate of increase in the mean fitness of any organism at any time ascribable to natural selection acting through changes in gene frequencies is exactly equal to its genetic variance in fitness at that time”.

Largely as a result of Fisher’s feud with the American geneticist Sewall Wright about adaptive landscapes, the theorem was widely misunderstood to mean that the average fitness of a population would always increase, even though models showed this not to be the case. In 1972, George R. Price showed that Fisher’s theorem was indeed correct (and that Fisher’s proof was also correct, given a typo or two), but did not find it to be of great significance. The sophistication that Price pointed out, and that had made understanding difficult, is that the theorem gives a formula for part of the change in gene frequency, and not for all of it. This is a part that can be said to be due to natural selection

Price’s paper is here:

• George R. Price, Fisher’s ‘fundamental theorem’ made clear, Annals of Human Genetics 36 (1972), 129–140.

I don’t find it very clear, perhaps because I didn’t spend enough time on it. But I think I get the idea.

My result is a theorem in the mathematical sense, though quite an easy one. I assume a population distribution evolves according to the replicator equation and derive an equation whose right-hand side matches that of Fisher’s original equation: the variance of the fitness.

But my left-hand side is different: it’s the square of the speed of the corresponding probability distribution, where speed is measured using the ‘Fisher information metric’. This metric was discovered by the same guy, Ronald Fisher, but I don’t think he used it in his work on the fundamental theorem!

Something a bit similar to my statement appears as Theorem 2 of this paper:

• Marc Harper, Information geometry and evolutionary game theory.

and for that theorem he cites:

• Josef Hofbauer and Karl Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.

However, his Theorem 2 really concerns the rate of increase of fitness, like Fisher’s fundamental theorem. Moreover, he assumes that the probability distribution p(t) flows along the gradient of a function, and I’m not assuming that. Indeed, my version applies to situations where the probability distribution moves round and round in periodic orbits!

Relative information and the Fisher information metric

The key to generalizing Fisher’s fundamental theorem is thus to focus on the speed at which p(t) moves, rather than the increase in fitness. Why do I call this speed the ‘rate at which the population learns information’? It’s because we’re measuring this speed using the Fisher information metric, which is closely connected to relative information, also known as relative entropy or the Kullback–Leibler divergence.

I explained this back in Part 7, but that explanation seems hopelessly technical to me now, so here’s a faster one, which I created while preparing my talk.

The information of a probability distribution q relative to a probability distribution p is

\displaystyle{     I(q,p) = \sum_{i =1}^n q_i \log\left(\frac{q_i}{p_i}\right) }

It says how much information you learn if you start with a hypothesis p saying that the probability of the ith situation was p_i, and then update this to a new hypothesis q.

Now suppose you have a hypothesis that’s changing with time in a smooth way, given by a time-dependent probability p(t). Then a calculation shows that

\displaystyle{ \left.\frac{d}{dt} I(p(t),p(t_0)) \right|_{t = t_0} = 0 }

for all times t_0. This seems paradoxical at first. I like to jokingly put it this way:

To first order, you’re never learning anything.

However, as long as the velocity \frac{d}{dt}p(t_0) is nonzero, we have

\displaystyle{ \left.\frac{d^2}{dt^2} I(p(t),p(t_0)) \right|_{t = t_0} > 0 }

so we can say

To second order, you’re always learning something… unless your opinions are fixed.

This lets us define a ‘rate of learning’—that is, a ‘speed’ at which the probability distribution p(t) moves. And this is precisely the speed given by the Fisher information metric!

In other words:

\displaystyle{ \left\|\frac{dp}{dt}(t_0)\right\|^2 =  \left.\frac{d^2}{dt^2} I(p(t),p(t_0)) \right|_{t = t_0} }

where the length is given by Fisher information metric. Indeed, this formula can be used to define the Fisher information metric. From this definition we can easily work out the concrete formula I gave earlier.

In summary: as a probability distribution moves around, the relative information between the new probability distribution and the original one grows approximately as the square of time, not linearly. So, to talk about a ‘rate at which information is learned’, we need to use the above formula, involving a second time derivative. This rate is just the speed at which the probability distribution moves, measured using the Fisher information metric. And when we have a probability distribution describing how many replicators are of different species, and it’s evolving according to the replicator equation, this speed is also just the variance of the fitness!


February 01, 2017

John BaezBiology as Information Dynamics

This is my talk for the workshop Biological Complexity: Can It Be Quantified?

• John Baez, Biology as information dynamics, 2 February 2017.

Abstract. If biology is the study of self-replicating entities, and we want to understand the role of information, it makes sense to see how information theory is connected to the ‘replicator equation’—a simple model of population dynamics for self-replicating entities. The relevant concept of information turns out to be the information of one probability distribution relative to another, also known as the Kullback–Leibler divergence. Using this we can get a new outlook on free energy, see evolution as a learning process, and give a clean general formulation of Fisher’s fundamental theorem of natural selection.

For more, read:

• Marc Harper, The replicator equation as an inference dynamic.

• Marc Harper, Information geometry and evolutionary game theory.

• Barry Sinervo and Curt M. Lively, The rock-paper-scissors game and the evolution of alternative male strategies, Nature 380 (1996), 240–243.

• John Baez, Diversity, entropy and thermodynamics.

• John Baez, Information geometry.

The last reference contains proofs of the equations shown in red in my slides.
In particular, Part 16 contains a proof of my updated version of Fisher’s fundamental theorem.


John BaezQuantifying Biological Complexity

Next week I’m going to this workshop:

Biological Complexity: Can It Be Quantified?, 1-3 February 2017, Beyond Center for Fundamental Concepts in Science, Arizona State University, Tempe Arizona. Organized by Paul Davies.

I haven’t heard that any of it will be made publicly available, but I’ll see if there’s something I can show you. Here’s the schedule:

Wednesday February 1st

9:00 – 9:30 am Paul Davies

Brief welcome address, outline of the subject and aims of the meeting

Session 1. Life: do we know it when we see it?

9:30 – 10:15 am: Chris McKay, “Mission to Enceladus”

10:15 – 10:45 am: Discussion

10:45– 11:15 am: Tea/coffee break

11:15 – 12:00 pm: Kate Adamala, “Alive but not life”

12:00 – 12:30 pm: Discussion

12:30 – 2:00 pm: Lunch

Session 2. Quantifying life

2:00 – 2:45 pm: Lee Cronin, “The living and the dead: molecular signatures of life”

2:45 – 3:30 pm: Sara Walker, “Can we build a life meter?”

3:30 – 4:00 pm: Discussion

4:00 – 4:30 pm: Tea/coffee break

4:30 – 5:15 pm: Manfred Laubichler, “Complexity is smaller than you think”

5:15 – 5:30 pm: Discussion

The Beyond Annual Lecture

7:00 – 8:30 pm: Sean Carroll, “Our place in the universe”

Thursday February 2nd

Session 3: Life, information and the second law of thermodynamics

9:00 – 9:45 am: James Crutchfield, “Vital bits: the fuel of life”

9:45 – 10:00 am: Discussion

10:00 – 10:45 pm: John Baez, “Information and entropy in biology”

10:45 – 11:00 am: Discussion

11:00 – 11:30 pm: Tea/coffee break

11:30 – 12:15 pm: Chris Adami, “What is biological information?”

12:15 – 12:30 pm: Discussion

12:30 – 2:00 pm: Lunch

Session 4: The emergence of agency

2:00 – 2:45 pm: Olaf Khang Witkowski, “When do autonomous agents act collectively?”

2:45 – 3:00 pm: Discussion

3:00 – 3:45 pm: William Marshall, “When macro beats micro”

3:45 – 4:00 pm: Discussion

4:00 – 4:30 am: Tea/coffee break

4:30 – 5:15pm: Alexander Boyd, “Biology’s demons”

5:15 – 5:30 pm: Discussion

Friday February 3rd

Session 5: New physics?

9:00 – 9:45 am: Sean Carroll, “Laws of complexity, laws of life?”

9:45 – 10:00 am: Discussion

10:00 – 10:45 am: Andreas Wagner, “The arrival of the fittest”

10:45 – 11:00 am: Discussion

11:00 – 11:30 am: Tea/coffee break

11:30 – 12:30 pm: George Ellis, “Top-down causation demands new laws”

12:30 – 2:00 pm: Lunch


Chad OrzelNew Book Alert: “Breakfast With Einstein”

So, I tweeted about this yesterday, but I also spent the entire day feeling achy and feverish, so didn’t have brains or time for a blog post with more details. I’m feeling healthier this morning, though time is still short, so I’ll give a quick summary of the details:

— As you can see in the photo (taken with my phone at Starbucks just before I took these to the post office to mail them), I signed a contract for a new book. Four copies, because lawyers.

— The contract is with Oneworld Publications in the UK, who had a best-seller on that side of the pond with How to Teach Quantum Physics to Your Dog.

— The working title of the book is Breakfast With Einstein, and the subject matter is basically this talk I gave at TEDxAlbany:

The basic idea is to use ordinary morning activities as a hook to talk about the quantum physics underlying everyday phenomena.

— The due date for the book is in December 2017, publication to be sometime in 2018. Probably. You know, publishing– everything is subject to change.

— There is not yet a US publisher for this book– we’ve had some interest, but to my great sorrow and annoyance, Eureka didn’t sell well, and that makes things difficult. We’re still working on getting a publisher for this side of the Atlantic, and when that happens, I’ll post another cell-phone photo of legal documents. If you’re a publisher and this sounds interesting, please drop me a line and I’ll put you in touch with my agent…

I’ve known about this for a while– we agreed to the deal on election day last November, so at least one good thing happened that day– but transatlantic business deals are more complicated, so it took a while to get everything set up. And now, I’ve signed the contracts, and I guess I need to write the book. Once I have brains and time for that, which is not this morning.

The traditional photo of a pile of signed contracts for a new book, just before mailing them.

The traditional photo of a pile of signed contracts for a new book, just before mailing them.

Tommaso DorigoA Slow-Motion Particle Collision In Anomaly!

Lubos Motl published the other day in his crazily active blog a very nice new review of "Anomaly! Collider Physics and the Quest for New Phenomena at Fermilab". The review is authored by Tristan du Pree, a colleague of mine who has worked in CMS until very recently - now he moved to a new job and changed to ATLAS! (BTW  thanks Lubos, and thanks Tristan!)
I liked a lot Tristan's commentary of my work, and since he mentions with quite appreciative terms the slow-motion description of a peculiar collision I offer in my book, I figured I'd paste that below.

read more

John PreskillHamiltonian: An American Musical (without Americana or music)

Author’s note: I intended to post this article three months ago. Other developments delayed the release. Thanks in advance for pardoning the untimeliness.

Critics are raving about it. Barak Obama gave a speech about it. It’s propelled two books onto bestseller lists. Committees have showered more awards on it than clouds have showered rain on California this past decade.

What is it? The Hamiltonian, represented by \hat{H}. It’s an operator (a mathematical object) that basically represents a system’s energy. Hamiltonians characterize systems classical and quantum, from a brick in a Broadway theater to the photons that form a spotlight. \hat{H} determines how a system evolves, or changes in time.

obama-h

I lied: Obama didn’t give a speech about the Hamiltonian. He gave a speech about Hamilton. Hamilton: An American Musical spotlights 18th-century revolutionary Alexander Hamilton. Hamilton conceived the United States’s national bank. He nurtured the economy as our first Secretary of the Treasury. The year after Alexander Hamilton died, William Rowan Hamilton was born. Rowan Hamilton conceived four-dimensional numbers called quaternions. He nurtured the style of physics, Hamiltonian mechanics, used to model quantum systems today.

2-hamiltons

Hamilton has enchanted audiences and critics. Ticket sell out despite costing over $1,000. Tonys, Grammys, and Pulitzers have piled up. Lawmakers, one newspaper reported, ridicule colleagues who haven’t seen the show. One political staff member confessed that “dodging ‘Hamilton’ barbs has affected her work—so much so that she hasn’t returned certain phone calls ‘because I couldn’t handle the anxiety’ of being harangued for her continued failure to see the show.”

Musical-theater fans across the country are applauding Alexander. Hamilton forbid that William Rowan should envy him. Let’s celebrate Hamiltonians.

playbill-h

I’ve been pondering the Hamiltonian

mbl-hamiltonian

It describes a chain of L sites. L ranges from 10 to 30 in most computer simulations. The cast consists of quantum particles. Each site houses one particle or none. \hat{n}_j represents the number of particles at site j. c_j represents the removal of a particle from site j, and c_j^\dag represents the adding of a particle.

The last term in \hat{H} represents the repulsion between particles that border each other. The “nn” in “E_{\rm nn}” stands for “nearest-neighbor.” The J term encodes particles’ hopping between sites. \hat{c}_j^\dag \hat{c}_{j+1} means, “A particle jumps from site j+1 to site j.”

The first term in \hat{H}, we call disorder. Imagine a landscape of random dips and hills. Imagine, for instance, crouching on the dirt and snow in Valley Forge. Boots and hooves have scuffed the ground. Zoom in; crouch lower. Imagine transplanting the row of sites into this landscape. h_j denotes the height of site j.

Say that the dips sink low and the hills rise high. The disorder traps particles like soldiers behind enemy lines. Particles have trouble hopping. We call this system many-body localized.

Imagine flattening the landscape abruptly, as by stamping on the snow. This flattening triggers a phase transition.  Phase transitions are drastic changes, as from colony to country. The flattening frees particles to hop from site to site. The particles spread out, in accordance with the Hamiltonian’s J term. The particles come to obey thermodynamics, a branch of physics that I’ve effused about.

The Hamiltonian encodes repulsion, hopping, localization, thermalization, and more behaviors. A richer biography you’ll not find amongst the Founding Fathers.

cast-h

As Hamiltonians constrain particles, politics constrain humans. A play has primed politicians to smile upon the name “Hamilton.” Physicists study Hamiltonians and petition politicians for funding. Would politicians fund us more if we emphasized the Hamiltonians in our science?

Gold star for whoever composes the most rousing lyrics about many-body localization. Or, rather, fifty white stars.


January 31, 2017

Terence TaoAnother problem about power series

By an odd coincidence, I stumbled upon a second question in as many weeks about power series, and once again the only way I know how to prove the result is by complex methods; once again, I am leaving it here as a challenge to any interested readers, and I would be particularly interested in knowing of a proof that was not based on complex analysis (or thinly disguised versions thereof), or for a reference to previous literature where something like this identity has occured. (I suspect for instance that something like this may have shown up before in free probability, based on the answer to part (ii) of the problem.)

Here is a purely algebraic form of the problem:

Problem 1 Let {F = F(z)} be a formal function of one variable {z}. Suppose that {G = G(z)} is the formal function defined by

\displaystyle G := \sum_{n=1}^\infty \left( \frac{F^n}{n!} \right)^{(n-1)}

\displaystyle = F + \left(\frac{F^2}{2}\right)' + \left(\frac{F^3}{6}\right)'' + \dots

\displaystyle = F + FF' + (F (F')^2 + \frac{1}{2} F^2 F'') + \dots,

where we use {f^{(k)}} to denote the {k}-fold derivative of {f} with respect to the variable {z}.

  • (i) Show that {F} can be formally recovered from {G} by the formula

    \displaystyle F = \sum_{n=1}^\infty (-1)^{n-1} \left( \frac{G^n}{n!} \right)^{(n-1)}

    \displaystyle = G - \left(\frac{G^2}{2}\right)' + \left(\frac{G^3}{6}\right)'' - \dots

    \displaystyle = G - GG' + (G (G')^2 + \frac{1}{2} G^2 G'') - \dots.

  • (ii) There is a remarkable further formal identity relating {F(z)} with {G(z)} that does not explicitly involve any infinite summation. What is this identity?

To rigorously formulate part (i) of this problem, one could work in the commutative differential ring of formal infinite series generated by polynomial combinations of {F} and its derivatives (with no constant term). Part (ii) is a bit trickier to formulate in this abstract ring; the identity in question is easier to state if {F, G} are formal power series, or (even better) convergent power series, as it involves operations such as composition or inversion that can be more easily defined in those latter settings.

To illustrate Problem 1(i), let us compute up to third order in {F}, using {{\mathcal O}(F^4)} to denote any quantity involving four or more factors of {F} and its derivatives, and similarly for other exponents than {4}. Then we have

\displaystyle G = F + FF' + (F (F')^2 + \frac{1}{2} F^2 F'') + {\mathcal O}(F^4)

and hence

\displaystyle G' = F' + (F')^2 + FF'' + {\mathcal O}(F^3)

\displaystyle G'' = F'' + {\mathcal O}(F^2);

multiplying, we have

\displaystyle GG' = FF' + F (F')^2 + F^2 F'' + F (F')^2 + {\mathcal O}(F^4)

and

\displaystyle G (G')^2 + \frac{1}{2} G^2 G'' = F (F')^2 + \frac{1}{2} F^2 F'' + {\mathcal O}(F^4)

and hence after a lot of canceling

\displaystyle G - GG' + (G (G')^2 + \frac{1}{2} G^2 G'') = F + {\mathcal O}(F^4).

Thus Problem 1(i) holds up to errors of {{\mathcal O}(F^4)} at least. In principle one can continue verifying Problem 1(i) to increasingly high order in {F}, but the computations rapidly become quite lengthy, and I do not know of a direct way to ensure that one always obtains the required cancellation at the end of the computation.

Problem 1(i) can also be posed in formal power series: if

\displaystyle F(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots

is a formal power series with no constant term with complex coefficients {a_1, a_2, \dots} with {|a_1|<1}, then one can verify that the series

\displaystyle G := \sum_{n=1}^\infty \left( \frac{F^n}{n!} \right)^{(n-1)}

makes sense as a formal power series with no constant term, thus

\displaystyle G(z) = b_1 z + b_2 z^2 + b_3 z^3 + \dots.

For instance it is not difficult to show that {b_1 = \frac{a_1}{1-a_1}}. If one further has {|b_1| < 1}, then it turns out that

\displaystyle F = \sum_{n=1}^\infty (-1)^{n-1} \left( \frac{G^n}{n!} \right)^{(n-1)}

as formal power series. Currently the only way I know how to show this is by first proving the claim for power series with a positive radius of convergence using the Cauchy integral formula, but even this is a bit tricky unless one has managed to guess the identity in (ii) first. (In fact, the way I discovered this problem was by first trying to solve (a variant of) the identity in (ii) by Taylor expansion in the course of attacking another problem, and obtaining the transform in Problem 1 as a consequence.)

The transform that takes {F} to {G} resembles both the exponential function

\displaystyle \exp(F) = \sum_{n=0}^\infty \frac{F^n}{n!}

and Taylor’s formula

\displaystyle F(z) = \sum_{n=0}^\infty \frac{F^{(n)}(0)}{n!} z^n

but does not seem to be directly connected to either (this is more apparent once one knows the identity in (ii)).


Filed under: 246A - complex analysis, math.CV, math.RA, question Tagged: identity, power series

January 30, 2017

Matt StrasslerPenny Wise, Pound Foolish

The cost to American science and healthcare of the administration’s attack on legal immigration is hard to quantify.  Maybe it will prevent a terrorist attack, though that’s hard to say.  What is certain is that American faculty are suddenly no longer able to hire the best researchers from the seven countries currently affected by the ban.  Numerous top scientists suddenly cannot travel here to share their work with American colleagues; or if already working here, cannot now travel abroad to learn from experts elsewhere… not to mention visiting their families.  Those caught outside the country cannot return, hurting the American laboratories where they are employed.

You might ask what the big deal is; it’s only seven countries, and the ban is temporary. Well (even ignoring the outsized role of Iran, whose many immigrant engineers and scientists are here because they dislike the ayatollahs and their alternative facts), the impact extends far beyond these seven.

The administration’s tactics are chilling.  Scientists from certain countries now fear that one morning they will discover their country has joined the seven, so that they too cannot hope to enter or exit the United States.  They will decide now to turn down invitations to work in or collaborate with American laboratories; it’s too risky.  At the University of Pennsylvania, I had a Pakistani postdoc, who made important contributions to our research effort. At the University of Washington we hired a terrific Pakistani mathematical physicist. Today, how could I advise someone like that to accept a US position?

Even those not worried about being targeted may decide the US is not the open and welcoming country it used to be.  Many US institutions are currently hiring people for the fall semester.  A lot of bright young scientists — not just Muslims from Muslim-majority nations — will choose instead to go instead to Canada, to the UK, and elsewhere, leaving our scientific enterprise understaffed.

Well, but this is just about science, yes?  Mostly elite academics presumably — it won’t affect the average person.  Right?

Wrong.  It will affect many of us, because it affects healthcare, and in particular, hospitals around the country.  I draw your attention to an article written by an expert in that subject:

http://www.cnn.com/2017/01/29/opinions/trump-ban-impact-on-health-care-vox/index.html

and I’d like to quote from the article (highlights mine):

“Our training hospitals posted job listings for 27,860 new medical graduates last year alone, but American medical schools only put out 18,668 graduates. International physicians percolate throughout the entire medical system. To highlight just one particularly intense specialty, fully 30% of American transplant surgeons started their careers in foreign medical schools. Even with our current influx of international physicians as well as steadily growing domestic medical school spots, the Association of American Medical Colleges estimates that we’ll be short by up to 94,700 doctors by 2025.

The President’s decision is as ill-timed as it was sudden. The initial 90-day order encompasses Match Day, the already anxiety-inducing third Friday in March when medical school graduates officially commit to their clinical training programs. Unless the administration or the courts quickly fix the mess President Trump just created, many American hospitals could face staffing crises come July when new residents are slated to start working.”

If you or a family member has to go into the hospital this summer and gets sub-standard care due to a lack of trained residents and doctors, you know who to blame.  Terrorism is no laughing matter, but you and your loved ones are vastly more likely to die due to a medical error than due to a terrorist.  It’s hard to quantify exactly, but it is clear that over the years since 2000, the number of Americans dying of medical errors is in the millions, while the number who died from terrorism is just over three thousand during that period, almost all of whom died on 9/11 in 2001. So addressing the terrorism problem by worsening a hospital problem probably endangers Americans more than it protects them.

Such is the problem of relying on alternative facts in place of solid scientific reasoning.


Filed under: Science and Modern Society Tagged: immigration

Doug NatelsonWhat is a crystal?

(I'm bringing this up because I want to write about "time crystals", and to do that....)

A crystal is a larger whole comprising a spatially periodic arrangement of identical building blocks.   The set of points that delineates the locations of those building blocks is called the lattice, and the minimal building block is called a basis.  In something like table salt, the lattice is cubic, and the basis is a sodium ion and a chloride ion.  This much you can find in a few seconds on wikipedia.  You can also have molecular crystals, where the building blocks are individual covalently bonded molecules, and the molecules are bound to each other via van der Waals forces.   Recently there has been a ton of excitement about graphene, transition metal dichalcogenides, and other van der Waals layered materials, where a 3d crystal is built up out of 2d covalently bonded crystals stacked periodically in the vertical direction.

The key physics points:   When placed together under the right conditions, the building blocks of a crystal spontaneously join together and assemble into the crystal structure.  While space has the same properties in every location ("invariance under continuous translation") and in every orientation ("invariance under continuous orientation"), the crystal environment doesn't.  Instead, the crystal has discrete translational symmetry (each lattice site is equivalent), and other discrete symmetries (e.g., mirror symmetry about some planes, or discrete rotational symmetries around some axes).   This kind of spontaneous symmetry breaking is so general that it happens in all kinds of systems, like plastic balls floating on reservoirs.  The spatial periodicity has all kinds of consequences, like band structure and phonon dispersion relations (how lattice vibration frequencies depend on vibration wavelengths and directions).

January 25, 2017

Sean CarrollWhat Happened at the Big Bang?

I had the pleasure earlier this month of giving a plenary lecture at a meeting of the American Astronomical Society. Unfortunately, as far as I know they don’t record the lectures on video. So here, at least, are the slides I showed during my talk. I’ve been a little hesitant to put them up, since some subtleties are lost if you only have the slides and not the words that went with them, but perhaps it’s better than nothing.

My assigned topic was “What We Don’t Know About the Beginning of the Universe,” and I focused on the question of whether there could have been space and time even before the Big Bang. Short answer: sure there could have been, but we don’t actually know.

So what I did to fill my time was two things. First, I talked about different ways the universe could have existed before the Big Bang, classifying models into four possibilities (see Slide 7):

  1. Bouncing (the universe collapses to a Big Crunch, then re-expands with a Big Bang)
  2. Cyclic (a series of bounces and crunches, extending forever)
  3. Hibernating (a universe that sits quiescently for a long time, before the Bang begins)
  4. Reproducing (a background empty universe that spits off babies, each of which begins with a Bang)

I don’t claim this is a logically exhaustive set of possibilities, but most semi-popular models I know fit into one of the above categories. Given my own way of thinking about the problem, I emphasized that any decent cosmological model should try to explain why the early universe had a low entropy, and suggested that the Reproducing models did the best job.

My other goal was to talk about how thinking quantum-mechanically affects the problem. There are two questions to ask: is time emergent or fundamental, and is Hilbert space finite- or infinite-dimensional. If time is fundamental, the universe lasts forever; it doesn’t have a beginning. But if time is emergent, there may very well be a first moment. If Hilbert space is finite-dimensional it’s necessary (there are only a finite number of moments of time that can possibly emerge), while if it’s infinite-dimensional the problem is open.

Despite all that we don’t know, I remain optimistic that we are actually making progress here. I’m pretty hopeful that within my lifetime we’ll have settled on a leading theory for what happened at the very beginning of the universe.

Doug NatelsonA book recommendation

I've been very busy lately, hence a slow down in posting, but in the meantime I wanted to recommend a book.  The Pope of Physics is the recent biography of Enrico Fermi from  Gino Segrè and Bettina Hoerlin.  The title is from Fermi's nickname as a young physicist in Italy - he and his colleagues (the "Via Panisperna boys", named for the address of the Institute of Physics in Rome) took to giving each other nicknames, and Fermi's was "the Pope" because of his apparent infallibility.  The book is compelling, gives insights into Fermi and his relationships, and includes stories about that wild era of physics that I didn't recall hearing before.   (For example, when trying to build the first critical nuclear pile at Stag Field in Chicago, there was a big contract dispute with Stone and Webster, the firm hired by the National Defense Research Council to do the job.  When it looked like the dispute was really going to slow things down, Fermi suggested that the physicists themselves just build the thing, and the put it together from something like 20000 graphite blocks in about two weeks.)

While it's not necessarily as page-turning as The Making of the Atomic Bomb, it's a very interesting biography that offers insights into this brilliant yet emotionally reserved person.  It's a great addition to the bookshelf.  For reference, other biographies that I suggest are True Genius:  The Life and Science of John Bardeen, and the more technical works No Time to be Brief:  A Scientific Biography of Wolfgang Pauli and Subtle is the Lord:  The Science and Life of Albert Einstein.

Andrew JaffeSOLE Survivor

I recently finished my last term lecturing our second-year Quantum Mechanics course, which I taught for five years. It’s a required class, a mathematical introduction to one of the most important set of ideas in all of physics, and really the basis for much of what we do, whether that’s astrophysics or particle physics or almost anything else. It’s a slightly “old-fashioned” course, although it covers the important basic ideas: the Schrödinger Equation, the postulates of quantum mechanics, angular momentum, and spin, leading almost up to what is needed to understand the crowning achievement of early quantum theory: the structure of the hydrogen atom (and other atoms).

A more modern approach might start with qubits: the simplest systems that show quantum mechanical behaviour, and the study of which has led to the revolution in quantum information and quantum computing.

Moreover, the lectures rely on the so-called Copenhagen interpretation, which is the confusing and sometimes contradictory way that most physicists are taught to think about the basic ontology of quantum mechanics: what it says about what the world is “made of” and what happens when you make a quantum-mechanical measurement of that world. Indeed, it’s so confusing and contradictory that you really need another rule so that you don’t complain when you start to think too deeply about it: “shut up and calculate”. A more modern approach might also discuss the many-worlds approach, and — my current favorite — the (of course) Bayesian ideas of QBism.

The students seemed pleased with the course as it is — at the end of the term, they have the chance to give us some feedback through our “Student On-Line Evaluation” system, and my marks have been pretty consistent. Of the 200 or so students in the class, only about 90 bother to give their evaluations, which is disappointingly few. But it’s enough (I hope) to get a feeling for what they thought.

SOLE 2016 Chart

So, most students Definitely/Mostly Agree with the good things, although it’s clear that our students are most disappointed in the feedback that they receive from us (this is a more general issue for us in Physics at Imperial and more generally, and which may partially explain why most of them are unwilling to feed back to us through this form).

But much more fun and occasionally revealing are the “free-text comments”. Given the numerical scores, it’s not too surprising that there were plenty of positive ones:

  • Excellent lecturer - was enthusiastic and made you want to listen and learn well. Explained theory very well and clearly and showed he responded to suggestions on how to improve.

  • Possibly the best lecturer of this term.

  • Thanks for providing me with the knowledge and top level banter.

  • One of my favourite lecturers so far, Jaffe was entertaining and cleary very knowledgeable. He was always open to answering questions, no matter how simple they may be, and gave plenty of opportunity for students to ask them during lectures. I found this highly beneficial. His lecturing style incorporates well the blackboards, projectors and speach and he finds a nice balance between them. He can be a little erratic sometimes, which can cause confusion (e.g. suddenly remembering that he forgot to write something on the board while talking about something else completely and not really explaining what he wrote to correct it), but this is only a minor fix. Overall VERY HAPPY with this lecturer!

But some were more mixed:

  • One of the best, and funniest, lecturers I’ve had. However, there are some important conclusions which are non-intuitively derived from the mathematics, which would be made clearer if they were stated explicitly, e.g. by writing them on the board.

  • I felt this was the first time I really got a strong qualitative grasp of quantum mechanics, which I certainly owe to Prof Jaffe’s awesome lectures. Sadly I can’t quite say the same about my theoretical grasp; I felt the final third of the course less accessible, particularly when tackling angular momentum. At times, I struggled to contextualise the maths on the board, especially when using new techniques or notation. I mostly managed to follow Prof Jaffe’s derivations and explanations, but struggled to understand the greater meaning. This could be improved on next year. Apart from that, I really enjoyed going to the lectures and thought Prof Jaffe did a great job!

  • The course was inevitably very difficult to follow.

And several students explicitly commented on my attempts to get students to ask questions in as public a way as possible, so that everyone can benefit from the answers and — this really is true! — because there really are no embarrassing questions!

  • Really good at explaining and very engaging. Can seem a little abrasive at times. People don’t like asking questions in lectures, and not really liking people to ask questions in private afterwards, it ultimately means that no questions really get answered. Also, not answering questions by email makes sense, but no one really uses the blackboard form, so again no one really gets any questions answered. Though the rationale behind not answering email questions makes sense, it does seem a little unnecessarily difficult.

  • We are told not to ask questions privately so that everyone can learn from our doubts/misunderstandings, but I, amongst many people, don’t have the confidence to ask a question in front of 250 people during a lecture.

  • Forcing people to ask questions in lectures or publically on a message board is inappropriate. I understand it makes less work for you, but many students do not have the confidence to ask so openly, you are discouraging them from clarifying their understanding.

Inevitably, some of the comments were contradictory:

  • Would have been helpful to go through examples in lectures rather than going over the long-winded maths to derive equations/relationships that are already in the notes.

  • Professor Jaffe is very good at explaining the material. I really enjoyed his lectures. It was good that the important mathematics was covered in the lectures, with the bulk of the algebra that did not contribute to understanding being left to the handouts. This ensured we did not get bogged down in unnecessary mathematics and that there was more emphasis on the physics. I liked how Professor Jaffe would sometimes guide us through the important physics behind the mathematics. That made sure I did not get lost in the maths. A great lecture course!

And also inevitably, some students wanted to know more about the exam:

  • It is a difficult module, however well covered. The large amount of content (between lecture notes and handouts) is useful. Could you please identify what is examinable though as it is currently unclear and I would like to focus my time appropriately?

And one comment was particularly worrying (along with my seeming “a little abrasive at times”, above):

  • The lecturer was really good in lectures. however, during office hours he was a bit arrogant and did not approach the student nicely, in contrast to the behaviour of all the other professors I have spoken to

If any of the students are reading this, and are willing to comment further on this, I’d love to know more — I definitely don’t want to seem (or be!) arrogant or abrasive.

But I’m happy to see that most students don’t seem to think so, and even happier to have learned that I’ve been nominated “multiple times” for Imperial’s Student Academic Choice Awards!

Finally, best of luck to my colleague Jonathan Pritchard, who will be taking over teaching the course next year.

January 24, 2017

Matt StrasslerAlternative Facts and Crying Wolf

My satire about “alternative facts” from yesterday took some flak for propagating the controversial photos of inaugurations that some say are real and some say aren’t. I don’t honestly care one bit about those photos. I think it is of absolutely no importance how many people went to Trump’s inauguration; it has no bearing on how he will perform as president, and frankly I don’t know why he’s making such a big deal out of it. Even if attendance was far less than he and his people claim, it could be for two very good reasons that would not reflect badly on him at all.

First, Obama’s inauguration was extraordinarily historic. For a nation with our horrific past —  with most of our dark-skinned citizens brought to this continent to serve as property and suffer under slavery for generations — it was a huge step to finally elect an African-American president. I am sure many people chose to go to the 2009 inauguration because it was special to them to be able to witness it, and to be able to say that they were there. Much as many people adore Trump, it’s not so historic to have an aging rich white guy as president.

Second, look at a map of the US, with its population distribution. A huge population with a substantial number of Obama’s supporters live within driving distance or train distance of Washington DC. From South Carolina to Massachusetts there are large left-leaning populations. Trump’s support was largest in the center of the US, but people would not have been able to drive from there or take a train. The cost of travel to Washington could have reduced Trump’s inauguration numbers without reflecting on his popularity.

So as far as I’m concerned, it really doesn’t make any difference if Trump’s inauguration numbers were small, medium or large. It doesn’t count in making legislation or in trade negotiations; it doesn’t count in anything except pride.

But what does count, especially in foreign affairs, is whether people listen to what a president says, and by extension to what his or her press secretary says. What bothers me is not the political spinning of facts. All politicians do that. What bothers me is the claim of having hosted “the best-attended inauguration ever” without showing any convincing evidence, and the defense of those claims (and we heard it again today) that this is because it’s ok to disagree with facts.

If facts can be chosen at will, even in principle, then science ceases to function. Science — a word that means “evidence-based reasoning applied logically to determine how reality really works” — depends on the existence and undeniability of evidence. It’s not an accident that physics, unlike some subjects, does not have a Republican branch and a Democratic branch; it doesn’t have a Muslim, Christian, Buddhist or Jewish branch;  there’s just one type.  I work with people from many countries and with many religious and political beliefs; we work together just fine, and we don’t have discussions about “alternative facts.”

If instead you give up evidence-based reasoning, then soon you have politics instead of science determining your decisions on all sorts of things that matter to people because it can hurt or kill them: food safety, road safety, airplane safety, medicine, energy policy, environmental protection, and most importantly, defense. A nation that abandons evidence is abandoning applied reason and logic; and the inevitable consequence is that people will die unnecessarily.  It’s not a minor matter, and it’s not outside the purview of scientists to take a stand on the issue.

Meanwhile, I find the context for this discussion almost as astonishing as the discussion itself. It’s one thing to say unbelievable things during a campaign, but it’s much worse once in power. For the press secretary on day two of a new administration to make an astonishing and striking claim, but provide unconvincing evidence, has the effect of completely undermining his function.  As every scientist knows by heart, extraordinary claims require extraordinary evidence.  Imagine the press office at the CERN laboratory announcing the discovery of the Higgs particle without presenting plots of its two experiments’ data; or imagine if the LIGO experimenters had claimed discovery of gravitational waves but shown no evidence.  Mistakes are going to happen, but they have to be owned: imagine if OPERA’s tentative suggestion of neutrinos-faster-than-light, which was an experimental blunder, or BICEP’s loud misinterpretation of their cosmological data, had not been publicly retracted, with a clear public explanation of what happened.  When an organization makes a strong statement but won’t present clear evidence in favor, and isn’t willing to retract the statement when shown evidence against it, it not only introduces immediate suspicion of the particular claim but creates a wider credibility problem that is extremely difficult to fix.

Fortunately, the Higgs boson has been observed by two different experiments, in two different data-taking runs of both experiments; the evidence is extraordinary.  And LIGO’s gravitational waves data is public; you can check it yourself, and moreover there will be plenty of opportunities for further verification as Advanced VIRGO comes on-line this year.    But the inauguration claim hasn’t been presented with extraordinary evidence in its favor, and there’s significant contradictory evidence (from train ridership and from local sales).    When something extraordinary is actually true, it’s true from all points of view, not subject to “alternative facts”; and the person claiming it has the responsibility to find evidence, of several different types, as soon as possible.  If firm evidence is lacking, the claim should only be made tentatively.  (A single photo isn’t convincing, one way or the other, especially nowadays.)

As any child knows, it’s like crying wolf.  If your loud claim isn’t immediately backed up, or isn’t later retracted with a public admission of error, then the next time you claim something exceptional, people will just laugh and ignore you.  And nothing’s worse than suggesting that “I have my facts and you have yours;” that’s the worst possible argument, used only when firm evidence simply isn’t available.

I can’t understand why a press secretary would blow his credibility so quickly on something of so little importance.  But he did it.  If the new standards are this low, can one expect truth on anything that actually matters?  It’s certainly not good for Russia that few outside the country believe a word that Putin says; speaking for myself, I would never invest a dollar there. Unfortunately, leaders and peoples around the world, learning that the new U.S. administration has “alternative facts” at its disposal, may already have drawn the obvious conclusion.    [The extraordinary claim that “3-5 million” non-citizens (up from 2-3 million, the previous version of the claim) voted in the last election, also presented without extraordinary evidence, isn’t helping matters.] There’s now already a risk that only the president’s core supporters will believe what comes from this White House, even in a time of crisis or war.

Of course all governments lie sometimes.  But it’s wise to tell the truth most of the time, so that your occasional lies will sometimes be thought to be true.  Governments that lie constantly, even pointlessly, aren’t believed even when they say something true.  They’ve cried wolf too often.

So what’s next?  Made-up numbers for inflation, employment, the budget deficit, tax revenue? Invented statistics for the number of people who have health insurance?  False information about the readiness of our armed forces and the cost of our self-defense?  How far will this go?  And how will we know?


Filed under: Science and Modern Society Tagged: facts, ScienceAndSociety

Steinn SigurðssonGlöggt er gests augað

The Aspen Art Museum is doing a series of interdisciplinary lectures, titled “Another Look”

Another Look Lecture: Gabriel Orozco & Cosmology – so this is a thing.

I did one of the lectures. The first one, I gather.
It was quite an interesting experience, for me at least.
Good fun, riffing on the perspective from physics on Orozco’s work, which is partially inspired by astronomy and thoughts on cosmology.

MoMA was very helpful in providing a perspective on Orozco’s work over the years.
The actual exhibition was very interesting. The central floor display piece was quite startling in person and gave me a new perspective.

Detail from central exhibit piece of Gabriel Orozco's exhibit at the Aspen Art Museum

Detail from central exhibit piece of Gabriel Orozco’s exhibit at the Aspen Art Museum

The second talk was on Friday:
Another Look: Obituaries & Adam McEwen with Bruce Weber
Wish I could have been there.
No, I really wish I could have been there… 🙂

There will be more “Another Look” lectures, I gather.

From From Unsafe Art

Scott AaronsonMy 116-page survey article on P vs. NP: better late than never

For those who just want the survey itself, not the backstory, it’s here. (Note: Partly because of the feedback I’ve gotten on this blog, it’s now expanded to 121 pages!)


Update (Jan. 23) By request, I’ve prepared a Kindle-friendly edition of this P vs. NP survey—a mere 260 pages!


Two years ago, I learned that John Nash—that John Nash—was, together with Michail Rassias, editing a volume about the great open problems in mathematics.  And they wanted me to write the chapter about the P versus NP question—a question that Nash himself had come close to raising, in a prescient handwritten letter that he sent to the National Security Agency in 1955.

On the one hand, I knew I didn’t have time for such an undertaking, and am such a terrible procrastinator that, in both previous cases where I wrote a book chapter, I singlehandedly delayed the entire volume by months.  But on the other hand, John Nash.

So of course I said yes.

What followed was a year in which Michail sent me increasing panicked emails (and then phone calls) informing me that the whole volume was ready for the printer, except for my P vs. NP thing, and is there any chance I’ll have it by the end of the week?  Meanwhile, I’m reading yet more papers about Karchmer-Wigderson games, proof complexity, time/space tradeoffs, elusive functions, and small-depth arithmetic circuits.  P vs. NP, as it turns out, is now a big subject.

And in the middle of it, on May 23, 2015, John Nash and his wife Alicia were tragically killed on the New Jersey Turnpike, on their way back from the airport (Nash had just accepted the Abel Prize in Norway), when their taxi driver slammed into a guardrail.

But while Nash himself sadly wouldn’t be alive to see it, the volume was still going forward.  And now we were effectively honoring Nash’s memory, so I definitely couldn’t pull out.

So finally, last February, after more months of struggle and delay, I sent Michail what I had, and it duly appeared in the volume Open Problems in Mathematics.

But I knew I wasn’t done: there was still sending my chapter out to experts to solicit their comments.  This I did, and massively-helpful feedback started pouring in, creating yet more work for me.  The thorniest section, by far, was the one about Geometric Complexity Theory (GCT): the program, initiated by Ketan Mulmuley and carried forward by a dozen or more mathematicians, that seeks to attack P vs. NP and related problems using a fearsome arsenal from algebraic geometry and representation theory.  The experts told me, in no uncertain terms, that my section on GCT got things badly wrong—but they didn’t agree with each other about how I was wrong.  So I set to work trying to make them happy.

And then I got sidetracked with the move to Austin and with other projects, so I set the whole survey aside: a year of sweat and tears down the toilet.  Soon after that, Bürgisser, Ikenmeyer, and Panova proved a breakthrough “no-go” theorem, substantially changing the outlook for the GCT program, meaning yet more work for me if and when I ever returned to the survey.

Anyway, today, confined to the house with my sprained ankle, I decided that the perfect was the enemy of the good, and that I should just finish the damn survey and put it up on the web, so readers can benefit from it before the march of progress (we can hope!) renders it totally obsolete.

So here it is!  All 116 pages, 268 bibliography entries, and 52,000 words.

For your convenience, here’s the abstract:

In 1955, John Nash sent a remarkable letter to the National Security Agency, in which—seeking to build theoretical foundations for cryptography—he all but formulated what today we call the P=?NP problem, considered one of the great open problems of science.  Here I survey the status of this problem in 2017, for a broad audience of mathematicians, scientists, and engineers.  I offer a personal perspective on what it’s about, why it’s important, why it’s reasonable to conjecture that P≠NP is both true and provable, why proving it is so hard, the landscape of related problems, and crucially, what progress has been made in the last half-century toward solving those problems.  The discussion of progress includes diagonalization and circuit lower bounds; the relativization, algebrization, and natural proofs barriers; and the recent works of Ryan Williams and Ketan Mulmuley, which (in different ways) hint at a duality between impossibility proofs and algorithms.

Thanks so much to everyone whose feedback helped improve the survey.  If you have additional feedback, feel free to share in the comments section!  My plan is to incorporate the next round of feedback by the year 2100, if not earlier.


Update (Jan. 4) Bill Gasarch writes to tell me that Lazslo Babai has posted an announcement scaling back his famous “Graph Isomorphism in Quasipolynomial Time” claim. Specifically, Babai says that, due to an error discovered by Harald Helfgott, his graph isomorphism algorithm actually runs in about 22^O(√log(n)) time, rather than the originally claimed npolylog(n). This still beats the best previously-known running time for graph isomorphism (namely, 2O(√(n log n))), and by a large margin, but not quite as large as before.

Babai pointedly writes:

I apologize to those who were drawn to my lectures on this subject solely because of the quasipolynomial claim, prematurely magnified on the internet in spite of my disclaimers.

Alas, my own experience has taught me the hard way that, on the Internet, it is do or do not. There is no disclaim.

In any case, I’ve already updated my P vs. NP survey to reflect this new development.


Another Update (Jan. 10) For those who missed it, Babai has another update saying that he’s fixed the problem, and the running time of his graph isomorphism algorithm is back to being quasipolynomial.


Update (Jan. 19): This moment—the twilight of the Enlightenment, the eve of the return of the human species back to the rule of thugs—seems like as good a time as any to declare my P vs. NP survey officially done. I.e., thanks so much to everyone who sent me suggestions for additions and improvements, I’ve implemented pretty much all of them, and I’m not seeking additional suggestions!

January 23, 2017

Matt StrasslerWhat’s all this fuss about having alternatives?

I don’t know what all the fuss is about “alternative facts.” Why, we scientists use them all the time!

For example, because of my political views, I teach physics students that gravity pulls down. That’s why the students I teach, when they go on to be engineers, put wheels on the bottom corners of cars, so that the cars don’t scrape on the ground. But in some countries, the physicists teach them that gravity pulls whichever way the country’s leaders instruct it to. That’s why their engineers build flying carpets as transports for their country’s troops. It’s a much more effective way to bring an army into battle, if your politics allows it.  We ought to consider it here.

Another example: in my physics class I claim that energy is “conserved” (in the physics sense) — it is never created out of nothing, nor is it ever destroyed. In our daily lives, energy is taken in with food, converted into special biochemicals for storage, and then used to keep us warm, maintain the pumping of our hearts, allow us to think, walk, breathe — everything we do. Those are my facts. But in some countries, the facts and laws are different, and energy can be created from nothing. The citizens of those countries never need to eat; it is a wonderful thing to be freed from this requirement. It’s great for their military, too, to not have to supply food for troops, or fuel for tanks and airplanes and ships. Our only protection against invasion from these countries is that if they crossed our borders they’d suddenly need fuel tanks.

Facts are what you make them; it’s entirely up to you. You need a good, well-thought-out system of facts, of course; otherwise they won’t produce the answers that you want. But just first figure out what you want to be true, and then go out and find the facts that make it true. That’s the way science has always been done, and the best scientists all insist upon this strategy.  As a simple illustration, compare the photos below.  Which picture has more people in it?   Obviously, the answer depends on what facts you’ve chosen to use.   [Picture copyright Reuters]  If you can’t understand that, you’re not ready to be a serious scientist!

A third example: when I teach physics to students, I instill in them the notion that quantum mechanics controls the atomic world, and underlies the transistors in every computer and every cell phone. But the uncertainty principle that arises in quantum mechanics just isn’t acceptable in some countries, so they don’t factualize it. They don’t use seditious and immoral computer chips there; instead they use proper vacuum tubes. One curious result is that their computers are the size of buildings. The CDC advises you not to travel to these countries, and certainly not to take electronics with you. Not only might your cell phone explode when it gets there, you yourself might too, since your own molecules are held together with quantum mechanical glue. At least you should bring a good-sized bottle of our local facts with you on your travels, and take a good handful before bedtime.

Hearing all the naive cries that facts aren’t for the choosing, I became curious about what our schools are teaching young people. So I asked a friend’s son, a bright young kid in fourth grade, what he’d been learning about alternatives and science. Do you know what he answered?!  I was shocked. “Alternative facts?”, he said. “You mean lies?” Sheesh. Kids these days… What are we teaching them? It’s a good thing we’ll soon have a new secretary of education.


Filed under: LHC News, Science and Modern Society, The Scientific Process Tagged: facts, science, Science&Society

Steinn SigurðssonA missing piece of the puzzle

I’ve been puzzling over the rationale for some recent events…

Exxon has a large contract to develop oil and natural gas resources in the Russia.
This can only go forward if sanctions on Russia are lifted, which seems likely to happen in the near future.

But, there is too much oil and capacity to surge produce more oil and gas on the market. If nothing else, the US has well developed capacity which is idling.
The problem, as it has been for the last few decades, is that Saudi Arabia can squeeze new producers out of the market, by increasing production and sharply dropping prices, for a while, which forces higher costs producers off the market.
Then the Saudis cut back.
Oil prices go up.
Profit.

So… for the Exxon deal to be really worth while, in the medium term, the Saudi capacity would have to be curtailed.
Or some other major producer removed from the market.

That, could of course be arranged, given the national security resources of a major power or two.

But that’d be totally evil.

January 22, 2017

Geraint F. LewisProton: a life story

Proton: a life story 
by Geraint F. Lewis


1035 years: I’ve lived a long and eventful life, but I know that death is almost upon me. Around me, my kind are slowly melting into the darkness that is now the universe, and my time will eventually come.

I’ve lived a long and eventful life…


10-43 seconds: A time of unbelievable light, unbelievable heat! I don’t remember the time before I was born, but I was there, disembodied, ethereal, part of the swirling, roaring fires of the universe coming in to being.

But the universe cooled. From the featureless inferno, its character crystalized into a seething sea of particles and forces. Electrons and quarks tore about, smashing and crashing into photons and neutrinos. The universe continued to cool.

1 second: The intensity of the heat steadily died away, and I was born. In truth, there was no precise moment of my birth, but as the universe cooled my innards, free quarks, bound together, and I was suddenly there! A proton!

But my existence seemed fleeting, and in this still crowded and superheated universe in an instant I was bumped and I transformed, changing from proton to neutron. And with another thump I was a proton again. Then neutron. Then proton. I can’t remember how many times I flipped and flopped from one to the other. But as the universe continued to cool, my identity eventually settled. I was a proton, and staying that way. At least for now!

10 seconds: The universe was now filled with jostling protons and neutrons. We crashed and collided, but I was drawn to the neutrons, and they to me. As one approached, we reached out to one another, but in the fleeting moment of a touch, the still immense heat of the universe tore us apart.

The universe cooled, and the jostling diminished. I held onto a passing neutron and we danced. Together we were something new, together we were a nucleus of deuterium. Around us, all of the neutrons found proton partners, although there were not enough to go around and many protons remained alone.

1 minute: And still the universe cooled. Things steadily slowed, and before I realised we had grabbed onto another pair, one more proton, one more neutron, and as the new group we were helium. And it was not just us! All around us in the universe, protons and neutrons were getting together. The first elements were being forged.

But as quickly as it begun, it was over. The temperature continued to drop as the universe expanded. Collisions calmed. Instead of eagerly joining together, us newly formed nuclei of atoms now avoided one another. I settled down into my life as helium.

380,000 years: After its superbly energetic start, the universe rapidly darkened. And in the darkness, other nuclei bounced around me. Electrons, still holding on to the fire of their birth, zipped between us. But the universe cooled and cooled, slowly robbing these speedy electrons of their energy, and they were inexorably drawn closer.

Two electrons joined, orbiting about us protons and neutrons. We had become a new thing entirely, an atom of helium! Other helium nuclei were doing the same, while lone protons, also grabbing at electrons, transformed into hydrogen! This excitement was fleeting, and very soon us atoms settled back into the darkness.

10 million years: The universe was still dark, but that didn’t mean that nothing was happening. Gravity was there, pulling on us atoms of hydrogen and helium, pooling us into clouds and clumps. It felt uncomfortable to be close to so many other atoms, and the constant bumping and grinding ripped off our electrons. Back to being just a nucleus of helium!

Throughout the universe, many massive lumps of hydrogen and helium were forming, with intense gravity squeezing hard at their very hearts. Temperatures soared, and protons again began to crash together, combining first into deuterium and then into helium, and then into carbon, oxygen and other elements not yet seen in the universe. And from these furnaces came heat and light, and the first stars shone and lit up the darkness.

2 billion years: I was spared the intensity at the stellar core, riding the plasma currents in the outer parts of a star. There was a lot of jostling and bumping, but it was relatively cool here, and I retained my identity of helium. But things were changing.

My star was aging quickly, and instead of the steady burning of its youth, it began to groan and wheeze, puffing and swelling as its nuclear reactor faltered and failed.  The stellar pulsing was becoming a wild ride, until eventually I was blown completely off the star and thrown back into the gas of interstellar space.

3 billion years: I swirled around for a while, bathed in the light of a hundred billion stars. But gravity does not sleep and I soon found myself back inside a newly born star. But this time it was different! No sedate atmospheric bobbing for me. I found myself in the intense blaze of the stellar core.

The temperature rose quickly, and nuclei smashed together. These collisions were violent, with a violence I had last seen at the start of the universe. And after a bruising series of collisions, I was helium no more. Now I resided with other protons and neutrons in a nucleus of carbon.

3.1 billion years: The stellar heart roared, and just beneath me the fires burnt unbelievably hot. Down there, at the very centre, carbon was forged into oxygen, neon and silicon, building heavier and heavier elements. Eventually the stellar furnace was producing iron, a nuclear dead-end that cannot fuel the burning that keeps a star alive.

As the fires at the stellar furnace continued to rage, more and more iron built up in the core. Until there was so much that the nuclear fires went out and the heart of the star suddenly stopped. With nothing to prevent gravity’s insatiable attraction, the star’s outer layers collapsed, and in an instant this crushing reignited the nuclear fires, now burning uncontrollably. The star exploded and ripped itself apart. In my new carbon identity, I found myself thrust again out into the universe.

5 billion years: Deep space is now different. Yes, there is plenty of hydrogen and helium out here, but there are lots of heavier atoms, like myself, bobbing about, the ashes from billions of dead and dying stars. We gather into immense clouds of gas and dust, illuminated by continuing generations of stars that shine.

In this cool environment, we can again collect some electrons and live as an atom, but this time an atom of carbon. Before long, we’re bumping into other atoms, linking together and forming long molecules, alcohols, formaldehydes and more. But gravity is at work again, tugging on the clouds and drawing us in. It looks like I’m heading for another journey inside a star.

8 billion years: Although this time it’s different. I find myself in the swish and swirl of material orbiting the forming star. And strange things are happening, as molecules build and intertwine, growing and clumping as the fetal star steadily grows. The heart of the star ignites, and the rush of radiation blows away the swirling disk, sending back into deep space.

But I remain, deep in a molecule bound with other molecules and held within a rock, a rock too large to billow in the solar wind. And these rocks are colliding and sticking, growing and forming a planet. In the blink of a cosmic eye, billions of tonnes have come together, which gravity has moulded into a ball. Initially hot from its birth, this newly built planet steadily cooled and solidified in the light of its host star.

10 billion years: For a brief while, this planet was dead and sterile, but water began to flow on its surface and an atmosphere wrapped and warmed it. I remained in the ground, although the rocks and the dirt continued to churn as the planet continued to cooled.

And then something amazing happed! Things moved on the surface! I didn’t see how it began, but collections of molecules were acting in unison. These bags of chemical processes slurped across the planet for billions of years, and then themselves begun to join and work together.

13 billion years: Eventually I found myself caught up in this stuff called life, with me, as carbon, integrated into the various pieces of changing and evolving creatures. But it was oh so transitory, being part of one bit of life, then back to the soil, and then part of another. Some times, as one type of life, other life consumed me, with molecules dismembered and reintegrated into other creatures.

Once I found myself in the fronds of a plant, a fern, waving gently in the breeze under a sunlit sky. But when this beautiful plant died, its body was pressed into the mud of the swamp in which it sat, and I was ripped evermore from the cycle of life. Pressures and temperatures grew as more and more material was pressed upon me, and I was buried deeper and deeper within the ground.

13.7 billion years: And there I lay, with the intense squeezing rapidly ripping away my molecular identity. Again, I was simply carbon. But here, deep in the planet, there were a lot of carbon atoms and slowly we found affinity for one another. Through soaring pressures, we bound together, pressed and shaped into a crystal, a crystal of diamond.

Suddenly I was torn from my underground home, gazed at by a living creature I had never seen before. Accompanied by some gold, I spent a mere moment of the cosmos adorning the finger of one of these living creatures, these humans. This was truly some of the strangest time of my existence, oh the world I saw, but before long I was lost and buried in the dark ground. And there I stayed as rocks shifted and moved, and the planet aged.

19 billion years: With many other carbon atoms, I was still locked in diamond as the planet started to melt around me. The star, whose birth I had witnessed, was now old. It glowed intensely, immense and swollen, so that its erratic atmosphere engulfed the planet. The heat and the raging wind of the dying star ripped at the planet’s surface, hurling it into space.

And so too I was gone, embedded in the dust of my long dead planet, thrown again into space between stars. The rocky core of the planet that had been my home for almost ten billion years continually dragged against the star’s immense atmosphere, and fell to complete annihilation in the last beats of the stellar heart.

100 billion years: All around me, the universe has continued to expand and cool, but the expansion, originally slowing, has been steadily accelerating. Immense groups of stars, the galaxies, are moving away from each other faster and faster. Their light, which blazed in the distant universe, has dimmed and diminished as they rushed away.

And by now the expansion is so rapid all of these distant galaxies have completely faded from view. Near me, stars continue to burn, but now set in the infinite blackness of a distant sky.

1 trillion years: The universe got older and my journey continued. Each time the story was the same; I’d swirl in space before gravity’s irresistible pull dragged me into a forming star. My diamond identity was rapidly lost on my first such plunge, with the immense pressures and temperatures ripping us into individual atoms of carbon. Eventually the star aged and died and I was spat back out into space.

While the story of stellar birth and stellar death repeated, I noticed that that there was steadily less and less hydrogen, and more and more other elements tumbling through interstellar space. And while I sometimes existed fleetingly in this molecule or that, I inevitably found myself pulled into flowing into the birth of a new star.

10 trillion years: I have passed through countless generations of stars, each time slightly different. Many of these have been relatively gentle affairs, but now and again I find myself caught up in a massive star, a star destined to explode when it dies.

And within this stellar forge, my identity was changed to heavier and heavier elements. But in the eventual cataclysm of stellar death, a supernova, the smashing of elements can create extraordinary heavy collections of protons and neutrons, nuclei of gold and lead. I emerged from one explosion in a nucleus of cobalt, 27 of us protons with 33 neutrons.

But this was not a happy nucleus, heaving and shaking. This instability cannot last. Relief came when one of many protons changed into a neutron, spitting out an electron and transforming us into nickel. But as nickel we did not settle, more heaves and shakes, and we continued to transform and transform again until we were iron, and then we are calm.

50 trillion years: The cycle continues, with endless eons in empty space punctuated with the burst of excitement spent within a star. But through each cycle, there is less and less gas was to be found in interstellar space, with many atoms locked up in the dead hearts of stars, dead hearts that simply sit in the darkness.  

And the stars are different too. Instead of the bright burning of immense young stars, the darkness is punctuated with an innumerable sea of feeble, small stars, lighting the universe with their faint, red glow.

85 trillion years: I am dragged once more into a forming star. While I don’t realise it, this is the end of the cycle for me as the puny star that is forming will never explode, will never shed its outer layers, never return me to the depths of deep space. More and more of my kin, the protons and neutrons, have an identical fate, a destiny to be locked seemingly forever in to the last generations of stars to shine. 

And deep within my final star, I am still hidden inside an iron nucleus. Around me, the nuclear furnace burns very slowly and very steadily, as some of the remaining hydrogen is burnt into helium, illuminating the universe with a watery glow.

100 trillion years: My star still gently shines, with many others scattered through the universe, but the raw fuel for the formation of stars, the gas between the stars, is gone. No more stars will ever be born.

The universe is a mix of the dead and the dying, the remnants of long dead stars, and the last of those still feebly burning, destined to join the graveyard when they exhaust the last of their nuclear fuel. From this point, only death and darkness face the universe.

110 trillion years: The inevitable has come, and my star has exhausted the last of its nuclear fuel. At its heart, the fires have gone out. My star has died, not with a bang, but with a very silent whimper.

And I, a single proton, am still locked inside my nucleus of iron, deep, deep within the star. It is still warm, a left over heat from when the fires burnt, and atoms bounce and jostle, but it’s a dying heat as the star cools, leaking its radiation into space.

120 trillion years: The last star, aged and exhausted, has died, and the universe is filled with little more than fading embers. The gentle glow continues for a short while, but darkness returns, a darkness not seen for more than a hundred trillion years.

The universe feels like it’s entering its end days, but in reality an infinite future stretches ahead. In the darkness, I still sit, locked within the corpse of my long dead star.

10 quadrillion years: The last heat in my star has gone, radiated away into space, and we are as cold and dark as space itself. Everything has slowed to a crawl as the universe continues to wind down.   

But in the darkness, monsters lurk. Black holes, the crushed cores of massive dead stars, have been slowly slurping matter and eating the stellar dead, and in the dark they continue to feed, continue to grow. My remnant home is lucky, avoiding this fate, but many dead stars are crushed out of existence within the black hole’s heart.

1031 years: Further countless eons have passed, eons where nothing happened, nothing changed. But now, in the darkness, something new is stirring, a slow, methodic activity as matter itself has started to melt. My kindred protons, protons that have existed since the birth of time, are vanishing from existence, decaying to be replaced with other small particles.

My own remnant star is slowly falling apart, as individual atoms decay and break down. My own iron home is also disintegrating around me, with protons steadily decaying away. All of the dead stars are steadily turning to dust.

1034 years:  The stars are gone, and I find myself alone, a single proton sitting in the blackness of space. In the darkness around me, protons are still decaying away, still ceasing to be. The universe is slowly becoming a featureless sea, with little more than electrons and photons in the darkness.

Looking back over the immense history of the universe, it is difficult to remember the brief glory days, the days where the stars shone, with planets, and at least some life. But that has all gone, and is gone forever.

1035 years: There are very few of us protons left, and I am amongst the last. I know the inevitable will come soon, and I too will cease to exist, and will return to the ephemeral state that existed before my birth.

I will be gone, but there are still things hidden in the darkness. Even the black holes eventually die, spitting decaying particles into the void. And after 10100 years, even this will end as the last black hole dies away. And as it does so, the universe will enter into the last night, a night that will truly last forever.
  


I’ve lived a long and eventful life…



January 21, 2017

Scott AaronsonA day to celebrate

Today—January 20, 2017—I have something cheerful, something that I’m celebrating.  It’s Lily’s fourth birthday. Happy birthday Lily!

As part of her birthday festivities, and despite her packed schedule, Lily has graciously agreed to field a few questions from readers of this blog.  You can ask about her parents, favorite toys, recent trip to Disney World, etc.  Just FYI: to the best of my knowledge, Lily doesn’t have any special insight about computational complexity, although she can write the letters ‘N’ and ‘P’ and find them on the keyboard.  Nor has she demonstrated much interest in politics, though she’s aware that many people are upset because a very bad man just became the president.  Anyway, if you ask questions that are appropriate for a real 4-year-old girl, rather than a blog humor construct, there’s a good chance I’ll let them through moderation and pass them on to her!

Meanwhile, here’s a photo I took of UT Austin students protesting Trump’s inauguration beneath the iconic UT tower.

January 18, 2017

January 17, 2017

Doug NatelsonWhat is the difference between science and engineering?

In my colleague Rebecca Richards-Kortum's great talk at Rice's CUWiP meeting this past weekend, she spoke about her undergrad degree in physics at Nebraska, her doctorate in medical physics from MIT, and how she ended up doing bioengineering.  As a former undergrad engineer who went the other direction, I think her story did a good job of illustrating the distinctions between science and engineering, and the common thread of problem-solving that connects them.

In brief, science is about figuring out the ground rules about how the universe works.   Engineering is about taking those rules, and then figuring out how to accomplish some particular task.   Both of these involve puzzle-like problem-solving.  As a physics example on the experimental side, you might want to understand how electrons lose energy to vibrations in a material, but you only have a very limited set of tools at your disposal - say voltage sources, resistors, amplifiers, maybe a laser and a microscope and a spectrometer, etc.  Somehow you have to formulate a strategy using just those tools.  On the theory side, you might want to figure out whether some arrangement of atoms in a crystal results in a lowest-energy electronic state that is magnetic, but you only have some particular set of calculational tools - you can't actually solve the complete problem and instead have to figure out what approximations would be reasonable, keeping the essentials and neglecting the extraneous bits of physics that aren't germane to the question.

Engineering is the same sort of process, but goal-directed toward an application rather than specifically the acquisition of new knowledge.  You are trying to solve a problem, like constructing a machine that functions like a CPAP, but has to be cheap and incredibly reliable, and because of the price constraint you have to use largely off-the-shelf components.  (Here's how it's done.)

People act sometimes like there is a vast gulf between scientists and engineers - like the former don't have common sense or real-world perspective, or like the latter are somehow less mathematical or sophisticated.  Those stereotypes even comes through in pop culture, but the differences are much less stark than that.  Both science and engineering involve creativity and problem-solving under constraints.   Often which one is for you depends on what you find most interesting at a given time - there are plenty of scientists who go into engineering, and engineers can pursue and acquire basic knowledge along the way.  Particularly in the modern, interdisciplinary world, the distinction is less important than ever before.

January 14, 2017

Andrew JaffeElectoral woes and votes

Like everyone else in my bubble, I’ve been angrily obsessing about the outcome of the US Presidential election for the last two weeks. I’d like to say that I’ve been channelling that obsession into action, but so far I’ve mostly been reading and hoping (and being disappointed). And trying to parse all the “explanations” for Trump’s election.

Mostly, it’s been about what the Democrats did wrong (imperfect Hillary, ignoring the white working class, not visiting Wisconsin, too much identity politics), and what the Republicans did right (imperfect Trump, dog whistles, focusing on economics and security).

But there has been an ongoing strain of purely procedural complaint: that the system is rigged, but (ironically?) in favour of Republicans. In fact, this is manifestly true: liberals (Democrats) are more concentrated — mostly in cities — than conservatives (Republicans) who are spread more evenly and dominate in rural areas. And the asymmetry is more true for the sticky ideologies than the fungible party affiliations, especially when “liberal” encompasses a whole raft of social issues rather than just left-wing economics. This has been exacerbated by a few decades of gerrymandering. So the House of Representatives, in particular, tilts Republican most of the time. And the Senate, with its non-proportional representation of two per state, regardless of size, favours those spread-out Republicans, too (although party dominance of the Senate is less of a stranglehold for the Republicans than that of the House).

But one further complaint that I’ve heard several times is that the Electoral College is rigged, above and beyond those reasons for Republican dominance of the House and Senate: as we know, Clinton has won the popular vote, by more than 1.5 million as of this writing — in fact, my own California absentee ballot has yet to be counted. The usual argument goes like this: the number of electoral votes allocated to a state is the sum of the number of members of congress (proportional to the population) and the number of senators (two), giving a total of five hundred and thirty-eight. For the most populous states, the addition of two electoral votes doesn’t make much of a difference. New Jersey, for example, has 12 representatives, and 14 electoral votes, about a 15% difference; for California it’s only about 4%. But the least populous states (North and South Dakota, Montana, Wyoming, Alaska) have only one congressperson each, but three electoral votes, increasing the share relative to population by a factor of 3 (i.e., 300%). In a Presidential election, the power of a Wyoming voter is more than three times that of a Californian.

This is all true, too. But it isn’t why Trump won the election. If you changed the electoral college to allocate votes equal to the number of congressional representatives alone (i.e., subtract two from each state), Trump would have won 245 to 191 (compared to the real result of 306 to 232).1 As a further check, since even the representative count is slightly skewed in favour of small states (since even the least populous state has at least one), I did another version where the electoral vote allocation is exactly proportional to the 2010 census numbers, but it gives the same result. (Contact me if you would like to see the numbers I use.)

Is the problem (I admit I am very narrowly defining “problem” as “responsible for Trump’s election”, not the more general one of fairness!), therefore, not the skew in vote allocation, but instead the winner-take-all results in each state? Maine and Nebraska already allocate their two “Senatorial” electoral votes to the statewide winner, and one vote for the winner of each congressional district, and there have been proposals to expand this nationally. Again, this wouldn’t solve the “problem”. Although I haven’t crunched the numbers myself, it appears that ticket-splitting (voting different parties for President and Congress) is relatively low. Since the Republicans retained control of Congress, their electoral votes under this system would be similar to their congressional majority of 239 to 194 (their are a few results outstanding), and would only get worse if we retain the two Senatorial votes per state. Indeed, with this system, Romney would have won in 2012.

So the “problem” really does go back to the very different geographical distribution of Democrats and Republicans. Almost any system which segregates electoral votes by location (especially if subjected to gerrymandering) will favour the more widely dispersed party. So perhaps the solution is to just to use nationwide popular voting for Presidential elections. This would also eliminate the importance of a small number of swing states and therefore require more national campaigning. (It could be enacted by a Constitutional amendment, or a scheme like the National Popular Vote Interstate Compact.) Alas, it ain’t gonna happen.


  1. I have assumed Trump wins Michigan, and I have allocated all of Maine to Clinton and all of Nebraska to Trump; see below. ↩︎