I’ve been spending some time with Simon Willerton’s paper Tight spans, Isbell completions and semi-tropical modules. In particular, I’ve been trying to understand tight spans.

The tight span of a metric space $A$ is another metric space $T(A)$, in which $A$ naturally embeds. For instance, the tight span of a two-point space is a line segment containing the original two points as its endpoints. Similarly, the tight span of a three-point space is a space shaped like the letter Y, with the original three points at its tips. Because of examples like this, some people like to think of the tight span as a kind of abstract convex hull.

Simon’s paper puts the tight span construction into the context of a categorical construction, Isbell conjugacy. I now understand these things better than I did, but there’s still a lot I don’t get. Here goes.

Simon wrote a blog post summarizing the main points of his paper, but I want to draw attention to slightly different aspects of it than he does. So, much as I recommend that post of his, I’ll make this self-contained.

We begin with Isbell conjugacy. For any small category $A$, there’s an adjunction

$$\check{\,\,}: [A^{op}, Set] \leftrightarrows [A, Set]^{op}: \hat{\,\,}$$

defined for $F: A^{op} \to Set$ and $b \in A$ by

$$\check{F}(b) = Hom(F, A(-, b))$$

and for $G: A \to Set$ and $a \in A$ by

$$\hat{G}(a) = Hom(G, A(a, -)).$$

We call $\check{F}$ the (Isbell) conjugate of $F$, and similarly $\hat{G}$ the (Isbell) conjugate of $G$.

Like any adjunction, it restricts to an equivalence of categories in a canonical way. Specifically, it’s an equivalence between

the full subcategory of $[A^{op}, Set]$ consisting of those objects $F$ such that the canonical map $F \to \hat{\check{F}}$ is an isomorphism

and

the full subcategory of $[A, Set]^{op}$ consisting of those objects $G$ such that the canonical map $G \to \check{\hat{G}}$ is an isomorphism.

I’ll call either of these equivalent categories the reflexive completion $R(A)$ of $A$. (Simon called it the Isbell completion, possibly with my encouragement, but “reflexive completion” is more descriptive and I prefer it now.) So, the reflexive completion of a category consists of all the “reflexive” presheaves on it — those canonically isomorphic to their double conjugate.

All of this categorical stuff generalizes seamlessly to an enriched context, at least if we work over a complete symmetric monoidal closed category.

For example, suppose we take our base category to be the category $Ab$ of abelian groups. Let $k$ be a field, viewed as a one-object $Ab$-category. Both $[k^{op}, Ab]$ and $[k, Ab]$ are the category of $k$-vector spaces, and both $\check{\,\,}$ and $\hat{\,\,}$ are the dual vector space construction. The reflexive completion $R(k)$ of $k$ is the category of $k$-vector spaces $V$ for which the canonical map $V \to V^{\ast\ast}$ is an isomorphism — in other words, the finite-dimensional vector spaces.

But that’s not the example that will matter to us here.

We’ll be thinking primarily about the case where the base category is the poset $([0, \infty], \geq)$ (the reverse of the usual order!) with monoidal structure given by addition. As Lawvere observed long ago, a $[0, \infty]$-category is then a “generalized metric space”: a set $A$ of points together with a distance function

$$d: A \times A \to [0, \infty]$$

satisfying the triangle inequality $d(a, b) + d(b, c) \geq d(a, c)$ and the equaiton $d(a, a) = 0$. These are looser structures than classical metric spaces, mainly because of the absence of the symmetry axiom $d(a, b) = d(b, a)$.

The enriched functors are distance-decreasing maps between metric spaces: those functions $f: A \to B$ satisfying $d_B(f(a_1), f(a_2)) \leq d_A(a_1, a_2)$. I’ll just call these “maps” of metric spaces.

If you work through the details, you find that Isbell conjugacy for metric spaces works as follows. Let $A$ be a generalized metric space. The conjugate of a map $f: A^{op} \to [0, \infty]$ is the map $\check{f}: A \to [0, \infty]$ defined by

$$\check{f}(b) = sup_{a \in A} max \{ d(a, b) - f(a), 0 \}$$

and the conjugate of a map $g: A \to [0, \infty]$ is the map $\hat{g}: A \to [0, \infty]$ defined by

$$\hat{g}(a) = sup_{b \in A} max \{ d(a, b) - g(b), 0 \}.$$

We always have $f \geq \hat{\check{f}}$, and the reflexive completion $R(A)$ of $A$ consists of all maps $f: A^{op} \to [0, \infty]$ such that $f = \hat{\check{f}}$. (You can write out an explicit formula for that, but I’m not convinced it’s much help.) The metric on $R(A)$ is the sup metric.

All that comes out of the general categorical machinery.

However, we can say something more that only makes sense because of the particular base category we’re using.

As we all know, symmetric metric spaces — the ones we’re most used to — are particular interesting. For a symmetric metric space $A$, the distinction between covariant and contravariant functors on $A$ vanishes. The two kinds of conjugate, $\hat{\,\,}$ and $\check{\,\,}$, are also the same. I’ll write ${\,\,}^\ast$ for them both.

The reflexive completion $R(A)$ consists of the functions $A \to [0, \infty]$ that are equal to their double conjugate. But because there is no distinction between covariant and contravariant, we can also consider the functions $A \to [0, \infty]$ equal to their single conjugate.

The set of such functions is — by definition, if you like — the tight span $T(A)$ of $A$. So

$$T(A) = \{ f : A \to [0, \infty] \,|\, f = f^\ast \},$$

while

$$R(A) = \{ f: A \to [0, \infty] \,|\, f = f^{\ast\ast} \}.$$

Both come equipped with the sup metric, and both contain $A$ as a subspace, via the Yoneda embedding $a \mapsto d(-, a)$. So $A \subseteq R(A) \subseteq T(A)$.

Example Let $A$ be the symmetric metric space consisting of two points distance $D$ apart. Its reflexive completion $R(A)$ is the set $[0, D] \times [0, D]$ with metric $$d((s_1, s_2), (t_1, t_2)) = max \{ t_1 - s_1, t_2 - s_2, 0 \}.$$ The Yoneda embedding identifies the two points of $A$ with the points $(0, D)$ and $(D, 0)$ of $R(A)$. The tight span $T(A)$ is the straight line between these two points of $R(A)$ (a diagonal of the square), which is isometric to the ordinary Euclidean line segment $[0, D]$.

As that example shows, the reflexive completion of a space needn’t be symmetric, even if the original space was symmetric. On the other hand, it’s not too hard to show that the tight span of a space is always symmetric. Simon’s Theorem 4.1.1 slots everything into place:

Theorem (Willerton) Let $A$ be a symmetric metric space. Then the tight span $T(A)$ is the largest symmetric subspace of $R(A)$ containing $A$.

Here “largest” means that if $B$ is another symmetric subspace of $R(A)$ containing $A$ then $B \subseteq T(A)$. It’s not even obvious that there is a largest one. For instance, given any non-symmetric metric space $C$, what’s the largest symmetric subspace of $C$? There isn’t one, just because every singleton subset of $C$ is symmetric.

Simon’s told the following story before, but I can’t resist telling it again. Tight spans have been discovered independently many times over. The first time they were discovered, they were called by the less catchy name of “injective envelope” (because $T(A)$ is the smallest injective metric space containing $A$). And the person who made that first discovery? Isbell — who, as far as anyone knows, never noticed that this had anything to do with Isbell conjugacy.

Let me finish with something I don’t quite understand.

Simon’s Theorem 3.1.4 says the following. Let $A$ be a symmetric metric space. (He doesn’t assume symmetry, but I will.) Then for any $a \in A$, $p \in R(A)$ and $\varepsilon \gt 0$, there exists $b \in A$ such that

$$d(a, p) + d(p, b) \leq d(a, b) + \varepsilon.$$

In other words, $a$, $p$ and $b$ are almost collinear.

A loose paraphrasing of this is that every point in the reflexive completion of $A$ is close to being on a geodesic between points of $A$. The theorem does imply this, but it says a lot more. Look at the quantification. We get to choose one end $a$ of the not-quite-geodesic, as well as the point $p$ in the reflexive completion, and we’re guaranteed that if we continue the not-quite-geodesic from $a$ on through $p$, then we’ll eventually meet another point of $A$ (or nearly).

Let’s get rid of those “not quite”s, and at the same time focus attention on the tight span rather than the reflexive completion. Back in Isbell’s original 1964 paper (cited by Simon, in case you want to look it up), it’s shown that if $A$ is compact then so is its tight span $T(A)$. Of course, Simon’s Theorem 3.1.4 applies in particular when $p \in T(A)$. But then compactness of $T(A)$ means that we can drop the $\varepsilon$.

In other words: let $A$ be a compact symmetric metric space. Then for any $a \in A$ and $p \in T(A)$, there exists $b \in A$ such that

$$d(a, p) + d(p, b) = d(a, b).$$

So, if you place your pencil at a point of $A$, draw a straight line from it to a point of $T(A)$, and keep going, you’ll eventually meet another point of $A$.

This leaves me wondering what tight spans of common geometric figures actually look like.

For example, take an arc $A$ of a circle — any size arc, just not the whole circle. Embed it in the plane and give it the Euclidean metric. I said originally that the tight span is sometimes thought of as a sort of abstract convex hull, and indeed, the introduction to Simon’s paper says that some authors have actually used this name instead of “tight span”. But the result I just stated makes this seem highly misleading. It implies that the tight span of $A$ is not its convex hull, and indeed, can’t be any subspace of the Euclidean plane (unless, perhaps, it’s $A$ itself, which I suspect is not the case). But what is it?

My new book is out!

Click the image for more information.

It’s an introductory category theory text, and I can prove it exists: there’s a copy right in front of me. (You too can purchase a proof.) Is it unique? Maybe. Here are three of its properties:

• It doesn’t assume much.
• It sticks to the basics.
• It’s short.

I want to thank the $n$-Café patrons who gave me encouragement during my last week of work on this. As I remarked back then, some aspects of writing a book — even a short one — require a lot of persistence.

But I also want to take this opportunity to make a suggestion. There are now quite a lot of introductions to category theory available, of various lengths, at various levels, and in various styles. I don’t kid myself that mine is particularly special: it’s just what came out of my individual circumstances, as a result of the courses I’d taught. I think the world has plenty of introductions to category theory now.

What would be really good is for there to be a nice second book on category theory. Now, there are already some resources for advanced categorical topics: for instance, in my book, I cite both the $n$Lab and Borceux’s three-volume Handbook of Categorical Algebra for this. But useful as those are, what we’re missing is a shortish book that picks up where Categories for the Working Mathematician leaves off.

Let me be more specific. One of the virtues of Categories for the Working Mathematician (apart from being astoundingly well-written) is that it’s selective. Mac Lane covers a lot in just 262 pages, and he does so by repeatedly making bold choices about what to exclude. For instance, he implicitly proves that for any finitary algebraic theory, the category of algebras has all colimits — but he does so simply by proving it for groups, rather than explicitly addressing the general case. (After all, anyone who knows what a finitary algebraic theory is could easily generalize the proof.) He also writes briskly: few words are wasted.

I’m imagining a second book on category theory of a similar length to Categories for the Working Mathematician, and written in the same brisk and selective manner. Over beers five years ago, Nicola Gambino and I discussed what this hypothetical book ought to contain. I’ve lost the piece of paper I wrote it down on (thus, Nicola is absolved of all blame), but I attempted to recreate it sometime later. Here’s a tentative list of chapters, in no particular order:

• Enriched categories
• 2-categories (and a bit on higher categories)
• Topos theory (obviously only an introduction) and categorical set theory
• Fibrations
• Bimodules, Morita equivalence, Cauchy completeness and absolute colimits
• Operads and Lawvere theories
• Categorical logic (again, just a bit) and internal category theory
• Derived categories
• Flat functors and locally presentable categories
• Ends and Kan extensions (already in Mac Lane’s book, but maybe worth another pass).

Someone else should definitely write such a book.

Hamas is trying to kill as many civilians as it can.

Israel is trying to kill as few civilians as it can.

Neither is succeeding very well.

Update (July 28): Please check out a superb essay by Sam Harris on the Israeli/Palestinian conflict.  While, as Harris says, the essay contains “something to offend everyone”—even me—it also brilliantly articulates many of the points I’ve been trying to make in this comment thread.

See also a good HuffPost article by Ali A. Rizvi, a “Pakistani-Canadian writer, physician, and musician.”

René Descartes (1596 – 1650) was an outstanding physicist, mathematician and philosopher. In physics, he laid the ground work for Isaac Newton’s (1642 – 1727) laws of motion by pioneering work on the concept of inertia. In mathematics, he developed the foundations of analytic geometry, as illustrated by the term Cartesian[1] coordinates. However, it is in his role as a philosopher that he is best remembered. Rather ironic, as his breakthrough method was a failure.

Descartes’s goal in philosophy was to develop a sound basis for all knowledge based on ideas that were so obvious they could not be doubted. His touch stone was that anything he perceived clearly and distinctly as being true was true. The archetypical example of this was the famous I think therefore I am.  Unfortunately, little else is as obvious as that famous quote and even it can be––and has been––doubted.

Euclidean geometry provides the illusionary ideal to which Descartes and other philosophers have strived. You start with a few self-evident truths and derive a superstructure built on them.  Unfortunately even Euclidean geometry fails that test. The infamous parallel postulate has been questioned since ancient times as being a bit suspicious and even other Euclidean postulates have been questioned; extending a straight line depends on the space being continuous, unbounded and infinite.

So how are we to take Euclid’s postulates and axioms?  Perhaps we should follow the idea of Sir Karl Popper (1902 – 1994) and consider them to be bold hypotheses. This casts a different light on Euclid and his work; perhaps he was the first outstanding scientist.  If we take his basic assumptions as empirical[2] rather than sure and certain knowledge, all we lose is the illusion of certainty. Euclidean geometry then becomes an empirically testable model for the geometry of space time. The theorems, derived from the basic assumption, are prediction that can be checked against observations satisfying Popper’s demarcation criteria for science. Do the angles in a triangle add up to two right angles or not? If not, then one of the assumptions is false, probably the parallel line postulate.

Back to Descartes, he criticized Galileo Galilei (1564 – 1642) for having built without having considered the first causes of nature, he has merely sought reasons for particular effects; and thus he has built without a foundation. In the end, that lack of a foundation turned out to be less of a hindrance than Descartes’ faulty one.  To a large extent, sciences’ lack of a foundation, such as Descartes wished to provide, has not proved a significant obstacle to its advance.

Like Euclid, Sir Isaac Newton had his basic assumptions—the three laws of motion and the law of universal gravity—but he did not believe that they were self-evident; he believed that he had inferred them by the process of scientific induction. Unfortunately, scientific induction was as flawed as a foundation as the self-evident nature of the Euclidean postulates. Connecting the dots between a falling apple and the motion of the moon was an act of creative genius, a bold hypothesis, and not some algorithmic derivation from observation.

It is worth noting that, at the time, Newton’s explanation had a strong competitor in Descartes theory that planetary motion was due to vortices, large circulating bands of particles that keep the planets in place.  Descartes’s theory had the advantage that it lacked the occult action at a distance that is fundamental to Newton’s law of universal gravitation.  In spite of that, today, Descartes vortices are as unknown as is his claim that the pineal gland is the seat of the soul; so much for what he perceived clearly and distinctly as being true.

Galileo’s approach of solving problems one at time and not trying to solve all problems at once has paid big dividends. It has allowed science to advance one step at a time while Descartes’s approach has faded away as failed attempt followed failed attempt. We still do not have a grand theory of everything built on an unshakable foundation and probably never will. Rather we have models of widespread utility. Even if they are built on a shaky foundation, surely that is enough.

Peter Higgs (b. 1929) follows in the tradition of Galileo. He has not, despite his Noble prize, succeeded, where Descartes failed, in producing a foundation for all knowledge; but through creativity, he has proposed a bold hypothesis whose implications have been empirically confirmed.  Descartes would probably claim that he has merely sought reasons for a particular effect: mass. The answer to the ultimate question about life, the universe and everything still remains unanswered, much to Descartes’ chagrin but as scientists we are satisfied to solve one problem at a time then move on to the next one.

To receive a notice of future posts follow me on Twitter: @musquod.

I spent my research time today (much of it on planes and trains) working on my seminar (to be given tomorrow) about Gaussian Processes. I am relying heavily on the approach advocated by Foreman-Mackey, which is to start with weighted least squares (and the Bayesian generalization), then show how a kernel function in the covariance matrix changes the story, and then show how the Gaussian Process is latent in this solution. Not ready! But Foreman-Mackey made me a bunch of great demonstration figures.

I took a short nap yesterday, and of course as soon as I lay down on the bed, Emmy erupted in the furious barking that signals the arrival of a package. When I went out to get it, I found shiny new bound galley proofs of Eureka: Discovering Your Inner Scientist:

The just-arrived galley proof for my forthcoming book. Laptop included for scale.

I knew these were coming, but didn’t expect them so quickly. I asked for several to take to the UK, on the off chance that we’re seated next to somebody really famous on the flight to London, who says “Gosh, I’ve always wanted to blurb a book about science…,” so I’ll have something to give them. I’ll also have them at my talk in Bristol on August 13, and my Kaffeeklatsch at Loncon on the 14th. I might consider raffling one off, or awarding it to someone who performs great feats of strength science, or some such. You’ll have to come to those events to find out.

Anyway, while I was aware that these existed– and I’ve been carrying around a big stack of page proofs and a red pen for a week or two– seeing them in the paper made the whole thing much more real and exciting. It’s a book! With cover art and everything!

Now I really need to finish off these page proofs. And my talk for Bristol. And the prep for SteelyKid’s sixth birthday party tomorrow (the social event of the season!). And…

But: I have a book!

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On the second day of the Moore Foundation meeting, I gave my talk (about flexible models for exoplanet populations, exoplanet transits, and exoplanet-discovering hardware calibration). After my talk, I had a great conversation with Emmanuel Candès (Stanford), who asked me very detailed questions about my prior beliefs. I realized in the conversation that I have been violating all my own rules: I have been setting my prior beliefs about hyper-parameters in the space of the hyper-parameters and not in the space of the data. That is, you can only assess the influence and consistency of the prior pdf (consistency with your actual beliefs) by flowing the prior through the probabilistic model and generating data from it. I bet if I did that for some of the problems I was showing, I would find that my priors are absurd. This is a great rule, which I often say to others but don't do myself: Always sample data from your prior (not just parameters). This is a rule for Bayes but also a rule for those of us who eschew realism! More generally, Candès's expressed the view that priors should derive from data—prior data—a view with which I agree deeply. Unfortunately, when it comes to exoplanet populations, there really aren't any prior data to speak of.

There were many excellent talks again today; again this is an incomplete set of highlights for me: Titus Brown (MSU) explained his work on developing infrastructure for biology and bioinformatics. He made a number of comments about getting customer (or user) stories right and developing with the current customer in mind. These resonated for me in my experiences of software development. He also said that his teaching and workshops and outreach are self-interested: They feed back deep and valuable information about the customer. Jeffrey Heer (UW) said similar things about his development of DataWrangler, d3.js, and other data visualization tools. (d3.js is github's fourth most popular repository!) He showed some beautiful visualizations. Heer's demo of DataWrangler simply blew away the crowd, and there were questions about it for the rest of the day.

Carl Kingsford (CMU) caused me (and others) to gasp when he said that the Sequence Read Archive of biological sequences cannot be searched by sequence. It turns out that searching for strings in enormous corpuses of strings is actually a very hard problem (who knew?). He is using a new structure called a Bloom Filter Tree, in which k-mers (length-k subsections) are stored in the nodes and the leaves contain the data sets that contain those k-mers. It is very clever and filled with all the lovely engineering issues that the Astrometry.net data structures were filled with lo so many years ago. Kingsford focuses on writing careful code, so the combination of clever data structures and well written code gets him orders of magnitude speed-ups over the competition.

Causal inference was an explicit or implicit component of many of the talks today. For example, Matthew Stephens (Chicago) is using natural genetic variations as a "randomized experiment" to infer gene expression and function. Laurel Larson (Berkeley) is looking for precursor events and predictors for abrupt ecological changes; since her work is being used to trigger interventions, she requires a causal model.

Blair Sullivan (NC State) spoke about performing inferences with provable properties on graphs. She noted that most interesting problems are NP hard on arbitrary graphs, but become easier on graphs that can be embedded (without crossing the edges) on a planar or low-genus space. This was surprising to me, but apparently the explanation is simple: Planar graphs are much more likely to have single nodes that split the graph into disconnected sub-graphs. Another surprising thing to me is that "motif counting" (which I think is searching for identical subgraphs within a graph) is very hard; it can only be done exactly and in general for very small subgraphs (six-ish nodes).

The day ended with Laura Waller (Berkeley) talking about innovative imaging systems for microscopy, including light-field cameras, and then a general set of cameras that do non-degenerate illumination sequences and infer many properties beyond single-plane intensity measurements. She showed some very impressive demonstrations of light-field inferences with her systems, which are sophisticated, but built with inexpensive hardware. Her work has a lot of conceptual overlap with astronomy, in the areas of adaptive optics and imaging with non-degenerate masks.

Today was the first day of a private finalist meeting for the new Moore Foundation Data Driven Discovery Individual Investigator grants. The format is a shootout of short talks and a few group activities. There were many extremely exciting talks at the meeting; here is just an incomplete smattering of highlights for me:

Ethan White (Utah State) showed amazing ecology data, with most data sources being people looking at things and counting things in the field, but then also some remote sensing data. He is using high-resolution time-series data on plants and animals to forecast changes in species and the ecosystem. He appeared to be checking his models "in the space of the data"—not in terms of reproduction of some external "truth"—which was nice.

Yaser Abu-Mostafa (Caltech) spoke about the practice and pitfalls of machine learning; in particular he is interested in new methods to thwart data "snooping" which is the name he gives to the problem "if you torture your data enough, it will confess".

Carey Priebe (JHU) opened his talk with the "Cortical Column Conjecture" which claims that the cortex is made up of many repeats of small network structures that are themselves, in some sense, computing primitives. This hypothesis is hard to test both because the graphs of neural connections in real brains are very noisy, and because inference on graphs (including finding repeated sub-graphs) is combinatorically hard.

Amit Singer (Princeton) is using micrographs of large molecules to infer three-dimensional structures. Each micrograph provides noisy data on a two-dimensional projection of each molecule; the collection of such projections provides enough information to both infer the Euler angles (three angles per molecule) and the three-dimensional structure (a very high-dimensional object). This project is very related to things LeCun, Barron, and I were talking about many years ago with galaxies.

Kim Reynolds (UT Dallas) is using genetic variation to determine which parts of a protein sequence are important for setting the structure and which are replaceable. She makes predictions about mutations that would not interrupt structure or function. She showed amazing structures of cellular flagella parts, and proposed that she might be able to create new kinds of flagella that would be structurally similar but different in detail.

Yesterday I gave a lecture at the 3rd International Conference on New Frontiers in Physics, which is going on in kolympari (Crete). I spoke critically about the five-sigma criterion that is nowadays the accepted standard in particle physics and astrophysics for discovery claims.

My slides, as usual, are quite heavily written, which is a nuisance if you are sitting at the conference trying to follow my speech, but it becomes an asset if you are reading them by yourself post-mortem. You can find them here (pdf) and here (ppt) .

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This article appeared in Fermilab Today on July 30, 2014.

Fermilab physicist Arden Warner revolutionizes oil spill cleanup with magnetizable oil invention. Photo: Hanae Armitage

Four years ago, Fermilab accelerator physicist Arden Warner watched national news of the BP oil spill and found himself frustrated with the cleanup response.

“My wife asked ‘Can you separate oil from water?’ and I said ‘Maybe I could magnetize it!’” Warner recalled. “But that was just something I said. Later that night while I was falling asleep, I thought, you know what, that’s not a bad idea.”

Sleep forgone, Warner began experimenting in his garage. With shavings from his shovel, a splash of engine oil and a refrigerator magnet, Warner witnessed the preliminary success of a concept that could revolutionize the process of oil spill damage control.

Warner has received patent approval on the cleanup method.

The concept is simple: Take iron particles or magnetite dust and add them to oil. It turns out that these particles mix well with oil and form a loose colloidal suspension that floats in water. Mixed with the filings, the suspension is susceptible to magnetic forces. At a barely discernible 2 to 6 microns in size, the particles tend to clump together, and it only takes a sparse dusting for them to bond with the oil. When a magnetic field is applied to the oil and filings, they congeal into a viscous liquid known as a magnetorheological fluid. The fluid’s viscosity allows a magnetic field to pool both filings and oil to a single location, making them easy to remove. (View a 30-second video of the reaction.)

“It doesn’t take long — you add the filings, you pull them out. The entire process is even more efficient with hydrophobic filings. As soon as they hit the oil, they sink in,” said Warner, who works in the Accelerator Division. Hydrophobic filings are those that don’t like to interact with water — think of hydrophobic as water-fearing. “You could essentially have a device that disperses filings and a magnetic conveyor system behind it that picks it up. You don’t need a lot of material.”

Warner tested more than 100 oils, including sweet crude and heavy crude. As it turns out, the crude oils’ natural viscosity makes it fairly easy to magnetize and clear away. Currently, booms, floating devices that corral oil spills, are at best capable of containing the spill; oil removal is an entirely different process. But the iron filings can work in conjunction with an electromagnetic boom to allow tighter constriction and removal of the oil. Using solenoids, metal coils that carry an electrical current, the electromagnetic booms can steer the oil-filing mixture into collector tanks.

Unlike other oil cleanup methods, the magnetized oil technique is far more environmentally sound. There are no harmful chemicals introduced into the ocean — magnetite is a naturally occurring mineral. The filings are added and, briefly after, extracted. While there are some straggling iron particles, the vast majority is removed in one fell, magnetized swoop — the filings can even be dried and reused.

“This technique is more environmentally benign because it’s natural; we’re not adding soaps and chemicals to the ocean,” said Cherri Schmidt, head of Fermilab’s Office of Partnerships and Technology Transfer. “Other ‘cleanup’ techniques disperse the oil and make the droplets smaller or make the oil sink to the bottom. This doesn’t do that.”

Warner’s ideas for potential applications also include wildlife cleanup and the use of chemical sensors. Small devices that “smell” high and low concentrations of oil could be fastened to a motorized electromagnetic boom to direct it to the most oil-contaminated areas.

“I get crazy ideas all the time, but every so often one sticks,” Warner said. “This is one that I think could stick for the benefit of the environment and Fermilab.”

—Hanae Armitage

In which Rhett and I talk about awful academic computing systems, Worldcon, our Wikipedia pages, and AAPT meeting envy.

Some links:

– Rhett’s Wikipedia entry

– My Wikipedia entry

– The 2014 AAPT Summer Meeting

– LonCon 3, this year’s Worldcon

– My puzzling Worldcon schedule

We have some ideas for what to do next time, when our little hangout is old enough to drink, but you need to watch all the way to the end to hear those.

Inspired by the event at the UNESCO headquarters in Paris that celebrated the anniversary of the signature of the CERN convention, Sophie Redford wrote about her impressions on joining CERN as a young researcher. A CERN fellow designing detectors for the future CLIC accelerator, she did her PhD at the University of Oxford, observing rare B decays with the LHCb experiment.

The “60 years of CERN” celebrations give us all the chance to reflect on the history of our organization. As a young scientist, the early years of CERN might seem remote. However, the continuity of CERN and its values connects this distant past to the present day. At CERN, the past isn’t so far away.

Of course, no matter when you arrive at CERN for the first time, it doesn’t take long to realize that you are in a place with a special history. On the surface, CERN can appear scruffy. Haphazard buildings produce a maze of long corridors, labelled with seemingly random numbers to test the navigation of newcomers. Auditoriums retain original artefacts: ashtrays and blackboards unchanged since the beginning, alongside the modern-day gadgetry of projectors and video-conferencing systems.

The theme of re-use continues underground, where older machines form the injection chain for new. It is here, in the tunnels and caverns buried below the French and Swiss countryside, where CERN spends its money. Accelerators and detectors, their immense size juxtaposed with their minute detail, constitute an unparalleled scientific experiment gone global. As a young scientist this is the stuff of dreams, and you can’t help but feel lucky to be a part of it.

If the physical situation of CERN seems unique, so is the sociological. The row of flags flying outside the main entrance is a colourful red herring, for aside from our diverse allegiances during international sporting events, nationality is meaningless inside CERN. Despite its location straddling international borders, despite our wallets containing two currencies and our heads many languages, scientific excellence is the only thing that matters here. This is a community driven by curiosity, where coffee and cooperation result in particle beams. At CERN we question the laws of our universe. Many answers are as yet unknown but our shared goal of discovery bonds us irrespective of age or nationality.

As a young scientist at CERN I feel welcome and valued; this is an environment where reason and logic rule. I feel privileged to profit from the past endeavour of others, and great pride to contribute to the future of that which others have started. I have learnt that together we can achieve extraordinary things, and that seemingly insurmountable problems can be overcome.

In many ways, the second 60 years of CERN will be nothing like the first. But by continuing to build on our past we can carry the founding values of CERN into the future, allowing the next generation of young scientists to pursue knowledge without borders.

By Sophie Redford

Complete show on YouTube.  In case you were wondering what the fuss was about.

Best source I could find for this image: IFLS.

The average size of an atom is an Angstrom, 10^-10 m. The typical interatomar distance in molecules is a nanometer, 10^-9 meter, or let that be a few nanometers if you wish. At room temperature and normal atmospheric pressure, electrostatic repulsion prevents you from pushing atoms any closer together. So the 10^-8 meter in the cartoon seem about correct.

Yes, no, maybe, wtf.

One interesting feature of the heuristics of Garton, Park, Poonen, Wood, Voight, discussed here previously: they predict there are fewer elliptic curves of rank 3 than there are of rank 2.  Is this what we believe?  On one hand, you might believe that having three independent points should be “harder” than having only two.  But there’s the parity issue.  All right-thinking people believe that there are equally many rank 0 and rank 1 elliptic curves, because 100% of curves with even parity have rank 0, and 100% of curves with odd parity have rank 1.  If a curve has even parity, all that has to happen to force it to have rank 2 is to have a non-torsion point.  And if a curve has odd parity, all that has to happen to force it to have rank 3 is to have one more non-torsion point you don’t know about it.  So in that sense, it seems “equally hard” to have rank 2 or rank 3, given that parity should be even half the time and odd half the time.

So my intuition about this question is very weak.  What’s yours?  Should rank 3 be less common than rank 2?  The same?  More common?

Hidden in my papers with Chip Sebens on Everettian quantum mechanics is a simple solution to a fun philosophical problem with potential implications for cosmology: the quantum version of the Sleeping Beauty Problem. It’s a classic example of self-locating uncertainty: knowing everything there is to know about the universe except where you are in it. (Skeptic’s Play beat me to the punch here, but here’s my own take.)

The setup for the traditional (non-quantum) problem is the following. Some experimental philosophers enlist the help of a subject, Sleeping Beauty. She will be put to sleep, and a coin is flipped. If it comes up heads, Beauty will be awoken on Monday and interviewed; then she will (voluntarily) have all her memories of being awakened wiped out, and be put to sleep again. Then she will be awakened again on Tuesday, and interviewed once again. If the coin came up tails, on the other hand, Beauty will only be awakened on Monday. Beauty herself is fully aware ahead of time of what the experimental protocol will be.

So in one possible world (heads) Beauty is awakened twice, in identical circumstances; in the other possible world (tails) she is only awakened once. Each time she is asked a question: “What is the probability you would assign that the coin came up tails?”

Modified from a figure by Stuart Armstrong.

(Some other discussions switch the roles of heads and tails from my example.)

The Sleeping Beauty puzzle is still quite controversial. There are two answers one could imagine reasonably defending.

• “Halfer” — Before going to sleep, Beauty would have said that the probability of the coin coming up heads or tails would be one-half each. Beauty learns nothing upon waking up. She should assign a probability one-half to it having been tails.
• “Thirder” — If Beauty were told upon waking that the coin had come up heads, she would assign equal credence to it being Monday or Tuesday. But if she were told it was Monday, she would assign equal credence to the coin being heads or tails. The only consistent apportionment of credences is to assign 1/3 to each possibility, treating each possible waking-up event on an equal footing.

The Sleeping Beauty puzzle has generated considerable interest. It’s exactly the kind of wacky thought experiment that philosophers just eat up. But it has also attracted attention from cosmologists of late, because of the measure problem in cosmology. In a multiverse, there are many classical spacetimes (analogous to the coin toss) and many observers in each spacetime (analogous to being awakened on multiple occasions). Really the SB puzzle is a test-bed for cases of “mixed” uncertainties from different sources.

Chip and I argue that if we adopt Everettian quantum mechanics (EQM) and our Epistemic Separability Principle (ESP), everything becomes crystal clear. A rare case where the quantum-mechanical version of a problem is actually easier than the classical version.

In the quantum version, we naturally replace the coin toss by the observation of a spin. If the spin is initially oriented along the x-axis, we have a 50/50 chance of observing it to be up or down along the z-axis. In EQM that’s because we split into two different branches of the wave function, with equal amplitudes.

Our derivation of the Born Rule is actually based on the idea of self-locating uncertainty, so adding a bit more to it is no problem at all. We show that, if you accept the ESP, you are immediately led to the “thirder” position, as originally advocated by Elga. Roughly speaking, in the quantum wave function Beauty is awakened three times, and all of them are on a completely equal footing, and should be assigned equal credences. The same logic that says that probabilities are proportional to the amplitudes squared also says you should be a thirder.

But! We can put a minor twist on the experiment. What if, instead of waking up Beauty twice when the spin is up, we instead observe another spin. If that second spin is also up, she is awakened on Monday, while if it is down, she is awakened on Tuesday. Again we ask what probability she would assign that the first spin was down.

This new version has three branches of the wave function instead of two, as illustrated in the figure. And now the three branches don’t have equal amplitudes; the bottom one is (1/√2), while the top two are each (1/√2)² = 1/2. In this case the ESP simply recovers the Born Rule: the bottom branch has probability 1/2, while each of the top two have probability 1/4. And Beauty wakes up precisely once on each branch, so she should assign probability 1/2 to the initial spin being down. This gives some justification for the “halfer” position, at least in this slightly modified setup.

All very cute, but it does have direct implications for the measure problem in cosmology. Consider a multiverse with many branches of the cosmological wave function, and potentially many identical observers on each branch. Given that you are one of those observers, how do you assign probabilities to the different alternatives?

Simple. Each observer O_i appears on a branch with amplitude ψ_i, and every appearance gets assigned a Born-rule weight w_i = |ψ_i|². The ESP instructs us to assign a probability to each observer given by

It looks easy, but note that the formula is not trivial: the weights w_i will not in general add up to one, since they might describe multiple observers on a single branch and perhaps even at different times. This analysis, we claim, defuses the “Born Rule crisis” pointed out by Don Page in the context of these cosmological spacetimes.

Sleeping Beauty, in other words, might turn out to be very useful in helping us understand the origin of the universe. Then again, plenty of people already think that the multiverse is just a fairy tale, so perhaps we shouldn’t be handing them ammunition.

Given the recent Feynman explosion (timeline of events), some people may be casting about looking for an alternative source of colorful-character anecdotes in physics. Fortunately, the search doesn’t need to go all that far– if you flip back a couple of pages in the imaginary alphabetical listing of physicists, you’ll find a guy who fits the bill very well: Enrico Fermi.

Fermi’s contributions to physics are arguably as significant as Feynman’s. He was the first to work out the statistical mechanics of particles obeying the Pauli exclusion principle, now called “fermions” in his honor (Paul Dirac did the same thing independently a short time later, so the mathematical function is the “Fermi-Dirac distrbution“). He was also the first to develop a theory for the weak nuclear interaction, placing Wolfgang Pauli’s desperate suggestion of the existence of the neutrino on a sound theoretical footing. Fermi’s theory was remarkably successful, and anticipated or readily incorporated the next thirty-ish years of discoveries.

More than that, he was a successful experimental physicist. He did pioneering experiments with neutrons, including demonstrating the fission of heavy elements (though he initially misinterpreted this as the creation of transuranic elements) and was the first to successfully construct a nuclear reactor, as part of the Manhattan Project. The US’s great particle physics lab is named Fermilab in his honor.

One of the difficult things about replacing Feynman is that a lot of the genuinely admirable things about his approach to physics are sort of bound up with his personality. Meaning that it’s easy to slide from approach-to-physics stuff– spinning plates at Cornell, etc.– to relatively wholesome anecdotes– dazzling off-the-cuff calculations, cracking safes at Los Alamos– into the strip clubs and other things that make Feynman a polarizing figure.

Fermi brings a lot of the same positive features without the baggage. He had a similarly playful approach to a lot of physics-related things– the whole notion of “Fermi problems” and back-of-the-envelope calculations is pretty much essential to the physics mindset. Wikipedia has a great secondhand quote:

I can calculate anything in physics within a factor 2 on a few sheets: to get the numerical factor in front of the formula right may well take a physicist a year to calculate, but I am not interested in that.

He was also a charming and witty guy, with a quirky sense of humor (the photo above has a mistake in one of the equations, and people have spent years debating whether that was deliberate, because it’s the kind of thing he might’ve done as a joke on the PR people). He even has his own great Manhattan Project anecdotes– he famously estimated the strength of the blast by dropping pieces of paper and pacing off how far they blew when the shock wave hit, and prior to the Trinity test was reprimanded by Oppenheimer for running a betting pool on whether the test would ignite the atmosphere and obliterate life on Earth.

He also has the wide-ranging interests going for him. His name pops up all over physics, from statistical mechanics to the astrophysics of cosmic rays. And just like Feynman is more likely to be cited in popular writing for inspiring either nanotechnology or quantum computing than for his work on QED, Fermi’s true source of popular immortality is that damn “paradox” about aliens.

While the personal anecdotes may not quite stack up to those about Feynman, there isn’t the same dark edge. Fermi was happily married, and in fact moved to the US because of his wife, who was Jewish and subject to the racist policies put in place by Mussolini (they left directly from the Nobel Prize ceremony in 1938, where Fermi picked up a prize for work done in Rome). I’m not aware of any salacious Fermi stories, so he’s much safer in that regard.

Given all that, why is Fermi so much less well-known than Feynman? Partly because a lot of his contributions to physics were excessively practical– experimentalists tend to be less mythologized than theorists, and his greatest theoretical contributions came through cleaning up ideas proposed by Pauli. Mostly, though, it’s because he was a generation older than Feynman and died younger, in 1954. He never got the chance to be a grand old man, and didn’t live into the era where the sort of colorful anecdotage that so inflates Feynman’s status became popular. Had he lived another twenty years, things might’ve been different.

(It’s also interesting to speculate about what Schrödinger’s reputation would be had he been closer to Feynman’s age than Einstein’s. Schrödinger would’ve loved the Sixties, between his fascination with Vedic philosophy and the whole “free love” thing. But had he been alive through then, the skeevy aspects of his personal life would probably be better known, because most of his more sordid activities took place in an era when people didn’t really talk about that sort of thing in public, let alone write best-selling autobiographies about it.)

Anyway, that’s my plug for Fermi. If you find Feynman too problematic to promote– and that’s an entirely reasonable decision– Fermi gets you a lot of the same good stuff (great physicist, playful approach to science, charming and personable guy), without the darker side. He should get more press.

——

(That said, Fermi is one of the figures I regret not being able to feature more prominently in the forthcoming book. The problem is, the focus of the book is on process, and Fermi’s one of those guys whose process of discovery consisted mostly of “be super smart, and work really hard.” I couldn’t come up with a way to fit him into the framework of the book, other than the notion of back-of-the-envelope estimation. But you can’t do that without bringing math into it, and that would’ve pushed the book into a different category…)

This is just a short update on the saga of the anomalous excess of W-boson-pair production that the ATLAS and CMS collaborations have reported in their 7-TeV and 8-TeV proton-proton collision data. A small bit of information which I was unaware of, and which can be added to the picture.

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For some reason I was thinking about pieces of culture that have departed from the world but which somehow didn’t “stick” well enough to persist even in the sphere of nostalgia.  Like when people think about the early 1990s, the years when I was in college, they might well say “oh yeah, grunge” or “oh yeah, wearing used gas station T-shirts with a name stitched on” or “oh yeah, Twin Peaks” or “oh yeah, OK Soda” or whatever.

But no one says “oh yeah, Fido Dido.”  So here I am doing it.

It is inherently hard to try to list things you’ve forgotten about.  My list right now consists of

• Fido Dido
• Saying “bite me”
• Smartfood
• Devil sticks (from Jason Starr)

That’s it.  What have you got?

Sterile Neutrinos in Under 500 Words

Hi Folks,

In the Standard Model, we have three groups of particles: (i) force carriers, like photons and gluons; (ii) matter particles, like electrons, neutrinos and quarks; and (iii) the Higgs. Each force carrier is associated with a force. For example: photons are associated with electromagnetism, the W and Z bosons are associated with the weak nuclear force, and gluons are associated with the strong nuclear force. In principle, all particles (matter, force carries, the Higgs) can carry a charge associated with some force. If this is ever the case, then the charged particle can absorb or radiate a force carrier.

Credit: Wikipedia

As a concrete example, consider electrons and top quarks. Electrons carry an electric charge of “-1″ and a top quark carries an electric charge of “+2/3″. Both the electron and top quark can absorb/radiate photons, but since the top quark’s electric charge is smaller than the electron’s electric charge, it will not absorb/emit a photon as often as an electron. In a similar vein, the electron carries no “color charge”, the charge associated with the strong nuclear force, whereas the top quark does carry color and interacts via the strong nuclear force. Thus, electrons have no idea gluons even exist but top quarks can readily emit/absorb them.

Neutrinos  possess a weak nuclear charge and hypercharge, but no electric or color charge. This means that neutrinos can absorb/emit W and Z bosons and nothing else.  Neutrinos are invisible to photons (particle of light) as well as gluons (particles of the color force).  This is why it is so difficult to observe neutrinos: the only way to detect a neutrino is through the weak nuclear interactions. These are much feebler than electromagnetism or the strong nuclear force.

Sterile neutrinos are like regular neutrinos: they are massive (spin-1/2) matter particles that do not possess electric or color charge. The difference, however, is that sterile neutrinos do not carry weak nuclear or hypercharge either. In fact, they do not carry any charge, for any force. This is why they are called “sterile”; they are free from the influences of  Standard Model forces.

Credit: somerandompearsonsblog.blogspot.com

The properties of sterile neutrinos are simply astonishing. For example: Since they have no charge of any kind, they can in principle be their own antiparticles (the infamous “sterile Majorana neutrino“). As they are not associated with either the strong nuclear scale or electroweak symmetry breaking scale, sterile neutrinos can, in principle, have an arbitrarily large/small mass. In fact, very heavy sterile neutrinos might even be dark matter, though this is probably not the case. However, since sterile neutrinos do have mass, and at low energies they act just like regular Standard Model neutrinos, then they can participate in neutrino flavor oscillations. It is through this subtle effect that we hope to find sterile neutrinos if they do exist.

Credit: Kamioka Observatory/ICRR/University of Tokyo

Until next time!

Happy Colliding,

Richard (@bravelittlemuon)

Lance Mannion has a really nice contrast between childhood now and back in the 1970′s that doesn’t go in the usual decline-of-society direction. He grew up not too far from where I now live, and after describing his free-ranging youth, points out some of the key factors distinguishing it from today, that need to be accounted for before lamenting the lack of kids running around outside:

– A lot of the houses in “the old neighborhood” are still owned by the people who owned them back in the day, so the only kids around are visiting grandkids,

– Those homes that are occupied by families with kids are usually occupied by families with fewer kids than back in the day, so there are fewer older siblings to keep tabs on younger kids and that kind of thing,

– Most importantly, back in the day, there were fewer two-career families. Those kids running around out in the neighborhood were always within shouting distance of multiple parents.

I’m a bit younger than Lance, and grew up way out in the country, but this rings pretty true to my experience. And as I said, we live in a neighborhood not all that far from the one he talks about, and the changes he describes also ring true. Our neighborhood is great, but it’s split between families with kids and empty-nesters. The “with kids” fraction is increasing, but there are probably two childless houses for every one with kids, and most of the families are on the smaller side compared to the 70′s– I can’t think of anyone in the immediate neighborhood with more than three kids.

So a lot of things have changed to make it less likely that you’ll see kids running around outside. Which doesn’t mean there aren’t kids running around outside– my computer sits in front of a window that faces the street, and when I’m home during the day, I regularly see kids running and playing outside. But the overall numbers are reduced to a point where it’s fairly likely that people driving through the neighborhood could reasonably be clucking their tongues and talking about how sad it is that no kids play outside any more. You have to live here to know that there are kids around, because the density is lower than it used to be.

And the lack of kids is more apparent during the day, for the economic reasons Lance notes. Basically all of the families with kids in the area are two-career families, which means that during work hours, the number of kids around drops to nearly zero. They’re all in day care, even in the summer. Not because parents are overly controlling, or afraid to let their kids roam, but because they’re at work, because they have to be to live in this neighborhood. This also feeds some of the “never away from parents” thing that people talk about, because the time parents get to spend with their kids is more limited and thus more valuable.

But there are pockets that seem a lot like the old days– down the block from us, there’s a cluster of three families all with kids about SteelyKid’s age (including one of her kindergarten classmates). Those kids are in and out of each others’ houses and yards all day long, often with no visible adult supervision. SteelyKid had a friend over yesterday, and we wandered down there after lunch, where the kids ran around a lot like it was back in the day. It’s a little too far to just send a six-year-old off there on her own (and none of the houses between here and there are families with kids), but within a few years, I can easily imagine pointing SteelyKid in their direction after school and on weekends, and having the kids from down there show up in our yard.

Anyway, it’s worth reading Lance’s post, because it’s a cut above most of the hand-wringing you see about the way we raise kids these days. It’s really not as dire a situation as a lot of cultural critics make out, if you look carefully at what’s changed from the “good old days.”

I was in Barriques and “Bra,” by Cymande came on, and I was like, cool song, cool of Barriques to be playing this song that I’m cool for knowing about, maybe I should go say something to show everyone that I already know this cool song, and then I thought, why do I know about this song anyway? and I remembered that it was because sometime last year it was playing in Barriques and I was like, what is this song, it’s cool? and I Shazammed it.

So I guess what I’m saying is, I’m probably going to the right coffee shop.  Also, this song is cool.  I’m sort of fascinated by the long instrumental break that starts around 2:50.  It doesn’t seem like very much is happening; why is it so captivating?  I think my confusion on this point has something to do with my lack of understanding of drums.

It’s Saturday, so I’m at the coffee shop working on my thesis again. It’s become a tradition over the last year that I meet a writer friend each week, we catch up, have something to drink, and sit down for a few hours of good-quality writing time.

The work desk at the coffee shop: laptop, steamed pork bun, and rosebud latte.

We’ve gotten to know the coffee shop really well over the course of this year. It’s pretty new in the neighborhood, but dark and hidden enough that business is slow, and we don’t feel bad keeping a table for several hours. We have our favorite menu items, but we’ve tried most everything by now. Some mornings, the owner’s family comes in, and the kids watch cartoons at another table.

I work on my thesis mostly, or sometimes I’ll work on analysis that spills over from the week, or I’ll check on some scheduled jobs running on the computing cluster.

My friend Jason writes short stories, works on revising his novel (magical realism in ancient Egypt in the reign of Rameses XI), or drafts posts for his blog about the puzzles of the British constitution. We trade tips on how to organize notes and citations, and how to stay motivated. So I’ve been hearing a lot about the cultural difference between academic work in the humanities and the sciences. One of the big differences is the level of citation that’s expected.

As a particle physicist, when I write a paper it’s very clear which experiment I’m writing about. I only write about one experiment at a time, and I typically focus on a very small topic. Because of that, I’ve learned that the standard for making new claims is that you usually make one new claim per paper, and it’s highlighted in the abstract, introduction, and conclusion with a clear phrase like “the new contribution of this work is…” It’s easy to separate which work you claim as your own and which work is from others, because anything outside “the new contribution of this work” belongs to others. A single citation for each external experiment should suffice.

For academic work in history, the standard is much different: the writing itself is much closer to the original research. As a start, you’ll need a citation for each quote, going to sources that are as primary as you can get your hands on. The stranger idea for me is that you also need a citation for each and every idea of analysis that someone else has come up with, and that a statement without a citation is automatically claimed as original work. This shows up in the difference between Jason’s posts about modern constitutional issues and historical ones: the historical ones have huge source lists, while the modern ones are content with a few hyperlinks.

In both cases, things that are “common knowledge” doesn’t need to be cited, like the fact that TeV cosmic rays exist (they do) or the year that Elizabeth I ascended the throne (1558).

There’s a difference in the number of citations between modern physics research and history research. Is that because of the timing (historical versus modern) or the subject matter? Do they have different amounts of common knowledge? For modern topics in physics and in history, the sources are available online, so a hyperlink is a perfect reference, even in formal post. By that standard, all Quantum Diaries posts should be ok with the hyperlink citation model. But even in those cases, Jason puts footnoted citations to modern articles in the JSTOR database, and uses more citations overall.

Another cool aspect of our coffee shop is that the music is sometimes ridiculous, and it interrupts my thoughts if I get stuck in some esoteric bog. There’s an oddly large sample of German covers of 30s and 40s showtunes. You haven’t lived until you’ve heard “The Lady is a Tramp” in German while calculating oscillation probabilities. I’m kidding. Mostly.

Jason has shown me a different way of handling citations, and I’ve taught him some of the basics of HTML, so now his citations can appear as hyperlinks to the references list!

As habits go, I’m proud of this social coffee shop habit. I default to getting stuff done, even if I’m feeling slightly off or uninspired.  The social reward of hanging out makes up for the slight activation energy of getting off my couch, and once I’m out of the house, it’s always easier to focus.  I miss prime Farmers’ Market time, but I could go before we meet. The friendship has been a wonderful supportive certainty over the last year, plus I get some perspective on my field compared to others.

Predictably, my last post attracted plenty of outrage (some of it too vile to let through), along with the odd commenter who actually agreed with what I consider my fairly middle-of-the-road, liberal Zionist stance.  But since the outrage came from both sides of the issue, and the two sides were outraged about the opposite things, I guess I should feel OK about it.

Still, it’s hard not to smart from the burns of vituperation, so today I’d like to blog about a very different political issue: one where hopefully almost all Shtetl-Optimized readers will actually agree with me (!).

I’ve learned from colleagues that, over the past year, foreign-born scientists have been having enormously more trouble getting visas to enter the US than they used to.  The problem, I’m told, is particularly severe for cryptographers: embassy clerks are now instructed to ask specifically whether computer scientists seeking to enter the US work in cryptography.  If an applicant answers “yes,” it triggers a special process where the applicant hears nothing back for months, and very likely misses the workshop in the US that he or she had planned to attend.  The root of the problem, it seems, is something called the Technology Alert List (TAL), which has been around for a while—the State Department beefed it up in response to the 9/11 attacks—but which, for some unknown reason, is only now being rigorously enforced.  (Being marked as working in one of the sensitive fields on this list is apparently called “getting TAL’d.”)

The issue reached a comical extreme last October, when Adi Shamir, the “S” in RSA, Turing Award winner, and foreign member of the US National Academy of Sciences, was prevented from entering the US to speak at a “History of Cryptology” conference sponsored by the National Security Agency.  According to Shamir’s open letter detailing the incident, not even his friends at the NSA, or the president of the NAS, were able to grease the bureaucracy at the State Department for him.

It should be obvious to everyone that a crackdown on academic cryptographers serves no national security purpose whatsoever, and if anything harms American security and economic competitiveness, by diverting scientific talent to other countries.  (As Shamir delicately puts it, “the number of terrorists among the members of the US National Academy of Science is rather small.”)  So:

1. Any readers who have more facts about what’s going on, or personal experiences, are strongly encouraged to share them in the comments section.
2. Any readers who might have any levers of influence to pull on this issue—a Congressperson to write to, a phone call to make, an Executive Order to issue (I’m talking to you, Barack), etc.—are strongly encouraged to pull them.

I’ve just uploaded to the arXiv the D.H.J. Polymath paper “Variants of the Selberg sieve, and bounded intervals containing many primes“, which is the second paper to be produced from the Polymath8 project (the first one being discussed here). We’ll refer to this latter paper here as the Polymath8b paper, and the former as the Polymath8a paper. As with Polymath8a, the Polymath8b paper is concerned with the smallest asymptotic prime gap

where denotes the prime, as well as the more general quantities

In the breakthrough paper of Goldston, Pintz, and Yildirim, the bound was obtained under the strong hypothesis of the Elliott-Halberstam conjecture. An unconditional bound on , however, remained elusive until the celebrated work of Zhang last year, who showed that

The Polymath8a paper then improved this to . After that, Maynard introduced a new multidimensional Selberg sieve argument that gave the substantial improvement

unconditionally, and on the Elliott-Halberstam conjecture; furthermore, bounds on for higher were obtained for the first time, and specifically that for all , with the improvements and on the Elliott-Halberstam conjecture. (I had independently discovered the multidimensional sieve idea, although I did not obtain Maynard’s specific numerical results, and my asymptotic bounds were a bit weaker.)

In Polymath8b, we obtain some further improvements. Unconditionally, we have and , together with some explicit bounds on ; on the Elliott-Halberstam conjecture we have and some numerical improvements to the bounds; and assuming the generalised Elliott-Halberstam conjecture we have the bound , which is best possible from sieve-theoretic methods thanks to the parity problem obstruction.

There were a variety of methods used to establish these results. Maynard’s paper obtained a criterion for bounding which reduced to finding a good solution to a certain multidimensional variational problem. When the dimension parameter was relatively small (e.g. ), we were able to obtain good numerical solutions both by continuing the method of Maynard (using a basis of symmetric polynomials), or by using a Krylov iteration scheme. For large , we refined the asymptotics and obtained near-optimal solutions of the variational problem. For the bounds, we extended the reach of the multidimensional Selberg sieve (particularly under the assumption of the generalised Elliott-Halberstam conjecture) by allowing the function in the multidimensional variational problem to extend to a larger region of space than was previously admissible, albeit with some tricky new constraints on (and penalties in the variational problem). This required some unusual sieve-theoretic manipulations, notably an “epsilon trick”, ultimately relying on the elementary inequality , that allowed one to get non-trivial lower bounds for sums such as even if the sum had no non-trivial estimates available; and a way to estimate divisor sums such as even if was permitted to be comparable to or even exceed , by using the fundamental theorem of arithmetic to factorise (after restricting to the case when is almost prime). I hope that these sieve-theoretic tricks will be useful in future work in the subject.

With this paper, the Polymath8 project is almost complete; there is still a little bit of scope to push our methods further and get some modest improvement for instance to the bound, but this would require a substantial amount of effort, and it is probably best to instead wait for some new breakthrough in the subject to come along. One final task we are performing is to write up a retrospective article on both the 8a and 8b experiences, an incomplete writeup of which can be found here. If anyone wishes to contribute some commentary on these projects (whether you were an active contributor, an occasional contributor, or a silent “lurker” in the online discussion), please feel free to do so in the comments to this post.

Filed under: math.NT, paper, polymath Tagged: polymath8

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Black holes are defined by the presence of an event horizon which is the boundary of a region from which nothing can escape, ever. The word black hole is also often used to mean something that looks for a long time very similar to a black hole and that traps light, not eternally but only temporarily. Such space-times are said to have an “apparent horizon.” That they are not strictly speaking black holes was origin of the recent Stephen Hawking quote according to which black holes may not exist, by which he meant they might have only an apparent horizon instead of an eternal event horizon.

Black hole fireworks: quantum-gravity effects outside the horizon spark black to white hole tunneling
Hal M. Haggard, Carlo Rovelli
arXiv: 1407.0989

Figure 4 from arXiv: 1407.0989

Richard Hanson and Coryn Bailer-Jones (both MPIA) and I met today to talk about spatial priors and extinction modeling for Gaia. I showed them what I have on spatial priors, and we talked about the differences between using extinction measurements to predict new extinctions, using extinction measurements to predict dust densities, and so on. A key difference between the way I am thinking about it and the way Hanson and Bailer-Jones are thinking about it is that I don't want to instantiate the dust density (latent parameters) unless I have to. I would rather use the magic of the Gaussian Process to marginalize it out. We developed a set of issues for the document that I am writing on the subject. At Galaxy Coffee, Girish Kulkarni (MPIA) gave a great talk about the physics of the intergalactic medium and observational constraints from the absorption lines in quasar spectra.

I spent a chunk of the day with Melissa Ness (MPIA), fitting empirical models to APOGEE infrared spectra of stars. The idea is to do a simple linear supervised classification or regression, in which we figure out the dependence of the spectra on key stellar parameters, using a "training set" of stars with good stellar parameters. We worked in the pair-coding mode. By the end of the day we could show that we are able to identify regions of the spectrum that might serve as good metallicity indicators, relatively insensitive to temperature and log-g. The hopes for this project range from empirical metallicity index identification to label de-noising to building a full data-driven (supervised) stellar parameter pipeline. We ended our coding day pretty optimistic.

My op/ed about math teaching and Little League coaching is the most emailed article in the New York Times today.  Very cool!

But here’s something interesting; it’s only the 14th most viewed article, the 6th most tweeted, and the 6th most shared on Facebook.  On the other hand, this article about child refugees from Honduras is

#14 most emailed

#1 most viewed

#1 most shared on Facebook

#1 most tweeted

while Paul Krugman’s column about California is

#4 most emailed

#3 most viewed

#4 most shared on Facebook

#7 most tweeted.

Why are some articles, like mine, much more emailed than tweeted, while others, like the one about refugees, much more tweeted than emailed, and others still, like Krugman’s, come out about even?  Is it always the case that views track tweets, not emails?  Not necessarily; an article about the commercial success and legal woes of conservative poo-stirrer Dinesh D’Souza is #3 most viewed, but only #13 in tweets (and #9 in emails.)  Today’s Gaza story has lots of tweets and views but not so many emails, like the Honduras piece, so maybe this is a pattern for international news?  Presumably people inside newspapers actually study stuff like this; is any of that research public?  Now I’m curious.

I should really know better than to click any tweeted link with a huff.to shortened URL, but for some reason, I actually followed one to an article with the limited-reach clickbait title Curious About Quantum Physics? Read These 10 Articles!. Which is only part one, because Huffington Post, so it’s actually five articles.

Three of the five articles are Einstein papers from 1905, which is sort of the equivalent of making a Ten Essential Rock Albums list that includes Revolver, Abbey Road, and the White Album. One of the goals of a well-done list of “essential” whatever is to give a sense of the breadth of a subject, not just focus on a single example, so this is a big failure right off the bat.

But it’s even worse than that, because none of the three 1905 articles is the photoelectric effect paper, which is the only one of the lot that has any quantum physics in it. There’s a fourth Einstein paper on the list, as well, the theory of general relativity, which is famous for not being compatible with quantum mechanics. So this is really like a list of Ten Essential Rock Albums that includes three country songs and a Bach concerto.

I thought about using this as an opportunity to generate a better Ten Essential Quantum Papers list, including stuff like Bell’s Theorem (the physics equivalent of the first Velvet Underground record) and the No-Cloning Theorem (the physics equivalent of punk rock) (brief pause to let those who know Bill Wootters try to reconcile that mental image). And if you would like to make suggestions of things that ought to be on such a list in the comments, feel free.

(I’m also open to suggestions of better musical analogies– maybe the EPR paper is the real Velvet Underground record? With Bell’s paper being punk rock, making Wootters and Zurek… Nirvana, maybe? Or maybe Shor’s algorithm is the “Smells Like Teen Spirit” of quantum physics (Again, a brief pause while those who know Peter Shor try to picture him as Kurt Cobain)…)

But, really, on reflection, the whole exercise is kind of silly even by the standards of clickbait blog topics, because that’s not how science works. In science, and particularly a highly mathematical science like physics, there’s not that much real benefit to reading the original source material. The best explanation of a central concept is rarely if ever found in the first paper to present it. This goes right back to the start of the discipline, with Newton’s Principia Mathematica, which nobody reads because it’s written in really opaque Latin, a move he claimed in a letter was deliberate so as to avoid “being baited by little smatterers in mathematics” (Newton was kind of a dick). Newton’s mathematical notation is also pretty awful, and I’ve heard it claimed that the reason physics advanced faster in mainland Europe than in England during the 1700s was that on the continent, they adopted Leibniz’s system, which was way more user-friendly and is the basis for modern calculus notation. Similarly, Maxwell’s original presentation of his eponymous equations is really difficult to follow, and it’s only after the work of folks like Heaviside that they become the clear, elegant, and bumper-sticker-friendly version we know today.

That’s not to say that there’s no value in reading old papers– I’ve had a lot of fun writing up old MS theses from our department, and older work can be fascinating to read. But unlike primary works of (pop) culture, they’re much better if you come to them already knowing what they’re about. The fascination comes from seeing how people fumbled their way toward ideas that we now know to be correct. It’s rare for a “classic” paper to get all the way to the modern understanding of things, or even most of the way there– most of the great original works contain what we now know to be errors of interpretation. Others are revered today for discoveries that were somewhat tangential to what the original author thought was the main point– the Cavendish experiment is thought of today as a measurement of “big G,” but he presents it as a determination of the density of the Earth, because that was of more pressing practical interest at the time.

If you want to learn science, you’re much better off looking up the best modern treatment than going back to the original papers. A good recent textbook will have the bugs worked out, and present it in something close to the language used by working scientists today. A good popular-audience treatment (ahem) will cover the basic concepts starting from a more complete understanding of the field as it has developed, and with an eye toward making those concepts accessible to a modern reader. It’s not foolproof, of course– the steady progress of science over a stretch of decades often means that newer books need to cover a huge amount of material to get to the sexy cutting-edge stuff, and sometimes scant the basics a bit. But by and large, if you’re curious about quantum physics, you’d be much better off hitting the physics section of your local bookstore or library than digging through archived journals for the original papers.

So, a list of “Ten Essential Papers on Quantum Physics” is a deeply flawed concept right from the start, at least if the goal is to learn something about quantum physics that you didn’t already know. The same is true of almost every science, with a few exception– Darwin’s On the Origin of Species is still a really good read, but it’s the exception, not the rule. Such a list can be useful as a sort of historical map, or for providing some insight into the thought processes of the great scientists of yesteryear, and those can be very rewarding. But if you’re curious and want to learn, I don’t think any original papers can really be considered “essential.”

How can we discuss all the kinds of matter described by the ten-fold way in a single setup?

It’s bit tough, because 8 of them are fundamentally ‘real’ while the other 2 are fundamentally ‘complex’. Yet they should fit into a single framework, because there are 10 super division algebras over the real numbers, and each kind of matter is described using a super vector space — or really a super Hilbert space — with one of these super division algebras as its ‘ground field’.

Combining physical systems is done by tensoring their Hilbert spaces… and there does seem to be a way to do this even with super Hilbert spaces over different super division algebras. But what sort of mathematical structure can formalize this?

Here’s my current attempt to solve this problem. I’ll start with a warmup case, the threefold way. In fact I’ll spend most of my time on that! Then I’ll sketch how the ideas should extend to the tenfold way.

Fans of lax monoidal functors, Deligne’s tensor product of abelian categories, and the collage of a profunctor will be rewarded for their patience if they read the whole article. But the basic idea is supposed to be simple: it’s about a multiplication table.

The $\mathbb{3}$-fold way

First of all, notice that the set

$$\mathbb{3} = \{1,0,-1\}$$

is a commutative monoid under ordinary multiplication:

$$\begin{array}{rrrr} \mathbf{\times} & \mathbf{1} & \mathbf{0} & \mathbf{-1} \\ \mathbf{1} & 1 & 0 & -1 \\ \mathbf{0} & 0 & 0 & 0 \\ \mathbf{-1} & -1 & 0 & 1 \end{array}$$

Next, note that there are three (associative) division algebras over the reals: $\mathbb{R}, \mathbb{C}$ or $\mathbb{H}$. We can equip a real vector space with the structure of a module over any of these algebras. We’ll then call it a real, complex or quaternionic vector space.

For the real case, this is entirely dull. For the complex case, this amounts to giving our real vector space $V$ a complex structure: a linear operator $i: V \to V$ with $i^2 = -1$. For the quaternionic case, it amounts to giving $V$ a quaternionic structure: a pair of linear operators $i, j: V \to V$ with

$$i^2 = j^2 = -1, \qquad i j = -j i$$

We can then define $k = i j$.

The terminology ‘quaternionic vector space’ is a bit quirky, since the quaternions aren’t a field, but indulge me. $\mathbb{H}^n$ is a quaternionic vector space in an obvious way. $n \times n$ quaternionic matrices act by multiplication on the right as ‘quaternionic linear transformations’ — that is, left module homomorphisms — of $\mathbb{H}^n$. Moreover, every finite-dimensional quaternionic vector space is isomorphic to $\mathbb{H}^n$. So it’s really not so bad! You just need to pay some attention to left versus right.

Now: I claim that given two vector spaces of any of these kinds, we can tensor them over the real numbers and get a vector space of another kind. It goes like this:

$$\begin{array}{cccc} \mathbf{\otimes} & \mathbf{real} & \mathbf{complex} & \mathbf{quaternionic} \\ \mathbf{real} & real & complex & quaternionic \\ \mathbf{complex} & complex & complex & complex \\ \mathbf{quaternionic} & quaternionic & complex & real \end{array}$$

You’ll notice this has the same pattern as the multiplication table we saw before:

$$\begin{array}{rrrr} \mathbf{\times} & \mathbf{1} & \mathbf{0} & \mathbf{-1} \\ \mathbf{1} & 1 & 0 & -1 \\ \mathbf{0} & 0 & 0 & 0 \\ \mathbf{-1} & -1 & 0 & 1 \end{array}$$

So:

• $\mathbb{R}$ acts like 1.
• $\mathbb{C}$ acts like 0.
• $\mathbb{H}$ acts like -1.

There are different ways to understand this, but a nice one is to notice that if we have algebras $A$ and $B$ over some field, and we tensor an $A$-module and a $B$-module (over that field), we get an $A \otimes B$-module. So, we should look at this ‘multiplication table’ of real division algebras:

$$\begin{array}{lrrr} \mathbf{\otimes} & \mathbf{\mathbb{R}} & \mathbf{\mathbb{C}} & \mathbf{\mathbb{H}} \\ \mathbf{\mathbb{R}} & \mathbb{R} & \mathbb{C} & \mathbb{H} \\ \mathbf{\mathbb{C}} & \mathbb{C} & \mathbb{C} \oplus \mathbb{C} & \mathbb{C}[2] \\ \mathbf{\mathbb{H}} & \mathbb{H} & \mathbb{C}[2] & \mathbb{R}[4] \end{array}$$

Here $\mathbb{C}[2]$ means the 2 × 2 complex matrices viewed as an algebra over $\mathbb{R}$, and $\mathbb{R}[4]$ means that 4 × 4 real matrices.

What’s going on here? Naively you might have hoped for a simpler table, which would have instantly explained my earlier claim:

$$\begin{array}{lrrr} \mathbf{\otimes} & \mathbf{\mathbb{R}} & \mathbf{\mathbb{C}} & \mathbf{\mathbb{H}} \\ \mathbf{\mathbb{R}} & \mathbb{R} & \mathbb{C} &\mathbb{H} \\ \mathbf{\mathbb{C}} & \mathbb{C} & \mathbb{C} & \mathbb{C} \\ \mathbf{\mathbb{H}} & \mathbb{H} & \mathbb{C} & \mathbb{R} \end{array}$$

This isn’t true, but it’s ‘close enough to true’. Why? Because we always have a god-given algebra homomorphism from the naive answer to the real answer! The interesting cases are these:

$$\mathbb{C} \to \mathbb{C} \oplus \mathbb{C}$$ $$\mathbb{C} \to \mathbb{C}[2]$$ $$\mathbb{R} \to \mathbb{R}[4]$$

where the first is the diagonal map $a \mapsto (a,a)$, and the other two send numbers to the corresponding scalar multiples of the identity matrix.

So, for example, if $V$ and $W$ are $\mathbb{C}$-modules, then their tensor product (over the reals! — all tensor products here are over $\mathbb{R}$) is a module over $\mathbb{C} \otimes \mathbb{C} \cong \mathbb{C} \oplus \mathbb{C}$, and we can then pull that back via $f$ to get a right $\mathbb{C}$-module.

What’s really going on here?

There’s a monoidal category $Alg_{\mathbb{R}}$ of algebras over the real numbers, where the tensor product is the usual tensor product of algebras. The monoid $\mathbb{3}$ can be seen as a monoidal category with 3 objects and only identity morphisms. And I claim this:

Claim. There is an oplax monoidal functor $F : \mathbb{3} \to Alg_{\mathbb{R}}$ with $$\begin{array}{ccl} F(1) &=& \mathbb{R} \\ F(0) &=& \mathbb{C} \\ F(-1) &=& \mathbb{H} \end{array}$$

What does ‘oplax’ mean? Some readers of the $n$-Category Café eat oplax monoidal functors for breakfast and are chortling with joy at how I finally summarized everything I’d said so far in a single terse sentence! But others of you see ‘oplax’ and get a queasy feeling.

The key idea is that when we have two monoidal categories $C$ and $D$, a functor $F : C \to D$ is ‘oplax’ if it preserves the tensor product, not up to isomorphism, but up to a specified morphism. More precisely, given objects $x,y \in C$ we have a natural transformation

$$F_{x,y} : F(x \otimes y) \to F(x) \otimes F(y)$$

If you had a ‘lax’ functor this would point the other way, and they’re a bit more popular… so when it points the opposite way it’s called ‘oplax’.

(In the lax case, $F_{x,y}$ should probably be called the laxative, but we’re not doing that case, so I don’t get to make that joke.)

This morphism $F_{x,y}$ needs to obey some rules, but the most important one is that using it twice, it gives two ways to get from $F(x \otimes y \otimes z)$ to $F(x) \otimes F(y) \otimes F(z)$, and these must agree.

Let’s see how this works in our example… at least in one case. I’ll take the trickiest case. Consider

$$F_{0,0} : F(0 \cdot 0) \to F(0) \otimes F(0),$$

that is:

$$F_{0,0} : \mathbb{C} \to \mathbb{C} \otimes \mathbb{C}$$

There are, in principle, two ways to use this to get a homomorphism

$$F(0 \cdot 0 \cdot 0 ) \to F(0) \otimes F(0) \otimes F(0)$$

or in other words, a homomorphism

$$\mathbb{C} \to \mathbb{C} \otimes \mathbb{C} \otimes \mathbb{C}$$

where remember, all tensor products are taken over the reals. One is

$$\mathbb{C} \stackrel{F_{0,0}}{\longrightarrow} \mathbb{C} \otimes \mathbb{C} \stackrel{1 \otimes F_{0,0}}{\longrightarrow} \mathbb{C} \otimes (\mathbb{C} \otimes \mathbb{C})$$

and the other is

$$\mathbb{C} \stackrel{F_{0,0}}{\longrightarrow} \mathbb{C} \otimes \mathbb{C} \stackrel{F_{0,0} \otimes 1}{\longrightarrow} (\mathbb{C} \otimes \mathbb{C})\otimes \mathbb{C}$$

I want to show they agree (after we rebracket the threefold tensor product using the associator).

Unfortunately, so far I have described $F_{0,0}$ in terms of an isomorphism

$$\mathbb{C} \otimes \mathbb{C} \cong \mathbb{C} \oplus \mathbb{C}$$

Using this isomorphism, $F_{0,0}$ becomes the diagonal map $a \mapsto (a,a)$. But now we need to really understand $F_{0,0}$ a bit better, so I’d better say what isomorphism I have in mind! I’ll use the one that goes like this:

$$\begin{array}{ccl} \mathbb{C} \otimes \mathbb{C} &\to& \mathbb{C} \oplus \mathbb{C} \\ 1 \otimes 1 &\mapsto& (1,1) \\ i \otimes 1 &\mapsto &(i,i) \\ 1 \otimes i &\mapsto &(i,-i) \\ i \otimes i &\mapsto & (1,-1) \end{array}$$

This may make you nervous, but it truly is an isomorphism of real algebras, and it sends $a \otimes 1$ to $(a,a)$. So, unraveling the web of confusion, we have

$$\begin{array}{rccc} F_{0,0} : & \mathbb{C} &\to& \mathbb{C}\otimes \mathbb{C} \\ & a &\mapsto & a \otimes 1 \end{array}$$

Why didn’t I just say that in the first place? Well, I suffered over this a bit, so you should too! You see, there’s an unavoidable arbitrary choice here: I could just have well used $a \mapsto 1 \otimes a$. $F_{0,0}$ looked perfectly god-given when we thought of it as a homomorphism from $\mathbb{C}$ to $\mathbb{C} \oplus \mathbb{C}$, but that was deceptive, because there’s a choice of isomorphism $\mathbb{C} \otimes \mathbb{C} \to \mathbb{C} \oplus \mathbb{C}$ lurking in this description.

This makes me nervous, since category theory disdains arbitrary choices! But it seems to work. On the one hand we have

$$\begin{array}{ccccc} \mathbb{C} &\stackrel{F_{0,0}}{\longrightarrow} &\mathbb{C} \otimes \mathbb{C} &\stackrel{1 \otimes F_{0,0}}{\longrightarrow}& \mathbb{C} \otimes \mathbb{C} \otimes \mathbb{C} \\ a &\mapsto & a \otimes 1 & \mapsto & a \otimes (1 \otimes 1) \end{array}$$

On the other hand, we have

$$\begin{array}{ccccc} \mathbb{C} &\stackrel{F_{0,0}}{\longrightarrow} & \mathbb{C} \otimes \mathbb{C} &\stackrel{F_{0,0} \otimes 1}{\longrightarrow} & \mathbb{C} \otimes \mathbb{C} \otimes \mathbb{C} \\ a &\mapsto & a \otimes 1 & \mapsto & (a \otimes 1) \otimes 1 \end{array}$$

So they agree!

I need to carefully check all the other cases before I dare call my claim a theorem. Indeed, writing up this case has increased my nervousness… before, I’d thought it was obvious.

But let me march on, optimistically!

Consequences

In quantum physics, what matters is not so much the algebras $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{H}$ themselves as the categories of vector spaces — or indeed, Hilbert spaces —-over these algebras. So, we should think about the map sending an algebra to its category of modules.

For any field $k$, there should be a contravariant pseudofunctor

$$Rep: Alg_k \to Rex_k$$

where $Rex_k$ is the 2-category of

• $k$-linear finitely cocomplete categories,

• $k$-linear functors preserving finite colimits,

• and natural transformations.

The idea is that $Rep$ sends any algebra $A$ over $k$ to its category of modules, and any homomorphism $f : A \to B$ to the pullback functor $f^* : Rep(B) \to Rep(A)$.

(Functors preserving finite colimits are also called right exact; this is the reason for the funny notation $Rex$. It has nothing to do with the dinosaur of that name.)

Moreover, $Rep$ gets along with tensor products. It’s definitely true that given real algebras $A$ and $B$, we have

$$Rep(A \otimes B) \simeq Rep(A) \boxtimes Rep(B)$$

where $\boxtimes$ is the tensor product of finitely cocomplete $k$-linear categories. But we should be able to go further and prove $Rep$ is monoidal. I don’t know if anyone has bothered yet.

(In case you’re wondering, this $\boxtimes$ thing reduces to Deligne’s tensor product of abelian categories given some ‘niceness assumptions’, but it’s a bit more general. Read the talk by Ignacio López Franco if you care… but I could have used Deligne’s setup if I restricted myself to finite-dimensional algebras, which is probably just fine for what I’m about to do.)

So, if my earlier claim is true, we can take the oplax monoidal functor

$$F : \mathbb{3} \to Alg_{\mathbb{R}}$$

and compose it with the contravariant monoidal pseudofunctor

$$Rep : Alg_{\mathbb{R}} \to Rex_{\mathbb{R}}$$

giving a guy which I’ll call

$$Vect: \mathbb{3} \to Rex_{\mathbb{R}}$$

I guess this guy is a contravariant oplax monoidal pseudofunctor! That doesn’t make it sound very lovable… but I love it. The idea is that:

• $Vect(1)$ is the category of real vector spaces

• $Vect(0)$ is the category of complex vector spaces

• $Vect(-1)$ is the category of quaternionic vector spaces

and the operation of multiplication in $\mathbb{3} = \{1,0,-1\}$ gets sent to the operation of tensoring any one of these three kinds of vector space with any other kind and getting another kind!

So, if this works, we’ll have combined linear algebra over the real numbers, complex numbers and quaternions into a unified thing, $Vect$. This thing deserves to be called a $\mathbb{3}$-graded category. This would be a nice way to understand Dyson’s threefold way.

What’s really going on?

What’s really going on with this monoid $\mathbb{3}$? It’s a kind of combination or ‘collage’ of two groups:

• The Brauer group of $\mathbb{R}$, namely $\mathbb{Z}_2 \cong \{-1,1\}$. This consists of Morita equivalence classes of central simple algebras over $\mathbb{R}$. One class contains $\mathbb{R}$ and the other contains $\mathbb{H}$. The tensor product of algebras corresponds to multiplication in $\{-1,1\}$.

• The Brauer group of $\mathbb{C}$, namely the trivial group $\{0\}$. This consists of Morita equivalence classes of central simple algebras over $\mathbb{C}$. But $\mathbb{C}$ is algebraically closed, so there’s just one class, containing $\mathbb{C}$ itself!

See, the problem is that while $\mathbb{C}$ is a division algebra over $\mathbb{R}$, it’s not ‘central simple’ over $\mathbb{R}$: its center is not just $\mathbb{R}$, it’s bigger. This turns out to be why $\mathbb{C} \otimes \mathbb{C}$ is so funny compared to the rest of the entries in our division algebra multiplication table.

So, we’ve really got two Brauer groups in play. But we also have a homomorphism from the first to the second, given by ‘tensoring with $\mathbb{C}$’: complexifying any real central simple algebra, we get a complex one.

And whenever we have a group homomorphism $\alpha: G \to H$, we can make their disjoint union $G \sqcup H$ into monoid, which I’ll call $G \sqcup_\alpha H$.

It works like this. Given $g,g' \in G$, we multiply them the usual way. Given $h, h' \in H$, we multiply them the usual way. But given $g \in G$ and $h \in H$, we define

$$g h := \alpha(g) h$$

and

$$h g := h \alpha(g)$$

The multiplication on $G \sqcup_\alpha H$ is associative! For example:

$$(g g')h = \alpha(g g') h = \alpha(g) \alpha(g') h = \alpha(g) (g'h) = g(g'h)$$

Moreover, the element $1_G \in G$ acts as the identity of $G \sqcup_\alpha H$. For example:

$$1_G h = \alpha(1_G) h = 1_H h = h$$

But of course $G \sqcup_\alpha H$ isn’t a group, since “once you get inside $H$ you never get out”.

This construction could be called the collage of $G$ and $H$ via $\alpha$, since it’s reminiscent of a similar construction of that name in category theory.

Question. What do monoid theorists call this construction?

Question. Can we do a similar trick for any field? Can we always take the Brauer groups of all its finite-dimensional extensions and fit them together into a monoid by taking some sort of collage? If so, I’d call this the Brauer monoid of that field.

The $\mathbb{10}$-fold way

If you carefully read Part 1, maybe you can guess how I want to proceed. I want to make everything ‘super’.

I’ll replace division algebras over $\mathbb{R}$ by super division algebras over $\mathbb{R}$. Now instead of 3 = 2 + 1 there are 10 = 8 + 2:

• 8 of them are central simple over $\mathbb{R}$, so they give elements of the super Brauer group of $\mathbb{R}$, which is $\mathbb{Z}_8$.

• 2 of them are central simple over $\mathbb{C}$, so they give elements of the super Brauer group of $\mathbb{C}$, which is $\mathbb{Z}_2$.

Complexification gives a homomorphism

$$\alpha: \mathbb{Z}_8 \to \mathbb{Z}_2$$

namely the obvious nontrivial one. So, we can form the collage

$$\mathbb{10} = \mathbb{Z}_8 \sqcup_\alpha \mathbb{Z}_2$$

It’s a commutative monoid with 10 elements! Each of these is the equivalence class of one of the 10 real super division algebras.

I’ll then need to check that there’s an oplax monoidal functor

$$G : \mathbb{10} \to SuperAlg_{\mathbb{R}}$$

sending each element of $\mathbb{10}$ to the corresponding super division algebra.

If $G$ really exists, I can compose it with a thing

$$SuperRep : SuperAlg_{\mathbb{R}} \to Rex_{\mathbb{R}}$$

sending each super algebra to its category of ‘super representations’ on super vector spaces. This should again be a contravariant monoidal pseudofunctor.

We can call the composite of $G$ with $SuperRep$

$$SuperVect: \mathbb{10} \to \Rex_{\mathbb{R}}$$

If it all works, this thing $SuperVect$ will deserve to be called a $\mathbb{10}$-graded category. It contains super vector spaces over the 10 kinds of super division algebras in a single framework, and says how to tensor them. And when we look at super Hilbert spaces, this setup will be able to talk about all ten kinds of matter I mentioned last time… and how to combine them.

So that’s the plan. If you see problems, or ways to simplify things, please let me know!

There are 10 of each of these things:

• Associative real super-division algebras.

• Classical families of compact symmetric spaces.

• Ways that Hamiltonians can get along with time reversal ($T$) and charge conjugation ($C$) symmetry.

• Dimensions of spacetime in string theory.

It’s too bad nobody took up writing This Week’s Finds in Mathematical Physics when I quit. Someone should have explained this stuff in a nice simple way, so I could read their summary instead of fighting my way through the original papers. I don’t have much time for this sort of stuff anymore!

Luckily there are some good places to read about this stuff:

• Todd Trimble, The super Brauer group and super division algebras, April 27, 2005.

• Shinsei Ryu, Andreas P. Schnyde, Akira Furusaki and Andreas W. W. Ludwig, Topological insulators and superconductors: tenfold way and dimensional hierarchy, June 15, 2010.

• Gregory Moore and Dan Freed, Twisted equivariant matter, January 7, 2013.

• Gregory Moore, Quantum symmetries and compatible Hamiltonians, December 15, 2013.

Let me start by explaining the basic idea, and then move on to more fancy aspects.

Ten kinds of matter

The idea of the ten-fold way goes back at least to 1996, when Altland and Zirnbauer discovered that substances can be divided into 10 kinds.

The basic idea is pretty simple. Some substances have time-reversal symmetry: they would look the same, even on the atomic level, if you made a movie of them and ran it backwards. Some don’t — these are more rare, like certain superconductors made of yttrium barium copper oxide! Time reversal symmetry is described by an antiunitary operator $T$ that squares to 1 or to -1: please take my word for this, it’s a quantum thing. So, we get 3 choices, which are listed in the chart under $T$ as 1, -1, or 0 (no time reversal symmetry).

Similarly, some substances have charge conjugation symmetry, meaning a symmetry where we switch particles and holes: places where a particle is missing. The ‘particles’ here can be rather abstract things, like phonons - little vibrations of sound in a substance, which act like particles — or spinons — little vibrations in the lined-up spins of electrons. Basically any way that something can wave can, thanks to quantum mechanics, act like a particle. And sometimes we can switch particles and holes, and a substance will act the same way!

Like time reversal symmetry, charge conjugation symmetry is described by an antiunitary operator $C$ that can square to 1 or to -1. So again we get 3 choices, listed in the chart under $C$ as 1, -1, or 0 (no charge conjugation symmetry).

So far we have 3 × 3 = 9 kinds of matter. What is the tenth kind?

Some kinds of matter don’t have time reversal or charge conjugation symmetry, but they’re symmetrical under the combination of time reversal and charge conjugation! You switch particles and holes and run the movie backwards, and things look the same!

In the chart they write 1 under the $S$ when your matter has this combined symmetry, and 0 when it doesn’t. So, “0 0 1” is the tenth kind of matter (the second row in the chart).

This is just the beginning of an amazing story. Since then people have found substances called topological insulators that act like insulators in their interior but conduct electricity on their surface. We can make 3-dimensional topological insulators, but also 2-dimensional ones (that is, thin films) and even 1-dimensional ones (wires). And we can theorize about higher-dimensional ones, though this is mainly a mathematical game.

So we can ask which of the 10 kinds of substance can arise as topological insulators in various dimensions. And the answer is: in any particular dimension, only 5 kinds can show up. But it’s a different 5 in different dimensions! This chart shows how it works for dimensions 1 through 8. The kinds that can’t show up are labelled 0.

If you look at the chart, you’ll see it has some nice patterns. And it repeats after dimension 8. In other words, dimension 9 works just like dimension 1, and so on.

If you read some of the papers I listed, you’ll see that the $\mathbb{Z}$’s and $\mathbb{Z}_2$’s in the chart are the homotopy groups of the ten classical series of compact symmetric spaces. The fact that dimension $n+8$ works like dimension $n$ is called Bott periodicity.

Furthermore, the stuff about operators $T$, $C$ and $S$ that square to 1, -1 or don’t exist at all is closely connected to the classification of associative real super division algebras. It all fits together.

Super division algebras

In 2005, Todd Trimble wrote a short paper called The super Brauer group and super division algebras.

In it, he gave a quick way to classify the associative real super division algebras: that is, finite-dimensional associative real $\mathbb{Z}_2$-graded algebras having the property that every nonzero homogeneous element is invertible. The result was known, but I really enjoyed Todd’s effortless proof.

However, I didn’t notice that there are exactly 10 of these guys. Now this turns out to be a big deal. For each of these 10 algebras, the representations of that algebra describe ‘types of matter’ of a particular kind — where the 10 kinds are the ones I explained above!

So what are these 10 associative super division algebras?

3 of them are purely even, with no odd part: the usual associative division algebras $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$.

7 of them are not purely even. Of these, 6 are Morita equivalent to the real Clifford algebras $Cl_1, Cl_2, Cl_3, Cl_5, Cl_6$ and $Cl_7$. These are the superalgebras generated by 1, 2, 3, 5, 6, or 7 odd square roots of -1.

Now you should have at least two questions:

• What’s ‘Morita equivalence’? — and even if you know, why should it matter here? Two algebras are Morita equivalent if they have equivalent categories of representations. The same definition works for superalgebras, though now we look at their representations on super vector spaces ($\mathbb{Z}_2$-graded vector spaces). For physics what we really care about is the representations of an algebra or superalgebra: as I mentioned, those are ‘types of matter’. So, it makes sense to count two superalgebras as ‘the same’ if they’re Morita equivalent.

• 1, 2, 3, 5, 6, and 7? That’s weird — why not 4? Well, Todd showed that $Cl_4$ is Morita equivalent to the purely even super division algebra $\mathbb{H}$. So we already had that one on our list. Similarly, why not 0? $Cl_0$ is just $\mathbb{R}$. So we had that one too.

Representations of Clifford algebras are used to describe spin-1/2 particles, so it’s exciting that 8 of the 10 associative real super division algebras are Morita equivalent to real Clifford algebras.

But I’ve already mentioned one that’s not: the complex numbers, $\mathbb{C}$, regarded as a purely even algebra. And there’s one more! It’s the complex Clifford algebra $\mathbb{C}\mathrm{l}_1$. This is the superalgebra you get by taking the purely even algebra $\mathbb{C}$ and throwing in one odd square root of -1.

As soon as you hear that, you notice that the purely even algebra $\mathbb{C}$ is the complex Clifford algebra $\mathbb{C}\mathrm{l}_0$. In other words, it’s the superalgebra you get by taking the purely even algebra $\mathbb{C}$ and throwing in no odd square roots of -1.

More connections

At this point things start fitting together:

• You can multiply Morita equivalence classes of algebras using the tensor product of algebras: $[A] \otimes [B] = [A \otimes B]$. Some equivalence classes have multiplicative inverses, and these form the Brauer group. We can do the same thing for superalgebras, and get the super Brauer group. The super division algebras Morita equivalent to $Cl_0, \dots , Cl_7$ serve as representatives of the super Brauer group of the real numbers, which is $\mathbb{Z}_8$. I explained this in week211 and further in week212. It’s a nice purely algebraic way to think about real Bott periodicity!

• As we’ve seen, the super division algebras Morita equivalent to $Cl_0$ and $Cl_4$ are a bit funny. They’re purely even. So they serve as representatives of the plain old Brauer group of the real numbers, which is $\mathbb{Z}_2$.

• On the other hand, the complex Clifford algebras $\mathbb{C}\mathrm{l}_0 = \mathbb{C}$ and $\mathbb{C}\mathrm{l}_1$ serve as representatives of the super Brauer group of the complex numbers, which is also $\mathbb{Z}_2$. This is a purely algebraic way to think about complex Bott periodicity, which has period 2 instead of period 8.

Meanwhile, the purely even $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$ underlie Dyson’s ‘three-fold way’, which I explained in detail here:

• John Baez, Division algebras and quantum theory.

Briefly, if you have an irreducible unitary representation of a group on a complex Hilbert space $H$, there are three possibilities:

• The representation is isomorphic to its dual via an invariant symmetric bilinear pairing $g : H \times H \to \mathbb{C}$. In this case it has an invariant antiunitary operator $J : H \to H$ with $J^2 = 1$. This lets us write our representation as the complexification of a real one.

• The representation is isomorphic to its dual via an invariant antisymmetric bilinear pairing $\omega : H \times H \to \mathbb{C}$. In this case it has an invariant antiunitary operator $J : H \to H$ with $J^2 = -1$. This lets us promote our representation to a quaternionic one.

• The representation is not isomorphic to its dual. In this case we say it’s truly complex.

In physics applications, we can take $J$ to be either time reversal symmetry, $T$, or charge conjugation symmetry, $C$. Studying either symmetry separately leads us to Dyson’s three-fold way. Studying them both together leads to the ten-fold way!

So the ten-fold way seems to combine in one nice package:

• real Bott periodicity,
• complex Bott periodicity,
• the real Brauer group,
• the real super Brauer group,
• the complex super Brauer group, and
• the three-fold way.

I could throw ‘the complex Brauer group’ into this list, because that’s lurking here too, but it’s the trivial group, with $\mathbb{C}$ as its representative.

There really should be a better way to understand this. Here’s my best attempt right now.

The set of Morita equivalence classes of finite-dimensional real superalgebras gets a commutative monoid structure thanks to direct sum. This commutative monoid then gets a commutative rig structure thanks to tensor product. This commutative rig — let’s call it $\mathfrak{R}$ — is apparently too complicated to understand in detail, though I’d love to be corrected about that. But we can peek at pieces:

• We can look at the group of invertible elements in $\mathfrak{R}$ — more precisely, elements with multiplicative inverses. This is the real super Brauer group $\mathbb{Z}_8$.

• We can look at the sub-rig of $\mathfrak{R}$ coming from semisimple purely even algebras. As a commutative monoid under addition, this is $\mathbb{N}^3$, since it’s generated by $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$. This commutative monoid becomes a rig with a funny multiplication table, e.g. $\mathbb{C} \otimes \mathbb{C} = \mathbb{C} \oplus \mathbb{C}$. This captures some aspects of the three-fold way.

We should really look at a larger chunk of the rig $\mathfrak{R}$, that includes both of these chunks. How about the sub-rig coming from all semisimple superalgebras? What’s that?

And here’s another question: what’s the relation to the 10 classical families of compact symmetric spaces? The short answer is that each family describes a family of possible Hamiltonians for one of our 10 kinds of matter. For a more detailed answer, I suggest reading Gregory Moore’s Quantum symmetries and compatible Hamiltonians. But if you look at this chart by Ryu et al, you’ll see these families involve a nice interplay between $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$, which is what this story is all about:

The families of symmetric spaces are listed in the column “Hamiltonian”.

All this stuff is fitting together more and more nicely! And if you look at the paper by Freed and Moore, you’ll see there’s a lot more involved when you take the symmetries of crystals into account. People are beginning to understand the algebraic and topological aspects of condensed matter much more deeply these days.

The list

Just for the record, here are all 10 associative real super division algebras. 8 are Morita equivalent to real Clifford algebras:

• $Cl_0$ is the purely even division algebra $\mathbb{R}$.

• $Cl_1$ is the super division algebra $\mathbb{R} \oplus \mathbb{R}e$, where $e$ is an odd element with $e^2 = -1$.

• $Cl_2$ is the super division algebra $\mathbb{C} \oplus \mathbb{C}e$, where $e$ is an odd element with $e^2 = -1$ and $e i = -i e$.

• $Cl_3$ is the super division algebra $\mathbb{H} \oplus \mathbb{H}e$, where $e$ is an odd element with $e^2 = 1$ and $e i = i e, e j = j e, e k = k e$.

• $Cl_4$ is $\mathbb{H}[2]$, the algebra of $2 \times 2$ quaternionic matrices, given a certain $\mathbb{Z}_2$-grading. This is Morita equivalent to the purely even division algebra $\mathbb{H}$.

• $Cl_5$ is $\mathbb{H}[2] \oplus \mathbb{H}[2]$ given a certain $\mathbb{Z}_2$-grading. This is Morita equivalent to the super division algebra $\mathbb{H} \oplus \mathbb{H}e$ where $e$ is an odd element with $e^2 = -1$ and $e i = i e, e j = j e, e k = k e$.

• $Cl_6$ is $\mathbb{C}[4] \oplus \mathbb{C}[4]$ given a certain $\mathbb{Z}_2$-grading. This is Morita equivalent to the super division algebra $\mathbb{C} \oplus \mathbb{C}e$ where $e$ is an odd element with $e^2 = 1$ and $e i = -i e$.

• $Cl_7$ is $\mathbb{R}[8] \oplus \mathbb{R}[8]$ given a certain $\mathbb{Z}_2$-grading. This is Morita equivalent to the super division algebra $\mathbb{R} \oplus \mathbb{R}e$ where $e$ is an odd element with $e^2 = 1$.

$Cl_{n+8}$ is Morita equivalent to $Cl_n$ so we can stop here if we’re just looking for Morita equivalence classes, and there also happen to be no more super division algebras down this road. It is nice to compare $Cl_n$ and $Cl_{8-n}$: there’s a nice pattern here.

The remaining 2 real super division algebras are complex Clifford algebras:

• $\mathbb{C}\mathrm{l}_0$ is the purely even division algebra $\mathbb{C}$.

• $\mathbb{C}\mathrm{l}_1$ is the super division algebra $\mathbb{C} \oplus \mathbb{C} e$, where $e$ is an odd element with $e^2 = -1$ and $e i = i e$.

In the last one we could also say “with $e^2 = 1$” — we’d get something isomorphic, not a new possibility.

Ten dimensions of string theory

Oh yeah — what about the 10 dimensions in string theory? Are they really related to the ten-fold way?

It seems weird, but I think the answer is “yes, at least slightly”.

Remember, 2 of the dimensions in 10d string theory are those of the string worldsheet, which is a complex manifold. The other 8 are connected to the octonions, which in turn are connected to the 8-fold periodicity of real Clifford algebra. So the 8+2 split in string theory is at least slightly connected to the 8+2 split in the list of associative real super division algebras.

This may be more of a joke than a deep observation. After all, the 8 dimensions of the octonions are not individual things with distinct identities, as the 8 super division algebras coming from real Clifford algebras are. So there’s no one-to-one correspondence going on here, just an equation between numbers.

Still, there are certain observations that would be silly to resist mentioning.

One of the most profound and mysterious principles in all of physics is the Born Rule, named after Max Born. In quantum mechanics, particles don’t have classical properties like “position” or “momentum”; rather, there is a wave function that assigns a (complex) number, called the “amplitude,” to each possible measurement outcome. The Born Rule is then very simple: it says that the probability of obtaining any possible measurement outcome is equal to the square of the corresponding amplitude. (The wave function is just the set of all the amplitudes.)

Born Rule:    

The Born Rule is certainly correct, as far as all of our experimental efforts have been able to discern. But why? Born himself kind of stumbled onto his Rule. Here is an excerpt from his 1926 paper:

That’s right. Born’s paper was rejected at first, and when it was later accepted by another journal, he didn’t even get the Born Rule right. At first he said the probability was equal to the amplitude, and only in an added footnote did he correct it to being the amplitude squared. And a good thing, too, since amplitudes can be negative or even imaginary!

The status of the Born Rule depends greatly on one’s preferred formulation of quantum mechanics. When we teach quantum mechanics to undergraduate physics majors, we generally give them a list of postulates that goes something like this:

1. Quantum states are represented by wave functions, which are vectors in a mathematical space called Hilbert space.
2. Wave functions evolve in time according to the Schrödinger equation.
3. The act of measuring a quantum system returns a number, known as the eigenvalue of the quantity being measured.
4. The probability of getting any particular eigenvalue is equal to the square of the amplitude for that eigenvalue.
5. After the measurement is performed, the wave function “collapses” to a new state in which the wave function is localized precisely on the observed eigenvalue (as opposed to being in a superposition of many different possibilities).

It’s an ungainly mess, we all agree. You see that the Born Rule is simply postulated right there, as #4. Perhaps we can do better.

Of course we can do better, since “textbook quantum mechanics” is an embarrassment. There are other formulations, and you know that my own favorite is Everettian (“Many-Worlds”) quantum mechanics. (I’m sorry I was too busy to contribute to the active comment thread on that post. On the other hand, a vanishingly small percentage of the 200+ comments actually addressed the point of the article, which was that the potential for many worlds is automatically there in the wave function no matter what formulation you favor. Everett simply takes them seriously, while alternatives need to go to extra efforts to erase them. As Ted Bunn argues, Everett is just “quantum mechanics,” while collapse formulations should be called “disappearing-worlds interpretations.”)

Like the textbook formulation, Everettian quantum mechanics also comes with a list of postulates. Here it is:

1. Quantum states are represented by wave functions, which are vectors in a mathematical space called Hilbert space.
2. Wave functions evolve in time according to the Schrödinger equation.

That’s it! Quite a bit simpler — and the two postulates are exactly the same as the first two of the textbook approach. Everett, in other words, is claiming that all the weird stuff about “measurement” and “wave function collapse” in the conventional way of thinking about quantum mechanics isn’t something we need to add on; it comes out automatically from the formalism.

The trickiest thing to extract from the formalism is the Born Rule. That’s what Charles (“Chip”) Sebens and I tackled in our recent paper:

Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics
Charles T. Sebens, Sean M. Carroll

A longstanding issue in attempts to understand the Everett (Many-Worlds) approach to quantum mechanics is the origin of the Born rule: why is the probability given by the square of the amplitude? Following Vaidman, we note that observers are in a position of self-locating uncertainty during the period between the branches of the wave function splitting via decoherence and the observer registering the outcome of the measurement. In this period it is tempting to regard each branch as equiprobable, but we give new reasons why that would be inadvisable. Applying lessons from this analysis, we demonstrate (using arguments similar to those in Zurek’s envariance-based derivation) that the Born rule is the uniquely rational way of apportioning credence in Everettian quantum mechanics. In particular, we rely on a single key principle: changes purely to the environment do not affect the probabilities one ought to assign to measurement outcomes in a local subsystem. We arrive at a method for assigning probabilities in cases that involve both classical and quantum self-locating uncertainty. This method provides unique answers to quantum Sleeping Beauty problems, as well as a well-defined procedure for calculating probabilities in quantum cosmological multiverses with multiple similar observers.

Chip is a graduate student in the philosophy department at Michigan, which is great because this work lies squarely at the boundary of physics and philosophy. (I guess it is possible.) The paper itself leans more toward the philosophical side of things; if you are a physicist who just wants the equations, we have a shorter conference proceeding.

Before explaining what we did, let me first say a bit about why there’s a puzzle at all. Let’s think about the wave function for a spin, a spin-measuring apparatus, and an environment (the rest of the world). It might initially take the form

(α[up] + β[down] ; apparatus says “ready” ; environment₀).             (1)

This might look a little cryptic if you’re not used to it, but it’s not too hard to grasp the gist. The first slot refers to the spin. It is in a superposition of “up” and “down.” The Greek letters α and β are the amplitudes that specify the wave function for those two possibilities. The second slot refers to the apparatus just sitting there in its ready state, and the third slot likewise refers to the environment. By the Born Rule, when we make a measurement the probability of seeing spin-up is |α|², while the probability for seeing spin-down is |β|².

In Everettian quantum mechanics (EQM), wave functions never collapse. The one we’ve written will smoothly evolve into something that looks like this:

α([up] ; apparatus says “up” ; environment₁)
     + β([down] ; apparatus says “down” ; environment₂).             (2)

This is an extremely simplified situation, of course, but it is meant to convey the basic appearance of two separate “worlds.” The wave function has split into branches that don’t ever talk to each other, because the two environment states are different and will stay that way. A state like this simply arises from normal Schrödinger evolution from the state we started with.

So here is the problem. After the splitting from (1) to (2), the wave function coefficients α and β just kind of go along for the ride. If you find yourself in the branch where the spin is up, your coefficient is α, but so what? How do you know what kind of coefficient is sitting outside the branch you are living on? All you know is that there was one branch and now there are two. If anything, shouldn’t we declare them to be equally likely (so-called “branch-counting”)? For that matter, in what sense are there probabilities at all? There was nothing stochastic or random about any of this process, the entire evolution was perfectly deterministic. It’s not right to say “Before the measurement, I didn’t know which branch I was going to end up on.” You know precisely that one copy of your future self will appear on each branch. Why in the world should we be talking about probabilities?

Note that the pressing question is not so much “Why is the probability given by the wave function squared, rather than the absolute value of the wave function, or the wave function to the fourth, or whatever?” as it is “Why is there a particular probability rule at all, since the theory is deterministic?” Indeed, once you accept that there should be some specific probability rule, it’s practically guaranteed to be the Born Rule. There is a result called Gleason’s Theorem, which says roughly that the Born Rule is the only consistent probability rule you can conceivably have that depends on the wave function alone. So the real question is not “Why squared?”, it’s “Whence probability?”

Of course, there are promising answers. Perhaps the most well-known is the approach developed by Deutsch and Wallace based on decision theory. There, the approach to probability is essentially operational: given the setup of Everettian quantum mechanics, how should a rational person behave, in terms of making bets and predicting experimental outcomes, etc.? They show that there is one unique answer, which is given by the Born Rule. In other words, the question “Whence probability?” is sidestepped by arguing that reasonable people in an Everettian universe will act as if there are probabilities that obey the Born Rule. Which may be good enough.

But it might not convince everyone, so there are alternatives. One of my favorites is Wojciech Zurek’s approach based on “envariance.” Rather than using words like “decision theory” and “rationality” that make physicists nervous, Zurek claims that the underlying symmetries of quantum mechanics pick out the Born Rule uniquely. It’s very pretty, and I encourage anyone who knows a little QM to have a look at Zurek’s paper. But it is subject to the criticism that it doesn’t really teach us anything that we didn’t already know from Gleason’s theorem. That is, Zurek gives us more reason to think that the Born Rule is uniquely preferred by quantum mechanics, but it doesn’t really help with the deeper question of why we should think of EQM as a theory of probabilities at all.

Here is where Chip and I try to contribute something. We use the idea of “self-locating uncertainty,” which has been much discussed in the philosophical literature, and has been applied to quantum mechanics by Lev Vaidman. Self-locating uncertainty occurs when you know that there multiple observers in the universe who find themselves in exactly the same conditions that you are in right now — but you don’t know which one of these observers you are. That can happen in “big universe” cosmology, where it leads to the measure problem. But it automatically happens in EQM, whether you like it or not.

Think of observing the spin of a particle, as in our example above. The steps are:

1. Everything is in its starting state, before the measurement.
2. The apparatus interacts with the system to be observed and becomes entangled. (“Pre-measurement.”)
3. The apparatus becomes entangled with the environment, branching the wave function. (“Decoherence.”)
4. The observer reads off the result of the measurement from the apparatus.

The point is that in between steps 3. and 4., the wave function of the universe has branched into two, but the observer doesn’t yet know which branch they are on. There are two copies of the observer that are in identical states, even though they’re part of different “worlds.” That’s the moment of self-locating uncertainty. Here it is in equations, although I don’t think it’s much help.

You might say “What if I am the apparatus myself?” That is, what if I observe the outcome directly, without any intermediating macroscopic equipment? Nice try, but no dice. That’s because decoherence happens incredibly quickly. Even if you take the extreme case where you look at the spin directly with your eyeball, the time it takes the state of your eye to decohere is about 10^-21 seconds, whereas the timescales associated with the signal reaching your brain are measured in tens of milliseconds. Self-locating uncertainty is inevitable in Everettian quantum mechanics. In that sense, probability is inevitable, even though the theory is deterministic — in the phase of uncertainty, we need to assign probabilities to finding ourselves on different branches.

So what do we do about it? As I mentioned, there’s been a lot of work on how to deal with self-locating uncertainty, i.e. how to apportion credences (degrees of belief) to different possible locations for yourself in a big universe. One influential paper is by Adam Elga, and comes with the charming title of “Defeating Dr. Evil With Self-Locating Belief.” (Philosophers have more fun with their titles than physicists do.) Elga argues for a principle of Indifference: if there are truly multiple copies of you in the world, you should assume equal likelihood for being any one of them. Crucially, Elga doesn’t simply assert Indifference; he actually derives it, under a simple set of assumptions that would seem to be the kind of minimal principles of reasoning any rational person should be ready to use.

But there is a problem! Naïvely, applying Indifference to quantum mechanics just leads to branch-counting — if you assign equal probability to every possible appearance of equivalent observers, and there are two branches, each branch should get equal probability. But that’s a disaster; it says we should simply ignore the amplitudes entirely, rather than using the Born Rule. This bit of tension has led to some worry among philosophers who worry about such things.

Resolving this tension is perhaps the most useful thing Chip and I do in our paper. Rather than naïvely applying Indifference to quantum mechanics, we go back to the “simple assumptions” and try to derive it from scratch. We were able to pinpoint one hidden assumption that seems quite innocent, but actually does all the heavy lifting when it comes to quantum mechanics. We call it the “Epistemic Separability Principle,” or ESP for short. Here is the informal version (see paper for pedantic careful formulations):

ESP: The credence one should assign to being any one of several observers having identical experiences is independent of features of the environment that aren’t affecting the observers.

That is, the probabilities you assign to things happening in your lab, whatever they may be, should be exactly the same if we tweak the universe just a bit by moving around some rocks on a planet orbiting a star in the Andromeda galaxy. ESP simply asserts that our knowledge is separable: how we talk about what happens here is independent of what is happening far away. (Our system here can still be entangled with some system far away; under unitary evolution, changing that far-away system doesn’t change the entanglement.)

The ESP is quite a mild assumption, and to me it seems like a necessary part of being able to think of the universe as consisting of separate pieces. If you can’t assign credences locally without knowing about the state of the whole universe, there’s no real sense in which the rest of the world is really separate from you. It is certainly implicitly used by Elga (he assumes that credences are unchanged by some hidden person tossing a coin).

With this assumption in hand, we are able to demonstrate that Indifference does not apply to branching quantum worlds in a straightforward way. Indeed, we show that you should assign equal credences to two different branches if and only if the amplitudes for each branch are precisely equal! That’s because the proof of Indifference relies on shifting around different parts of the state of the universe and demanding that the answers to local questions not be altered; it turns out that this only works in quantum mechanics if the amplitudes are equal, which is certainly consistent with the Born Rule.

See the papers for the actual argument — it’s straightforward but a little tedious. The basic idea is that you set up a situation in which more than one quantum object is measured at the same time, and you ask what happens when you consider different objects to be “the system you will look at” versus “part of the environment.” If you want there to be a consistent way of assigning credences in all cases, you are led inevitably to equal probabilities when (and only when) the amplitudes are equal.

What if the amplitudes for the two branches are not equal? Here we can borrow some math from Zurek. (Indeed, our argument can be thought of as a love child of Vaidman and Zurek, with Elga as midwife.) In his envariance paper, Zurek shows how to start with a case of unequal amplitudes and reduce it to the case of many more branches with equal amplitudes. The number of these pseudo-branches you need is proportional to — wait for it — the square of the amplitude. Thus, you get out the full Born Rule, simply by demanding that we assign credences in situations of self-locating uncertainty in a way that is consistent with ESP.

We like this derivation in part because it treats probabilities as epistemic (statements about our knowledge of the world), not merely operational. Quantum probabilities are really credences — statements about the best degree of belief we can assign in conditions of uncertainty — rather than statements about truly stochastic dynamics or frequencies in the limit of an infinite number of outcomes. But these degrees of belief aren’t completely subjective in the conventional sense, either; there is a uniquely rational choice for how to assign them.

Working on this project has increased my own personal credence in the correctness of the Everett approach to quantum mechanics from “pretty high” to “extremely high indeed.” There are still puzzles to be worked out, no doubt, especially around the issues of exactly how and when branching happens, and how branching structures are best defined. (I’m off to a workshop next month to think about precisely these questions.) But these seem like relatively tractable technical challenges to me, rather than looming deal-breakers. EQM is an incredibly simple theory that (I can now argue in good faith) makes sense and fits the data. Now it’s just a matter of convincing the rest of the world!

read more

So, the Templeton Foundation invited me to write a 1500-word essay on the above question.  It’s like a blog post, except they pay me to do it!  My essay is now live, here.  I hope you enjoy my attempt at techno-futurist prose.  You can comment on the essay either here or over at Templeton’s site.  Thanks very much to Ansley Roan for commissioning the piece.

The following concept seems to have been reinvented a bunch of times by a bunch of people, and every time they give it a different name.

Definition: Let $C$ be a category with pullbacks and a class of weak equivalences. A morphism $f:A\to B$ is a [insert name here] if the pullback functor $f^\ast:C/B \to C/A$ preserves weak equivalences.

In a right proper model category, every fibration is one of these. But even in that case, there are usually more of these than just the fibrations. There is of course also a dual notion in which pullbacks are replaced by pushouts, and every cofibration in a left proper model category is one of those.

What should we call them?

The names that I’m aware of that have so far been given to these things are:

1. sharp map, by Charles Rezk. This is a dualization of the terminology flat map used for the dual notion by Mike Hopkins (I don’t know a reference, does anyone?). I presume that Hopkins’ motivation was that a ring homomorphism is flat if tensoring with it (which is the pushout in the category of commutative rings) is exact, hence preserves weak equivalences of chain complexes.

   However, “flat” has the problem of being a rather overused word. For instance, we may want to talk about these objects in the canonical model structure on $Cat$ (where in fact it turns out that every such functor is a cofibration), but flat functor has a very different meaning. David White has pointed out that “flat” would also make sense to use for the monoid axiom in monoidal model categories.

2. right proper, by Andrei Radulescu-Banu. This is presumably motivated by the above-mentioned fact that fibrations in right proper model categories are such. Unfortunately, proper map also has another meaning.

3. $h$-fibration, by Berger and Batanin. This is presumably motivated by the fact that “$h$-cofibration” has been used by May and Sigurdsson for an intrinsic notion of cofibration in topologically enriched categories, that specializes in compactly generated spaces to closed Hurewicz cofibrations, and pushouts along the latter preserve weak homotopy equivalences. However, it makes more sense to me to keep “$h$-cofibration” with May and Sigurdsson’s original meaning.

4. Grothendieck $W$-fibration (where $W$ is the class of weak equivalences on $C$), by Ara and Maltsiniotis. Apparently this comes from unpublished work of Grothendieck. Here I guess the motivation is that these maps are “like fibrations” and are determined by the class $W$ of weak equivalences.

Does anyone know of other references for this notion, perhaps with other names? And any opinions on what the best name is? I’m currently inclined towards “$W$-fibration” mainly because it doesn’t clash with anything else, but I could be convinced otherwise.

Physicists are not known for finesse. “Even if it cost us our funding,” I’ve heard a physicist declare, “we’d tell you what we think.” Little wonder I irked the porter who directed me toward central Cambridge.

The University of Cambridge consists of colleges as the US consists of states. Each college has a porter’s lodge, where visitors check in and students beg for help after locking their keys in their rooms. And where physicists ask for directions.

Last March, I ducked inside a porter’s lodge that bustled with deliveries. The woman behind the high wooden desk volunteered to help me, but I asked too many questions. By my fifth, her pointing at a map had devolved to jabbing.

Read the subtext, I told myself. Leave.

Or so I would have told myself, if not for that afternoon.

That afternoon, I’d visited Cambridge’s CMS, which merits every letter in “Centre for Mathematical Sciences.” Home to Isaac Newton’s intellectual offspring, the CMS consists of eight soaring, glass-walled, blue-topped pavilions. Their majesty walloped me as I turned off the road toward the gatehouse. So did the congratulatory letter from Queen Elizabeth II that decorated the route to the restroom.

I visited Nilanjana Datta, an affiliated lecturer of Cambridge’s Faculty of Mathematics, and her student, Felix Leditzky. Nilanjana and Felix specialize in entropies and one-shot information theory. Entropies quantify uncertainties and efficiencies. Imagine compressing many copies of a message into the smallest possible number of bits (units of memory). How few bits can you use per copy? That number, we call the optimal compression rate. It shrinks as the number of copies compressed grows. As the number of copies approaches infinity, that compression rate drops toward a number called the message’s Shannon entropy. If the message is quantum, the compression rate approaches the von Neumann entropy.

Good luck squeezing infinitely many copies of a message onto a hard drive. How efficiently can we compress fewer copies? According to one-shot information theory, the answer involves entropies other than Shannon’s and von Neumann’s. In addition to describing data compression, entropies describe the charging of batteries, the concentration of entanglement, the encrypting of messages, and other information-processing tasks.

Speaking of compressing messages: Suppose one-shot information theory posted status updates on Facebook. Suppose that that panel on your Facebook page’s right-hand side showed news weightier than celebrity marriages. The news feed might read, “TRENDING: One-shot information theory: Second-order asymptotics.”

Second-order asymptotics, I learned at the CMS, concerns how the optimal compression rate decays as the number of copies compressed grows. Imagine compressing a billion copies of a quantum message ρ. The number of bits needed about equals a billion times the von Neumann entropy H_vN(ρ). Since a billion is less than infinity, 1,000,000,000 H_vN(ρ) bits won’t suffice. Can we estimate the compression rate more precisely?

The question reminds me of gas stations’ hidden pennies. The last time I passed a station’s billboard, some number like $3.65 caught my eye. Each gallon cost about $3.65, just as each copy of ρ costs about H_vN(ρ) bits. But a 9/10, writ small, followed the $3.65. If I’d budgeted $3.65 per gallon, I couldn’t have filled my tank. If you budget H_vN(ρ) bits per copy of ρ, you can’t compress all your copies.

Suppose some station’s owner hatches a plan to promote business. If you buy one gallon, you pay $3.654. The more you purchase, the more the final digit drops from four. By cataloguing receipts, you calculate how a tank’s cost varies with the number of gallons, n. The cost equals $3.65 × n to a first approximation. To a second approximation, the cost might equal $3.65 × n + a√n, wherein a represents some number of cents. Compute a, and you’ll have computed the gas’s second-order asymptotics.

Nilanjana and Felix computed a’s associated with data compression and other quantum tasks. Second-order asymptotics met information theory when Strassen combined them in nonquantum problems. These problems developed under attention from Hayashi, Han, Polyanski, Poor, Verdu, and others. Tomamichel and Hayashi, as well as Li, introduced quantumness.

In the total-cost expression, $3.65 × n depends on n directly, or “linearly.” The second term depends on √n. As the number of gallons grows, so does √n, but √n grows more slowly than n. The second term is called “sublinear.”

Which is the word that rose to mind in the porter’s lodge. I told myself, Read the sublinear text.

Little wonder I irked the porter. At least—thanks to quantum information, my mistake, and facial expressions’ contagiousness—she smiled.

With thanks to Nilanjana Datta and Felix Leditzky for explanations and references; to Nilanjana, Felix, and Cambridge’s Centre for Mathematical Sciences for their hospitality; and to porters everywhere for providing directions.

During my first semester I coincidentally found out that the guy who often sat next to me, one of the better students, believed the Earth was only 15,000 years old. Once on the topic, he produced stacks of colorful leaflets which featured lots of names, decorated by academic titles, claiming that scientific evidence supports the scripture. I laughed at him, initially thinking he was joking, but he turned out to be dead serious and I was clearly going to roast in hell until future eternity.

“Any mathematical model, no matter how complicated, consists of a set of assumptions, from whichj are deduced a set of conclusions. The technical machinery specific to each flavor of model is concerned with deducing the latter from the former. This deduction comes with a guarantee, which, unlike other guarantees, can never be invalidated. Provided the model is correct, if you accept its assumptions, you must as a matter of logic also accept its conclusions.”

The title of this post is a nod to Terry Tao’s four mini-polymath discussions, in which IMO questions were solved collaboratively online. As the beginning of what I hope will be a long exercise in gathering data about how humans solve these kinds of problems, I decided to have a go at one of this year’s IMO problems, with the idea of writing down my thoughts as I went along. Because I was doing that (and doing it directly into a LaTeX file rather than using paper and pen), I took quite a long time to solve the problem: it was the first question, and therefore intended to be one of the easier ones, so in a competition one would hope to solve it quickly and move on to the more challenging questions 2 and 3 (particularly 3). You get an average of an hour and a half per question, and I think I took at least that, though I didn’t actually time myself.

What I wrote gives some kind of illustration of the twists and turns, many of them fruitless, that people typically take when solving a problem. If I were to draw a moral from it, it would be this: when trying to solve a problem, it is a mistake to expect to take a direct route to the solution. Instead, one formulates subquestions and gradually builds up a useful bank of observations until the direct route becomes clear. Given that we’ve just had the football world cup, I’ll draw an analogy that I find not too bad (though not perfect either): a team plays better if it patiently builds up to an attack on goal than if it hoofs the ball up the pitch or takes shots from a distance. Germany gave an extraordinary illustration of this in their 7-1 defeat of Brazil.

I imagine that the rest of this post will be much more interesting if you yourself solve the problem before reading what I did. I in turn would be interested in hearing about other people’s experiences with the problem: were they similar to mine, or quite different? I would very much like to get a feel for how varied people’s experiences are. If you’re a competitor who solved the problem, feel free to join the discussion!

If I find myself with some spare time, I might have a go at doing the same with some of the other questions.

What follows is exactly what I wrote (or rather typed), with no editing at all, apart from changing the LaTeX so that it compiles in WordPress and adding two comments that are clearly marked in red.

Problem Let be an infinite sequence of positive integers. Prove that there exists a unique integer such that

First thought.

Slight bafflement.

Observation.

The expression in the middle is not an average. If we were to replace it by an average we would have the second inequality automatically.

Proof discovery technique.

Try looking at simple cases. Here we could consider what happens when , for example. Then the inequality says

Here we automatically have the first inequality, but there is no reason for the second inequality to be true.

.

Putting those observations together, we see that the first inequality is true when , and the second inequality is “close to being true” as gets large, since it is true if we replace by in the denominator.

If the inequality holds for a unique , then a plausible guess is that the first inequality fails at some and if is minimal such that it fails, then both inequalities are true for . I shall investigate that in due course, but I have another idea.

Question.

It is clear that WLOG . Can we now choose in such a way that we always get equality for the second inequality? We can certainly solve the equations, so the question is whether the resulting sequence will be increasing.

We get , so I’d better set and then continue constructing a sequence.

So , , , and so on. Thus all the with are equal, which they are not supposed to be. This feels significant.

.

Out of interest, what happens to the inequalities when we (illegally) take the above sequence? We get , so we get equality on both sides except when when we get .

Proof discovery technique.

Try to disprove the result.

Proof discovery subtechnique.

Try to find the simplest counterexample you can.

An obvious thing to do is to try to make the inequality true when and when . So let’s go. Without loss of generality , . We now need .

For we need . That can be rearranged to , exactly contradicting what we had before.

.

That doesn’t solve the problem but it looks interesting. In particular, it suggests rearranging the first inequality in the general case, to

That’s quite nice because the right hand side is a genuine average this time.

Actually, if getting an average is what we care about, we could also rearrange the first inequality by simply multiplying through by , which gives

.

I think it is time to revisit that guess, in order to try to prove at least that there exists a solution. So we know that the first inequality holds when , since all it says then is that . Can it always hold? If so, then again WLOG and , and after that we get , , etc.

Let’s write and for . Then we have , , , etc. We also require .

Let’s set . Now the first condition becomes but . Is that possible?

Is it possible with equality? WLOG . Then we have , , , etc.

.

I’m starting to wonder whether the integer must be something like 1 or 2. Let’s think about it. We know that . If then we have our . Suppose instead that . Then , so . Now if then we are again done, so suppose that .

But since , we can simply insert in between the two. Why can’t we continue doing that kind of thing? Let me try.

If , then , so we can insert in between the two.

.

I seem to have disproved the result, so I’d better see where I’m going wrong. I’ll try to construct a sequence explicitly. I’ll take , . I need , so I’ll take . Now I need , so I’ll take . Now I need , so I’ll take .

I don’t seem to be getting stuck, so let me try to prove that I can always continue. Suppose I’ve already chosen . Then the condition I need is that

.

By induction we already have that , from which it follows that and therefore that . We may therefore find between these two numbers, as desired.

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You idiot Gowers, read the question: the have to be positive integers.

Fortunately, the work I’ve done so far is not a complete waste of time. [The half-conscious thought in the back of my mind here, which is clearer in retrospect, was that the successive differences in the example I had just constructed were getting smaller and smaller. So it seemed highly likely that using the same general circle of thoughts I would be able to prove that I couldn't take the to be integers.]

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Here’s a trivial observation: if the second inequality fails, then . So if , then . How long can we keep that going with positive integers? Answer: for ever, since we can take .

.

Never mind about that. I want to go back to an earlier idea. [It isn't obvious what I mean by "earlier idea" here. Actually, I had earlier had the idea of defining the as below, but got distracted by something else and ended up not writing it down. So a small part of the record of my journey to the proof is missing.] It is simply to define and for . Then for if the first inequality holds we have

.

So each new is less than the average of the up to that , and hence less than the average of the before that . But that means that the average of the forms a decreasing sequence. That also means that the are bounded above by , something I could have observed ages ago. So they can’t be an increasing sequence of integers.

.

I’ve now shown that the first inequality must fail at some point. Suppose is the first point at which it fails. Then we have

and

The second inequality tells us that exceeds the average of , which implies that it exceeds the average of . That gives us the inequality

.

So now I’ve proved that there exists an integer such that the inequalities both hold. It remains to prove uniqueness. This formulation with the ought to help. We’ve picked the first point at which is at least as big as the average of . Does that imply that is at least as big as the average of ? Yes, because is at least as big as that average, and is bigger than . In other words, we can prove easily that if the first inequality fails for then it fails for , and hence by induction for all .

UPDATE: I HAVE NOW GONE BACK TO MODERATING COMMENTS ONLY IF THEY ARE FROM PEOPLE WHO HAVE NOT HAD A COMMENT ACCEPTED IN THE PAST. SO IF YOU HAVE A SUGGESTION TO MAKE FOR AN ECM2016 SPEAKER, PLEASE EMAIL A MEMBER OF THE COMMITTEE DIRECTLY RATHER THAN COMMENTING HERE.

Just before I start this post, let me say that I do still intend to write a couple of follow-up posts to my previous one about journal prices. But I’ve been busy with a number of other things, so it may still take a little while.

This post is about the next European Congress of Mathematics, which takes place in Berlin in just over two years’ time. I have agreed to chair the scientific committee, which is responsible for choosing approximately 10 plenary speakers and approximately 30 invited lecturers, the latter to speak in four or five parallel sessions.

The ECM is less secretive than the ICM when it comes to drawing up its scientific programme. In particular, the names of the committee members were made public some time ago, and you can read them here.

I am all in favour of as much openness as possible, so I am very pleased that this is the way that the European Mathematical Society operates. But what is the maximum reasonable level of openness in this case? Clearly, public discussion of the merits of different candidates is completely out of order, but I think anything else goes. In particular, and this is the main point of the post, I would very much welcome suggestions for potential speakers. If you know of a mathematician who is European (and for these purposes Europe includes certain not obviously European countries such as Russia and Israel), has done exciting work (ideally recently), and will not already be speaking about that work at the International Congress of Mathematicians in Seoul, then we would like to hear about it. Our main aim is that the congress should be rewarding for its participants, so we will take some account of people’s ability to give a good talk. This applies in particular to plenary speakers.

I shall moderate all comments on this post. If you suggest a possible speaker, I will not publish your comment, but will note the suggestion. More general comments are also welcome and will be published, assuming that they are the kinds of comments I would normally allow.

[In parentheses, let me say what my comment policy now is. The volume of spam I get on this blog has reached a level where I have decided to implement a feature that WordPress allows, where if you have never had a comment accepted, then your comment will automatically be moderated. I try to check the moderation queue quite frequently. If you have had a comment accepted in the past, then your comments will appear as normal.

I am very reluctant to delete comments, but I do delete obvious spam, and I also delete any comment that tries to use this blog as a form of self-promotion (such as using a comment to draw attention to the author's proof of the Riemann hypothesis, or to the author's fascinating blog, etc. etc.). I sometimes delete pingbacks as well -- it depends whether I think readers of my blog might conceivably be interested in the post from which the pingback originates.]

Going back to the European Congress, if you would prefer to make your suggestion by getting in contact directly with a committee member, then that is obviously fine too. The list of committee members includes email addresses.

However you make your suggestions, it would be very helpful if you could give not just a name but a brief reason for the suggestion: what the work is that you think should be recognised, and why it is important.

The main other thing I am happy to be open about is the stage that the committee has reached in its deliberations, and the plans for how it will carry out its work. Right now, we are at the stage of trying to put together a longlist of possible speakers. I have asked the other committee members to suggest to me at least six potential speakers each, of whom at least six should be broadly in their area. I hope that will give us enough candidates to make it possible to achieve a reasonable subject balance. We will of course also strive for other forms of balance, such as gender and geographical balance, to the extent that we can. Once we have a decent-sized longlist, we will cut it down to the right sort of size.

We are aiming to produce a near-complete list of speakers by around November. This is rather a long time in advance of the Congress itself, which worried me a bit, but I have permission from the EMS to leave open a few slots so that if somebody does something spectacular after November, then we will have the option of inviting them to speak.