In my online course we’re now into the third chapter of Fong and Spivak’s book Seven Sketches. Now we’re talking about databases!

To some extent this is just an excuse to (finally) introduce categories, functors, natural transformations, adjoint functors and Kan extensions. Great stuff, and databases are a great source of easy examples.

But it’s also true that Spivak helps run a company called Categorical Informatics that actually helps design databases using category theory! And his partner, Ryan Wisnesky, would be happy to talk to people about it. If you’re interested, click the link: he’s attending my course.

To read and join discussions on Chapter 3 go here:

• Chapter 3

You can also do exercises and puzzles, and see other people’s answers to these.

Here are the lectures I’ve given so far:

• Lecture 34 – Chapter 3: Categories
• Lecture 35 – Chapter 3: Categories versus Preorders
• Lecture 36 – Chapter 3: Categories from Graphs
• Lecture 37 – Chapter 3: Presentations of Categories
• Lecture 38 – Chapter 3: Functors
• Lecture 39 – Chapter 3: Databases
• Lecture 40 – Chapter 3: Relations
• Lecture 41 – Chapter 3: Composing Functors
• Lecture 42 – Chapter 3: Transforming Databases
• Lecture 43 – Chapter 3: Natural Transformations
• Lecture 44 – Chapter 3: Categories, Functors and Natural Transformations
• Lecture 45 – Chapter 3: Composing Natural Transformations
• Lecture 46 – Chapter 3: Isomorphisms
• Lecture 47 – Chapter 3: Adjoint Functors
• Lecture 48 – Chapter 3: Adjoint Functors

It was my great pleasure to sit on the PhD defense committee for the successful defense of Sarah Pearson. She wrote a thesis about low-mass galaxies and globular clusters, considering both their interactions with each other, and with the bigger galaxies into which they later fall. She has some nice analyses of the Palomar 5 tidal stream, and what it's morphology might tell us about the Milky Way halo and bar. And also nice results on gas bridges and streams around pairs of dwarf galaxies.

I was most interested in her stellar-stream results, including several things I hadn't thought about before: One is that prograde streams are more affected by the bar and spiral arms in the disk than retrograde streams. Another is that we might be able to find globular-cluster streams around other galaxies nearby. That would be incredible! And since (as she showed) you can learn a lot about a galaxy just from the shape of a stream, we might not need to do much more than detect streams around other galaxies to learn a lot. It was a pleasure to serve on the committee, and it is a beautiful body of work.

Two more students in the Applied Category Theory 2018 school wrote a blog article about a paper they read:

• Tai-Danae Bradley and Brad Theilman, Cognition, convexity and category theory, The n-Category Café, 10 March 2018.

Tai-Danae Bradley is a mathematics PhD student at the CUNY Graduate Center and well-known math blogger. Brad Theilman is a grad student in neuroscience at the Gentner Lab at U. C. San Diego. I was happy to get to know both of them when the school met in Leiden.

In their blog article, they explain this paper:

• Joe Bolt, Bob Coecke, Fabrizio Genovese, Martha Lewis, Dan Marsden, and Robin Piedeleu, Interacting conceptual spaces I.

Fans of convex sets will enjoy this!

Today’s installment in the ongoing project to sketch the $\infty$-elephant: atomic geometric morphisms.

Chapter C3 of Sketches of an Elephant studies various classes of geometric morphisms between toposes. Pretty much all of this chapter has been categorified, except for section C3.5 about atomic geometric morphisms. To briefly summarize the picture:

• Sections C3.1 (open geometric morphisms) and C3.3 (locally connected geometric morphisms) are steps $n=-1$ and $n=0$ on an infinite ladder of locally n-connected geometric morphisms, for $-1 \le n \le \infty$. A geometric morphism between $(n+1,1)$-toposes is locally $n$-connected if its inverse image functor is locally cartesian closed and has a left adjoint. More generally, a geometric morphism between $(m,1)$-toposes is locally $n$-connected, for $n\lt m$, if it is “locally” locally $n$-connected on $n$-truncated maps.

• Sections C3.2 (proper geometric morphisms) and C3.4 (tidy geometric morphisms) are likewise steps $n=-1$ and $n=0$ on an infinite ladder of n-proper geometric morphisms.

• Section C3.6 (local geometric morphisms) is also step $n=0$ on an infinite ladder: a geometric morphism between $(n+1,1)$-toposes is $n$-local if its direct image functor has an indexed right adjoint. Cohesive toposes, which have attracted a lot of attention around here, are both locally $\infty$-connected and $\infty$-local. (Curiously, the $n=-1$ case of locality doesn’t seem to be mentioned in the 1-Elephant; has anyone seen it before?)

So what about C3.5? An atomic geometric morphism between elementary 1-toposes is usually defined as one whose inverse image functor is logical. This is an intriguing prospect to categorify, because it appears to mix the “elementary” and “Grothendieck” aspects of topos theory: a geometric morphisms are arguably the natural morphisms between Grothendieck toposes, while logical functors are more natural for the elementary sort (where “natural” means “preserves all the structure in the definition”). So now that we’re starting to see some progress on elementary higher toposes (my post last year has now been followed by a preprint by Rasekh), we might hope be able to make some progress on it.

Unfortunately, the definitions of elementary $(\infty,1)$-topos currently under consideration have a problem when it comes to defining logical functors. A logical functor between 1-toposes can be defined as a cartesian closed functor that preserves the subobject classifier, i.e. $F(\Omega) \cong \Omega$. The higher analogue of the subobject classifier is an object classifier — but note the switch from definite to indefinite article! For Russellian size reasons, we can’t expect to have one object classifer that classifies all objects, only a tower of “universes” each of which classifies some subcollection of “small” objects.

What does it mean for a functor to “preserve” the tower of object classifiers? If an $(\infty,1)$-topos came equipped with a specified tower of object classifiers (indexed by $\mathbb{N}$, say, or maybe by the ordinal numbers), then we could ask a logical functor to preserve them one by one. This would probably be the relevant kind of “logical functor” when discussing categorical semantics of homotopy type theory: since type theory does have a specified tower of universe types $U_0 : U_1 : U_2 : \cdots$, the initiality conjecture for HoTT should probably say that the syntactic category is an elementary $(\infty,1)$-topos that’s initial among logical functors of this sort.

However, Grothendieck $(\infty,1)$-topoi don’t really come equipped with such a tower. And even if they did, preserving it level by level doesn’t seem like the right sort of “logical functor” to use in defining atomic geometric morphisms; there’s no reason to expect such a functor to “preserve size” exactly.

What do we want of a logical functor? Well, glancing through some of the theorems about logical functors in the 1-Elephant, one result that stands out to me is the following: if $F:\mathbf{S}\to \mathbf{E}$ is a logical functor with a left adjoint $L$, then $L$ induces isomorphisms of subobject lattices $Sub_{\mathbf{E}}(A) \cong Sub_{\mathbf{S}}(L A)$. This is easy to prove using the adjointness $L\dashv F$ and the fact that $F$ preserves the subobject classifier:

$$Sub_{\mathbf{E}}(A) \cong \mathbf{E}(A,\Omega_{\mathbf{E}}) \cong \mathbf{E}(A,F \Omega_{\mathbf{S}}) \cong \mathbf{E}(L A,\Omega_{\mathbf{S}})\cong Sub_{\mathbf{S}}(L A).$$

What would be the analogue for $(\infty,1)$-topoi? Well, if we imagine hypothetically that we had a classifier $U$ for all objects, then the same argument would show that $L$ induces an equivalence between entire slice categories $\mathbf{E}/A \simeq \mathbf{S}/L A$. (Actually, I’m glossing over something here: the direct arguments with $\Omega$ and $U$ show only an equivalence between sets of subobjects and cores of slice categories. The rest comes from the fact that $F$ preserves local cartesian closure as well as the (sub)object classifier, so that we can enhance $\Omega$ to an internal poset and $U$ to an internal full subcategory and both of these are preserved by $F$ as well.)

In fact, the converse is true too: reversing the above argument shows that $F$ preserves $\Omega$ if and only if $L$ induces isomorphisms of subobject lattices, and similarly $F$ preserves $U$ if and only if $L$ induces equivalences of slice categories. The latter condition, however, is something that can be said without reference to the nonexistent $U$. So if we have a functor $F:\mathbf{E}\to \mathbf{S}$ between $(\infty,1)$-toposes that has a left adjoint $L$, then I think it’s reasonable to define $F$ to be logical if it is locally cartesian closed and $L$ induces equivalences $\mathbf{E}/A \simeq \mathbf{S}/L A$.

Furthermore, a logical functor between 1-toposes has a left adjoint if and only if it has a right adjoint. (This follows from the monadicity of the powerset functor $P : \mathbf{E}^{op} \to \mathbf{E}$ for 1-toposes, which we don’t have an analogue of (yet) in the $\infty$-case.) In particular, if the inverse image functor in a geometric morphism is logical, then it automatically has a left adjoint, so that the above characterization of logical-ness applies. And since a logical functor is locally cartesian closed, this geometric morphism is automatically locally connected as well. This suggests the following:

Definition: A geometric morphism $p:\mathbf{E}\to \mathbf{S}$ between $(\infty,1)$-topoi is $\infty$-atomic if

1. It is locally $\infty$-connected, i.e. $p^\ast$ is locally cartesian closed and has a left adjoint $p_!$, and
2. $p_!$ induces equivalences of slice categories $\mathbf{E}/A \simeq \mathbf{S}/p_! A$ for all $A\in \mathbf{E}$.

This seems natural to me, but it’s very strong! In particular, taking $A=1$ we get an equivalence $\mathbf{E}\simeq \mathbf{E}/1 \simeq \mathbf{S}/p_! 1$, so that $\mathbf{E}$ is equivalent to a slice category of $\mathbf{S}$. In other words, $\infty$-atomic geometric morphisms coincide with local homeomorphisms!

Is that really reasonable? Actually, I think it is. Consider the simplest example of an atomic geometric morphism of 1-topoi that is not a local homeomorphism: $[G,Set] \to Set$ for a group $G$. The corresponding geometric morphism of $(\infty,1)$-topoi $[G,\infty Gpd] \to \infty Gpd$ is a local homeomorphism! Specifically, we have $[G,\infty Gpd] \simeq \infty Gpd / B G$. So in a sense, the difference between atomic and locally-homeomorphic vanishes in the limit $n\to \infty$.

To be sure, there are other atomic geometric morphisms of 1-topoi that do not extend to local homeomorphisms of $(\infty,1)$-topoi, such as $Cont(G) \to Set$ for a topological group $G$. But it seems reasonable to me to regard these as “1-atomic morphisms that are not $\infty$-atomic” — a thing which we should certainly expect to exist, just as there are locally 0-connected morphisms that are not locally $\infty$-connected, and 0-proper morphisms that are not $\infty$-proper.

We can also “see” how the difference gets “pushed off to $\infty$” to vanish, in terms of sites of definition. In C3.5.8 of the 1-Elephant it is shown that every atomic Grothendieck topos has a site of definition in which (among other properties) all morphisms are effective epimorphisms. If we trace through the proof, we see that this effective-epi condition comes about as the “dual” class to the monomorphisms that the left adjoint of a logical functor induces an equivalence on. Since an $(n+1,1)$-topos has classifiers for $n$-truncated objects, we would expect an atomic one to have a site of definition in which all morphisms belong to the dual class of the $n$-truncated morphisms, i.e. the $n$-connected morphisms. So as $n\to \infty$, we get stronger and stronger conditions on the morphisms in our site, until in the limit we have a classifier for all morphisms, and the morphisms in our site are all required to be equivalences. In other words, the site is itself an $\infty$-groupoid, and thus the topos of (pre)sheaves on it is a slice of $\infty Gpd$.

However, it could be that I’m missing something and this is not the best categorification of atomic geometric morphisms. Any thoughts from readers?

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Today was my last day and fifth lecture at TASI. This lecture was crowd-sourced in content! I spoke about Fisher information, linear algebra tips and tricks, and decision theory and model selection. On the latter I strongly advocated engineering methods like cross-validation!

Over lunch I had a great set of conversations with Zach Berta-Thompson about precise measurement for exoplanets, and also hack weeks like the #GaiaSprint. We went deep into the limits on ultra-precise photometry from the ground. We wondered at the point that the best imaging systems get the best precision (on photometry, of point sources) by de-focusing. That has always struck me as somehow absurd, though it's true that you don't have to understand your system nearly so well when you are out of focus (for many reasons).

We had one very good idea: Instead of de-focusing, put in an objective prism! You could get many of the benefits of de-focus but also get far more information about the atmosphere and speckle and scintillation and so on. In principle, you might beat the best measurements made to date. And it is a cheap experiment to perform.

For the last two weeks I’ve been at Les Houches, a village in the French Alps, for the Summer School on Structures in Local Quantum Field Theory.

To assist, we have a view of some very large structures in local quantum field theory

Les Houches has a long history of prestigious summer schools in theoretical physics, going back to the activity of Cécile DeWitt-Morette after the second world war. This was more of a workshop than a “school”, though: each speaker gave one talk, and they weren’t really geared for students.

The workshop was organized by Dirk Kreimer and Spencer Bloch, who both have a long track record of work on scattering amplitudes with a high level of mathematical sophistication. The group they invited was an even mix of physicists interested in mathematics and mathematicians interested in physics. The result was a series of talks that managed to both be thoroughly technical and ask extremely deep questions, including “is quantum electrodynamics really an asymptotic series?”, “are there simple graph invariants that uniquely identify Feynman integrals?”, and several talks about something called the Spine of Outer Space, which still sounds a bit like a bad sci-fi novel. Along the way there were several talks showcasing the growing understanding of elliptic polylogarithms, giving me an opportunity to quiz Johannes Broedel about his recent work.

While some of the more mathematical talks went over my head, they spurred a lot of productive dialogues between physicists and mathematicians. Several talks had last-minute slides, added as a result of collaborations that happened right there at the workshop. There was even an entire extra talk, by David Broadhurst, based on work he did just a few days before.

We also had a talk by Jaclyn Bell, a former student of one of the participants who was on a BBC reality show about training to be an astronaut. She’s heavily involved in outreach now, and honestly I’m a little envious of how good she is at it.

A lot happened at Applied Category Theory 2018. Even as it’s still winding down, we’re already starting to plan a followup in 2019, to be held in Oxford. Here are some notes Joshua Tan sent out:

1. Discussions: Minutes from the discussions can be found here.

2. Photos: Ross Duncan took some very glamorous photos of the conference, which you can find here.

3. Videos: Videos of talks are online here: courtesy of Jelle Herold and Fabrizio Genovese.

4. Next year’s workshop: Bob Coecke will be organizing ACT 2019, to be hosted in Oxford sometime spring/summer. There will be a call for papers.

5. Next year’s school: Daniel Cicala is helping organize next year’s ACT school. Please contact him at if you would like to get involved.

6. Look forward to the official call for submissions, coming soon, for the first issue of Compositionality!

The minutes mentioned above contain interesting thoughts on these topics:

• Day 1: Causality
• Day 2: AI & Cognition
• Day 3: Dynamical Systems
• Day 4: Systems Biology
• Day 5: Closing

As seen in this comment on Polymath and explicated further in Fernando de Oliveira Filho’s thesis, section 4.4.

I actually spent much of today thinking about this so let me try to explain it in a down-to-earth way, because it involved me thinking about Bessel functions for the first time ever, surely a life event worthy of recording.

So here’s what we’re going to do.  As I mentioned last week, you can express this problem as follows:  suppose you have a map h: R^2 -> V, for some normed vector space V, which is a unit-distance embedding; that is, if |x-x’|_{R^2} = 1, then |h(x)-h(x’)|_V = 1.  (We don’t ask that h is an isometry, only that it preserves the distance-1 set.)

Then let t be the radius of the smallest hypersphere in V containing h(R^2).

Then any graph embeddable in R^2 with all edges of length 1 is sent to a unit-distance graph in V contained in the hyperplane of radius t; this turns out to be equivalent to saying the Lovasz number of G (ok, really I mean the Lovasz number of the complement of G) is at most 1/(1-2t).  So we want to show that t is bounded below 1, is the point.  Or rather:  we can find a V and a map from R^2 to V to make this the case.

So here’s one!  Let V be the space of L^2 functions on R^2 with the usual inner product.  Choose a square-integrable function F on R^2 — in fact let’s normalize to make F^2 integrate to 1 — and for each a in R^2 we let h(a) be the function F(x-a).

We want the distance between F(x-a) and F(x-b) to be the same for every pair of points at distance 1 from each other; the easiest way to arrange that is to insist that F(x) be a radially symmetric function F(x) = f(|x|); then it’s easy to see that the distance between F(x-a) and F(x-b) in V is a function G(a-b) which depends only on |a-b|.  We write

so that the squared distance between F(x) and F(x-r) is

.

In particular, if two points in R^2 are at distance 1, the squared distance between their images in V is 2(1-g(1)).  Note also that g(0) is the square integral of F, which is 1.

What kind of hypersphere encloses all the points F(x-a) in V?  We can just go ahead and take the “center” of our hypersphere to be 0; since |F| = 1, every point in h(R^2) lies in (indeed, lies on) the sphere of radius 1 around the origin.

Hey but remember:  we want to study a unit-distance embedding of R^2 in V.  Right now, h sends unit distances to the distance 2(1-g(1)), whatever that is.  We can fix that by scaling h by the square root of that number.  So now h sends unit distances to unit distances, and its image is enclosed in a hypersphere of radius

2(1-g(1))^{-1}

The more negative g(1) is, the smaller this sphere is, which means the more we can “fold” R^2 into a small space.  Remember, the relationship between hypersphere number and Lovasz theta is

and plugging in the above bound for the hypersphere number, we find that the Lovasz theta number of R^2, and thus the Lovasz theta number of any unit-distance graph in R^2, is at most

1-1/g(1).

So the only question is — what is g(1)?

Well, that depends on what g is.

Which depends on what F is.

Which depends on what f is.

And of course we get to choose what f is, in order to make g(1) as negative as possible.

How do we do this?  Well, here’s the trick.  The function G is not arbitrary; if it were, we could make g(1) whatever we wanted.  It’s not hard to see that G is what’s called a positive definite function on R^2.  And moreover, if G is positive definite, there exists some f giving rise to it.  (Roughly speaking, this is the fact that a positive definite symmetric matrix has a square root.)  So we ask:  if G is a positive definite (radially symmetric) function on R^2, and g(0) = 1, how small can g(1) be?

And now there’s an old theorem of (Wisconsin’s own!) Isaac Schoenberg which helpfully classifies the positive definite functions on R^2; they are precisely the functions G(x) = g(|x|) where g is a mixture of scalings of the Bessel function $J_0$:

for some everywhere nonnegative A(u).  (Actually it’s more correct to say that A is a distribution and we are integrating J_0(ur) against a non-decreasing measure.)

So g(1) can be no smaller than the minimum value of J_0 on [0,infty], and in fact can be exactly that small if you let A become narrowly supported around the minimum argument.  This is basically just taking g to be a rescaled version of J_0 which achieves its minimum at 1.  That minimum value is about -0.4, and so the Lovasz theta for any unit-distance subgraph on the plane is bounded above by a number that’s about 1 + 1/0.4 = 3.5.

To sum up:  I give you a set of points in the plane, I connect every pair that’s at distance 1, and I ask how you can embed that graph in a small hypersphere keeping all the distances 1.  And you say:  “Oh, I know what to do, just assign to each point a the radially symmetrized Bessel function J_0(|x-a|) on R^2, the embedding of your graph in the finite-dimensional space of functions spanned by those Bessel translates will do the trick!”

That is cool!

Remark: Oliveira’s thesis does this for Euclidean space of every dimension (it gets more complicated.)  And I think (using analysis I haven’t really tried to understand) he doesn’t just give an upper bound for the Lovasz number of the plane as I do in this post, he really computes that number on the nose.

Update:  DeCorte, Oliveira, and Vallentin just posted a relevant paper on the arXiv this morning!

As part of the Applied Category Theory 2018 school, Maru Sarazola wrote a blog article on open dynamical systems and their steady states. Check it out:

• Maru Sarazola, Dynamical systems and their steady states, 2 April 2018.

She compares two papers:

• David Spivak, The steady states of coupled dynamical systems compose according to matrix arithmetic.

• John Baez and Blake Pollard, A compositional framework for reaction networks, Reviews in Mathematical Physics 29 (2017), 1750028.
(Blog article here.)

It’s great, because I’d never really gotten around to understanding the precise relationship between these two approaches. I wish I knew the answers to the questions she raises at the end!

Two more students in the Applied Category Theory 2018 school wrote a blog article about something they read:

• Eliana Lorch and Joshua Tan, The behavioral approach to systems theory, 15 June 2018.

Eliana Lorch is a mathematician based in San Francisco. Joshua Tan is a grad student in computer science at the University of Oxford and one of the organizers of Applied Category Theory 2018.

They wrote a great summary of this paper, which has been an inspiration to me and many others:

• Jan Willems, The behavioral approach to open and interconnected systems, IEEE Control Systems 27 (2007), 46–99.

They also list many papers influenced by it, and raise a couple of interesting problems with Willems’ idea, which can probably be handled by generalizing it.

guest post by Eliana Lorch and Joshua Tan

As part of the Applied Category Theory seminar, we discussed an article commonly cited as an inspiration by many papers¹ taking a categorical approach to systems theory, The Behavioral Approach to Open and Interconnected Systems. In this sprawling monograph for the IEEE Control Systems Magazine, legendary control theorist Jan Willems poses and answers foundational questions like how to define the very concept of mathematical model, gives fully-worked examples of his approach to modeling from physical first principles, provides various arguments in favor of his framework versus others, and finally proves several theorems about the special case of linear time-invariant differential systems.

In this post, we’ll summarize the behavioral approach, Willems’ core definitions, and his “systematic procedure” for creating behavioral models; we’ll also examine the limitations of Willems’ framework, and conclude with a partial reference list of Willems-inspired categorical approaches to understanding systems.

The behavioral approach

Here’s the view from 10,000 feet of the behavioral approach in contrast with the traditional signal-flow approach:

Willems’ approach breaks down into: (1) considering a dynamical system as a ‘behavior,’ and (2) defining interconnection as variable sharing.

Dynamical system as behavior

Willems goes so far as to claim: “It is remarkable that the idea of viewing a system in terms of inputs and outputs, in terms of cause and effect, kept its central place in systems and control theory throughout the 20th century. Input/output thinking is not an appropriate starting point in a field that has modeling of physical systems as one of its main concerns.”

To get a sense of the inappropriateness of input/output-based modeling: consider a freely swinging pendulum in 2-dimensional space with a finite-sized bob. Now consider adding to this system a model representing the right-hand half-plane being filled with cement. With soft-contact mechanics, we could determine what force the cement exerts on the pendulum when it bounces against it — that is, when the pendulum bob’s center of mass comes within its radius of the right half-plane.

Traditionally, we might define a function that takes the pendulum’s position as input and produces a force as output. But this is insufficient to model the effect of the wall, which also prevents the pendulum bob’s center of mass from ever being in the right-hand half-plane; the wall imposes a constraint on the possible states of the world. How can we can capture this kind of constraint? In this case, we can extend the state model with inequalities delineating the feasible region of the state space.²

Willems’ insight is that the entire modeling framework can be subsumed by a sufficiently broad notion of “feasible region.” A dynamical system is simply a relation on the variables, forming a subset of all conceivable trajectories — a ‘behavior.’

Interconnection as variable sharing

The signal-flow approach to systems modeling requires labeling the terminals of a system as inputs or outputs before a model can be formulated; Willems argues this does not respect the actual physics. Most physical terminals — electrical wires, mechanical linkages or gears, thermal couplings, etc. — do not have an intrinsic, a priori directionality, and may permit “signals” to “flow” in either or both directions. Rather, physical interconnections constrain certain variables on either side to be equal (or equal-and-opposite). After having modeled a system, one may be able to prove that it obeys a certain partitioning of variables into inputs and outputs, but assuming this up front obscures the underlying physical reality. This paradigm shift amounts to moving from functional composition (given $a$ we compute $b=f(a)$, then $c=g(b)$) to relational composition: $(a,c)\in(R;S)$ iff $\exists ((a,b_0),(b_1,c))\in(R\times S).\;b_0=b_1$, which can be read as “variable sharing” between $b_0$ and $b_1$. This is a way of restoring symmetry to composition — giving no precedence either between the two entities being composed, nor between each entity’s domain and codomain.

Examples of systems to model within the behavioral framework.

Core definitions

Given some natural phenomenon we wish to model mathematically, the first step is to establish the universum, the set of all a priori feasible outcomes, notated $\mathbb{V}$. Then, Willems asserts a mathematical model to be a restriction of possible outcomes to a subset of $\mathbb{V}$.³ This subset itself is called the behavior of the model, and is written $\mathcal{B}$. This concept is, as the name suggests, at the center of Willems’ “behavioral approach”: he asserts that “equivalence of models, properties of models, model representations, and system identification must refer to the behavior.”

A dynamical system is a model in which elements of the universum $\mathbb{V}$ are functions of time, that is, a triple

$$\Sigma = \left(\mathbb{T}, \mathbb{W}, \mathcal{B}\right)$$

in which $\mathcal{B} \subseteq \mathbb{V} := \mathbb{W}^\mathbb{T}$. $\mathbb{T}$ is referred to as the time set (which may be discrete or continuous), and $\mathbb{W}$ is referred to as the signal space. The elements of $\mathcal{B}$ are trajectories $w: \mathbb{T}\rightarrow\mathbb{W}$.

A dynamical system with latent variables is one whose signal space is a Cartesian product of manifest variables and latent variables: the “full” system is a tuple

$$\Sigma_{full}= \left(\mathbb{T}, \mathbb{M}, \mathbb{L}, \mathcal{B}_{full}\right)$$

where the behavior $\mathcal{B}_{{full}}\subseteq \mathbb{V} := \left(\mathbb{M} \times \mathbb{L}\right)^\mathbb{T}$. Here $\mathbb{M}$ is the set of manifest values and $\mathbb{L}$ is the set of latent values.

A full behavior $\Sigma_{{full}}$ is said to induce or represent a manifest dynamical system $\Sigma=\left(\mathbb{T}, \mathbb{M}, \mathcal{B}\right)$, with the manifest behavior $\mathcal{B}$ defined by

$$\mathcal{B}:=\left\{m: \mathbb{T} \rightarrow \mathbb{M}\;|\;\exists \ell: \mathbb{T} \rightarrow \mathbb{L}. \left\langle m,\ell\right\rangle \in \mathcal{B}_{{full}}\right\}$$

The behavior of all the variables is determined by the equations specifying the first-principles physical laws, together with the equations expressing all the constraints from interconnection. Willems treats interconnection as simply “variable sharing,” that is, restricting behavior such that the trajectories assigned to the interconnected variables are constrained to be equal (or sometimes “equal and opposite,” depending on sign conventions).

An interconnection architecture is a sort of wiring diagram (analogous to operadic wiring diagrams) that describes the way in which a collection of systems is interconnected. Willems formalises this as a graph with leaves: a set $V$ of vertices (which are systems or modules), a set $E$ of edges (terminals), and a set $L$ of leaves (open wires), with an assignment map $\mathcal{A}$: to each edge an unordered pair of vertices and to each leaf a single vertex. A leaf is depicted as an open half-edge emanating from the graph, like a wire sticking out of a circuit — the “open” part of “open and interconnected systems.” (Note that this interpretation of a “leaf” differs from the usual graph-theoretic “vertex with degree one”; here, a leaf is like an “edge with degree one.”) There’s a type-checking condition that the set of leaves and internal half-edges emanating from a vertex must be completely matched to the set of terminals of the associated module, so that you don’t have hidden dangling wires.

Finally, the interconnection architecture requires a module embedding that specifies how each vertex is interpreted as a module: either a primitive model made of physical laws, or a sub-model within which there’s a further module embedding. Here we get a sense of the “zooming” nature of the modeling procedure.

Tearing, Zooming, Linking

Willems proposes a “systematic procedure” for generating models in this behavioral form: first decompose (tear) the system under investigation into smaller subsystems, then recursively apply the modeling process (zoom in) to each subsystem, and finally compose (link) the resulting submodels together into an overall system model. Rendered in pseudocode, it looks like this:

define makeModel(System system) => Model {
  if (system is directly governed by known physics) {
    return knownModel(system)
  } else {
    WiringDiagram<System> decomposition := tear(system)
    List<Model> submodels := decomposition.listSubsystems().fmap(makeModel)
    return decomposition.link(submodels)
  }
}

As an example, Willems analyzes an open hydraulic system made of two tanks interconnected by a pipe:

In the “tear” step, he breaks the system apart into three subsystems: the two tanks, (1) and (3) in the figure, and the pipe (2). In the “zoom” step, each of the three subsystems is “simple enough to be modeled using first-principles physical laws,” so he fills in the known model for each one (reaching the recursive base case, rather than starting again from “tear”). For the pipe, flows on each end are equal and opposite, and the difference in pressure is proportional to the flow; for each of the tanks, conservation of mass and Bernoulli’s laws relate the pressures, flows, and height of water in the tank.

Then, in the “link” step, he starts with, for each subsystem, a copy of the corresponding model (initially each relating a completely separate set of variables), then combines the models according to the links between subsystems, using the appropriate “interconnection laws” for each pair of connected terminals. In this example, the interconnection laws consist of setting connected pressures to be equal and connected flows to be equal and opposite.

An essential claim of Willems’ philosophy is that for physical systems for which we can use his modeling procedure of breaking systems into subsystems with interconnections, the hierarchical structure of our model will match reality closely enough that there will be straightforward physical principles governing the interactions at the interfaces (which tend to correspond to partitions of physical space). Many engineered systems deliberately have their important interfaces clearly delineated, but he explicitly disclaims that there are forms of interaction, such as gravitational interaction and colliding objects, which do not perfectly fit this framework.

Limitations of Willems’ framework

Not all interconnections fit. This framework assumes that the interface of a module can be specified—as a finite set of terminals—prior to composition with other modules, and Willems identifies three situations where this assumption fails:

• $n$-body problems exhibiting “virtual terminals” which are not so much a property of each module, but of each pair of modules. The classic example of this phenomenon is the $n$-body problem in gravitational (or electrostatic, etc.) dynamics: an $n$-body system has $O(n^2)$ interactions, but the combination of an $n$-body system and an $m$-body system has more than $O(n^2+m^2)$ interactions.

• “Distributed interconnections” in which a terminal has continuous spatial extent (e.g. heat conduction along a surface), calling for partial differential equations involving coordinate variables.

• Contact mechanics such as rolling, sliding, bouncing, collisions, etc., in which interconnections appear and vanish depending on the values of certain position variables, as objects come into and out of contact.

Directional components and systems. In contrast to an a posteriori partitioning of variables into inputs and outputs (meaning that any setting of the “inputs” uniquely determines the trajectories of the “outputs”), some components fundamentally exhibit a priori input/output behavior (that is, they cannot be back-driven), and Willems’ framework can’t accommodate these.

• Ideal amplifier. The behavior of an ideal amplifier with gain $K$, input $x$ and output $y$ would be $\left\{ (x,y) | y=K x\right\}$ (constant-gain model), yet Willems’ approach here would make the incorrect prediction that we could back-drive terminal $y$ by interconnecting it to a signal source and expect to observe the signal scaled by $1/K$ at terminal $x$. However, an “ideal amplifier” is not a first-principles physical law; the modeling procedure might suggest we “tear” an amplifier further into its component parts, and then “tear” the constituent transistors with a deconstruction (such as the Gummel-Poon model) into passive circuit primitives. This might result in a more realistic model of actual amplifier behavior, though it would have at least an order of magnitude more components than the constant-gain model.

• Humans, etc. Willems lists additional signal-flow systems for which the behavioral approach is not quite adequate: actuator inputs and sensor outputs interconnected via a controller, reactions of humans or animals, devices that respond to external commands, or switches and other logical devices.

Cartesian latent/manifest partitioning. Among Willems’ arguments against mandatory input/output partitioning is the simple and compelling example of a particle moving on a sphere, whose position is truly an output and whose velocity is truly an input—yet even in this seemingly favorable setup, the full state space (the tangent bundle of the sphere) cannot be decomposed as a Cartesian product of positions on the sphere with any vector space. However, Willems uses exactly the same hidden assumption in his definition of a dynamical system with latent variables. If a dynamical system’s full state space can’t be written as a Cartesian product, then its behavior can’t be represented in the way Willems defines.

Probability. Willems’ non-deterministic approach to behaviors is a kind of unquantified uncertainty; it doesn’t natively give us a way of associating probabilities with elements of a “behavior” (although behaviors could be considered as always having an implicitly uniform distribution). Nontrivial distributions could also be modeled by defining the universum $\mathbb{V}:=P\!\left(\mathbb{W}^\mathbb{T}\right)$ (where $P$ is probability, namely the Giry monad), but the non-determinism of $\mathcal{B}\subseteq \mathbb{V}$ introduces “Knightian uncertainty”, in that models are now sets of distributions, with no probabilities specified at the top level—and it’s unclear how such models should compose with non-stochastic models.

Categorical approaches to systems

As mentioned earlier, Willems has been an inspiration for many papers in applied category theory. One common feature is that many take a relational approach to semantics, providing a functor into (some subcategory of) $\mathbf{Rel}_\times$. Here is a (non-exhaustive!) reference list of these and related works.

Applications to specific domains:

• Passive linear networks: Baez and Fong, 2015. Constructs a “black-boxing” functor from a decorated-cospan category of passive linear circuits (composed of resistors, inductors and capacitors) to a behavior category of Lagrangian relations.

• Generalized circuit networks: Baez, Coya and Rebro, 2018. Generalizes the black-boxing functor to potentially nonlinear components/circuits.

• Reversible Markov processes: Baez, Fong and Pollard, 2016. Constructs a “black-boxing” functor from a decorated-cospan category of reversible Markov processes to a category of linear relations describing steady states.

• Petri nets / reaction networks: Baez and Pollard, 2017. Constructs a “black-boxing” functor from a decorated-cospan category of Petri nets to a category of semi-algebraic relations describing steady states, with an intermediate stop at a “grey-box” category of algebraic vector fields.

• Digital circuits: Ghica, Jung and Lopez, 2017. Defines a symmetric monoidal theory of circuits including a discrete-delay operator and feedback, with operational semantics.

• Discrete linear time-invariant dynamical systems (LTIDS): Fong, Sobocinski, and Rapisarda, 2016. Constructs a full and faithful functor from a freely generated symmetric monoidal theory into the PROP of LTIDSs and characterizes controllability in terms of spans, among other things—not only using Willems’ definitions and philosophy, but even some of his theorems.

General frameworks:

• Algebra of Open and Interconnected Systems: Brendan Fong’s 2016 doctoral thesis. Covers his technique of decorated cospans as well as more recent work on decorated corelations, both of which are especially useful to construct syntactic categories for various kinds of non-categorical diagrams.

• Topos of behavior types: Schultz and Spivak, 2017. Constructs a temporal type theory, as the internal logic of the category of sheaves on an interval domain, in which every object represents a behavior that seems to be essentially in Willems’ sense of the word.

• Operad of wiring diagrams: Vagner, Spivak and Lerman, 2015. Formalises construction of systems of differential equations on manifolds using Spivak’s “operad of wiring diagrams” approach to composition, which is conceptually similar to Willems’ notion of hierarchical zooming into modules.

• Signal flow graphs: Bonchi, Sobocinski and Zanasi, 2017. Sound and complete axiomatization of signal flow graphs, arguably the primary incumbent against which Willems’ behavioral approach contends.

• Bond graphs: Brandon Coya, 2017. Defines a category of bond graphs (an older general modeling framework which Willems acknowledges as a step in the right direction toward a behavioral approach) with functorial semantics as Lagrangian relations.

• Cospan/Span(Graph): Gianola, Kasangian and Sabadini in 2017 review a line of work mostly done in the 90’s by Sabadini, Walters and collaborators on what are essentially open and interconnected labeled-transition systems.

Many thanks to Pawel Sobocinski and Brendan Fong for feedback on this post, and to Sophie Raynor and other members of the seminar for thoughts and discussions.

¹ see e.g. Spivak and Schultz’s Temporal type theory; Fong, Sobocinski, and Rapisarda’s Categorical approach to open and interconnected dynamical systems; Bonchi, Sobocinski, and Zanasi’s Categorical semantics of signal flow graphs

² Depending on the formalisation, a system of differential equations could technically contain equality constraints without any actual derivatives, such as $x^2 + y^2 - 1 = 0$, which can restrict the feasible region without augmenting the modeling framework to include inequalities. We could even impose the constraint $x\leq 0$ by using a non-analytic function: $$0 = \begin{cases}e^{-1/x}& if x\gt 0\\ 0& if x\leq 0\end{cases}$$

³ Note: From a computer science perspective, this says that any “mathematical model” must be a non-deterministic model, as opposed to, on the one hand, a deterministic model (which would pick out an element of $\mathbb{V}$), or, on the other hand, a probabilistic model (which would give a distribution over $\mathbb{V}$). If we are given free choice of $\mathbb{V}$, any of these kinds of model is encodable as any other, but the choice is significant when it comes to composition.

Today my lecture was crowd-sourced! In response to popular opinions from the students, I spoke about cosmological large-scale structure experiments. I spoke about how the large surveys are collapsed to symmetry-respecting mean, variance, and three-point functions, and how simulations of large-scale structure are used to build surrogate likelihood functions for these summary statistics.

The fate of the current Wisconsin Assembly district map, precision-engineered to maintain a Republican majority in the face of anything short of a major Democratic wave election, is in the hands of the Supreme Court, which could announce a decision in Gill v. Whitford any day.

One theory of gerrymandering is that the practice isn’t much of a problem, because the power of a gerrymandered map “decays” with time — a map that suits a party in 2010 may, due to shifting demographics, be reasonably fair a few years later.

How’s the Wisconsin gerrymander doing in 2018?  We just had a statewide election in which Rebecca Dallet, the more liberal candidate, beat her conservative rival by 12 points, an unusually large margin for a Wisconsin statewide race.

The invaluable J. Miles Coleman broke the race down by Assembly district:

Dallet won in 58% of seats while getting 56% of the vote.  That sounds fair, but in fact a candidate who wins by 12 points is typically going to win in more seats than that.  (That’s why the courts are right to say proportional representation isn’t a reasonable expectation!)

Here’s the breakdown by Assembly district, shown a little bigger:

Dallet won by 2 points or less in 8 of the Assembly districts.  So, as a rough estimate, if she’d gotten 2% of the vote less, and won 54-46 instead of 56-44, you might guess she’d have won 49 out of 99 seats.  That’s consistent with the analysis of Herschlag, Ravier, and Mattingly conducted last year, which estimates that under current maps Democrats would need an 8-12 point statewide lead in order to win half the Assembly seats. (Figure 5 in the linked paper.)

I don’t think the gerrymander is decaying very much.  I think it’s robust enough to make GOP legislative control very likely through 2020, at which point it can be updated to last another ten years, and so on and so on.  This isn’t the same kind of softcore gerrymandering the Supreme Court allowed to stand in 1986, and I hope the 2018 Supreme Court decides to do something about it.

Today was my second day of lecturing at TASI, and I gave one (morning) lecture, on the use of MCMC sampling. In the afternoon, I looked at (for the first time) the GALAH data on detailed element abundances of stars. I looked at the question of whether the chemical abundances could be used to predict the Galactocentric radii. The idea is: If the gas involved in star formation is azimuthally mixed, there ought to be relationships between radius in the disk and chemical abundances. They didn't jump out! I have various ideas about why, but for now this will be back-burner.

Today was the sixth day (but my first day) of the Theory Advanced Study Institute summer school at CU Boulder. I gave two 75-minute lectures on data analysis, my first two of five lectures this week. In the first, I tried to boil down data-analysis to a set of over-arching principles. I got 8 principles. Maybe this is the introduction to the book I will never write! In the second lecture I spoke about fitting a model, from a frequentist perspective, but with a focus on the likelihood function. I am loving the interactive audience. See the wiki for a (constantly updating) description of the lectures I am giving.

There’s a been a lot of progress on the ‘field with one element’ since I discussed it back in “week259”. I’ve been starting to learn more about it, and especially its possible connections to the Riemann Hypothesis. This is a great place to start:

• Oliver Lorscheid, $\mathbb{F}_1$ for everyone.

Abstract. This text serves as an introduction to $\mathbb{F}_1$-geometry for the general mathematician. We explain the initial motivations for $\mathbb{F}_1$-geometry in detail, provide an overview of the different approaches to $\mathbb{F}_1$ and describe the main achievements of the field.

Lorscheid’s paper describes various approaches. Since I’m hoping the key to $\mathbb{F}_1$-mathematics is something very simple and natural that we haven’t fully explored yet, I’m especially attracted to these:

• Deitmar’s approach, which generalizes algebraic geometry by replacing commutative rings with commutative monoids. A lot of stuff in algebraic geometry, like the ideals and spectra of commutative rings, or the theory of schemes, doesn’t really require the additive aspect of a ring! So, for many purposes we can get away with commutative monoids, where we think of the monoid operation as multiplication. Sometimes it’s good to use commutative monoids equipped with a ‘zero’ element. The main problem is that algebraic geometry without addition seems to be approximately the same as toric geometry — a wonderful subject, but not general enough to handle everything we want from schemes over $\mathbb{F}_1$.

• Toën and Vaquié’s approach, which goes further and replaces commutative rings by commutative monoid objects in symmetric monoidal categories (which work best when they’re complete, cocomplete and cartesian closed). If our symmetric monoidal category is $(\mathbf{AbGp}, \otimes)$ we’re back to commutative rings, if it’s $(\mathbf{Set}, \times)$ we’ve got commutative monoids, but there are plenty of other nice choices: for example if it’s $(\mathbf{CommMon}, \otimes)$ we get commutative rigs, which are awfully nice.

One can also imagine ‘homotopy-coherent’ or ‘$\infty$-categorical’ analogues of these two approaches, which might provide a good home for certain ways the sphere spectrum shows up in this business as a substitute for the integers. For example, one could imagine that the ultimate replacement for a commutative ring is an $E_\infty$ algebra inside a symmetric monoidal $(\infty,1)$-category.

However, it’s not clear to me that homotopical thinking is the main thing we need to penetrate the mysteries of $\mathbb{F}_1$. There seem to be some other missing ideas….

Lorscheid’s own approach uses ‘blueprints’. A blueprint $(R,S)$ is a commutative rig $R$ equipped with a subset $S \subseteq R$ that’s closed under multiplication, contains $0$ and $1$, and generates $R$ as a rig.

I have trouble, just on general aesthetic grounds, believing that blueprints are final ultimate solution to the quest for a generalization of commutative rings that can handle the “field with one element”. They just don’t seem ‘god-given’ the way commutative monoids or commutative objects are. But they do various nice things.

Maybe someone has answered this already, since it’s a kind of obvious question:

Question. Is there a symmetric monoidal category $\mathbf{C}$ in which blueprints are the commutative monoid objects?

Maybe something like the category of ‘abelian groups equipped with a set of generators’?

Of course you should want to know what morphisms of blueprints are, because really we should want the category of commutative monoid objects in $\mathbf{C}$ to be equivalent to the category of blueprints. Luckily Lorscheid’s morphisms of blueprints are the obvious thing: a morphism $f : (R,S) \to (R',S')$ is a morphism of commutative rigs $f: R \to R'$ with $f(S) \subseteq S'$.

Anyway, there’s a lot more to say about $\mathbb{F}_1$, but Lorscheid’s paper is a great way to get into this subject.

In completely separate video news, here are videos of lectures I gave at CERN several years ago: “Cosmology for Particle Physicists” (May 2005). These are slightly technical — at the very least they presume you know calculus and basic physics — but are still basically accurate despite their age.

1. Introduction to Cosmology
2. Dark Matter
3. Dark Energy
4. Thermodynamics and the Early Universe
5. Inflation and Beyond

Update: I originally linked these from YouTube, but apparently they were swiped from this page at CERN, and have been taken down from YouTube. So now I’m linking directly to the CERN copies. Thanks to commenters Bill Schempp and Matt Wright.

This post is to announce that a new journal, Advances in Combinatorics, has just opened for submissions. I shall also say a little about the journal, about other new journals, about my own experiences of finding journals I am happy to submit to, and about whether we are any nearer a change to more sensible systems of dissemination and evaluation of scientific papers.

Advances in Combinatorics

Advances in Combinatorics is set up as a combinatorics journal for high-quality papers, principally in the less algebraic parts of combinatorics. It will be an arXiv overlay journal, so free to read, and it will not charge authors. Like its cousin Discrete Analysis (which has recently published its 50th paper) it will be run on the Scholastica platform. Its minimal costs are being paid for by the library at Queen’s University in Ontario, which is also providing administrative support. The journal will start with a small editorial board. Apart from me, it will consist of Béla Bollobás, Reinhard Diestel, Dan Kral, Daniela Kühn, James Oxley, Bruce Reed, Gabor Sarkozy, Asaf Shapira and Robin Thomas. Initially, Dan Kral and I will be the managing editors, though I hope to find somebody to replace me in that role once the journal is established. While I am posting this, Dan is simultaneously announcing the journal at the SIAM conference in Discrete Mathematics, where he has just given a plenary lecture. The journal is also being announced by COAR, the Confederation of Open Access Repositories. This project aligned well with what they are trying to do, and it was their director, Kathleen Shearer, who put me in touch with the library at Queen’s.

As with Discrete Analysis, all members of the editorial board will be expected to work: they won’t just be lending their names to give the journal bogus prestige. Each paper will be handled by one of the editors, who, after obtaining external opinions (when the paper warrants them) will make a recommendation to the rest of the board. All decisions will be made collectively. The job of the managing editors will be to make sure that this process runs smoothly, but when it comes to decisions, they will have no more say than any other editor.

The rough level that the journal is aiming at is that of a top specialist journal such as Combinatorica. The reason for setting it up is that there is a gap in the market for an “ethical” combinatorics journal at that level — that is, one that is not published by one of the major commercial publishers, with all the well known problems that result. We are not trying to destroy the commercial combinatorial journals, but merely to give people the option of avoiding them if they would prefer to submit to a journal that is not complicit in a system that uses its monopoly power to ruthlessly squeeze library budgets.

We are not the first ethical journal in combinatorics. Another example is The Electronic Journal of Combinatorics, which was set up by Herb Wilf back in 1994. The main difference between EJC and Advances in Combinatorics is that we plan to set a higher bar for acceptance, even if it means that we accept only a small number of papers. (One of the great advantages of a fully electronic journal is that we do not have a fixed number of issues per year, so we will not have to change our standards artificially in order to fill issues or clear backlogs.) We thus hope that EJC and AIC will between them offer suitable potential homes for a wide range of combinatorics papers. And on the more algebraic side, one should also mention Algebraic Combinatorics, which used to be the Springer journal The Journal of Algebraic Combinatorics (which officially continues with an entirely replaced editorial board — I don’t know whether it’s getting many submissions though), and also the Australasian Journal of Combinatorics.

So if you’re a combinatorialist who is writing up a result that you think is pretty good, then please consider submitting it to us. What do we mean by “pretty good”? My personal view — that is, I am not speaking for the rest of the editorial board — is that the work in a good paper should have a clear reason for others to be interested in it (so not, for example, incremental progress in some pet project of the author) and should have something about it that makes it count as a significant achievement, such as solving a well-known problem, clearing a difficult technical hurdle, inventing a new and potentially useful technique, or giving a beautiful and memorable proof.

What other ethical journals are there?

Suppose that you want to submit an article to a journal that is free to read and does not charge authors. What are your options? I don’t have a full answer to this question, so I would very much welcome feedback from other people, especially in areas of mathematics far from my own, about what the options are for them. But a good starting point is to consult the list of current member journals in the Free Journal Network, which Advances in Combinatorics hopes to join in due course.

Three notable journals not on that list are the following.

1. Acta Mathematica. This is one of a tiny handful of the very top journals in mathematics. Last year it became fully open access without charging author fees. So for a really good paper it is a great option.
2. Annales Henri Lebesgue. This is a new journal that has not yet published any articles, but is open for submissions. Like Acta Mathematica, it covers all of mathematics. It aims for a very high standard, but it is not yet clear what that means in practice: I cannot say that it will be roughly at the level of Journal X. But perhaps it will turn out to be suitable for a very good paper that is just short of the level of Annals, Acta, or JAMS.
3. Algebra and Number Theory. I am told that this is regarded as the top specialist journal in number theory. From a glance at the article titles, I don’t see much analytic number theory, but there are notable analytic number theorists on the editorial board, so perhaps I have just not looked hard enough.

Added later: I learn from Benoît Kloeckner and Emmanuel Kowalski in the comments below that my information about Algebra and Number Theory was wrong, since articles in that journal are not free to read until they are five years old. However, it is published by MSP, which is a nonprofit organization, so as subscription journals go it is at the ethical end of the spectrum.

Further update: I have heard from the editors of Annales Henri Lebesgue that they have had a number of strong submissions and expect the level of the journal to be at least as high as that of journals such as Advances in Mathematics, Mathematische Annalen and the Israel Journal of Mathematics, and perhaps even slightly higher.

In what areas are ethical journals most needed?

I would very much like to hear from people who would prefer to avoid the commercially published journals, but can’t, because there are no ethical journals of a comparable standard in their area. I hope that combinatorialists will no longer have that problem. My impression is that there is a lack of suitable journals in analysis and I’m told that the same is true of logic. I’m not quite sure what the situation is in geometry or algebra. (In particular, I don’t know whether Algebra and Number Theory is also considered as the top specialist journal for algebraists.) Perhaps in some areas there are satisfactory choices for papers of some standards but not of others: that too would be interesting to know. Where do you think the gaps are? Let me know in the comments below.

Starting a new journal.

I want to make one point loud and clear, which is that the mechanics of starting a new, academic-run journal are now very easy. Basically, the only significant obstacle is getting together an editorial board with the right combination of reputation in the field and willingness to work. What’s more, unless the journal grows large, the work is quite manageable — all the more so if it is spread reasonably uniformly amongst the editorial board. Creating the journal itself can be done on one of a number of different platforms, either for no charge or for a very small charge. Some examples are the Mersenne platform, which hosts the Annales Henri Lebesgue, the Episciences platform, which hosts the Epijournal de Géométrie Algébrique, and Scholastica, which, as I mentioned above, hosts Discrete Analysis and Advances in Combinatorics.

Of these, Scholastica charges a submission fee of $10 per article and the other two are free. There are a few additional costs — for example, Discrete Analysis pays a subscription to CrossRef in order to give DOIs to articles — but the total cost of running a new journal that isn’t too large is of the order of a few hundred dollars per year, as long as nobody is paid for what they do. (Discrete Analysis, like Advances in Combinatorics, gets very useful assistance from librarians, provided voluntarily, but even if they were paid the going rate, the total annual costs would be of the same order of magnitude as one “article processing charge” of the traditional publishers, which is typically around $1500 per article.)

What’s more, those few hundred dollars are not an obstacle either. For example, I know of a fund that is ready to support at least one other journal of a similar size to Discrete Analysis, there are almost certainly other libraries that would be interested in following the enlightened example of Queen’s University Library and supporting a journal (if you are a librarian reading this, then I strongly recommend doing so, as it will be helping to weaken the hold of the system that is currently costing you orders of magnitude more money), and I know various people who know about other means of obtaining funding. So if you are interested in starting a journal and think you can put together a credible editorial board, then get in touch: I can offer advice, funding (if the proposal looks a good one), and contact with several other people who are knowledgeable and keen to help.

A few remarks about my own relationship with mathematical publishing.

My attitudes to journals and the journal system have evolved quite a lot in the last few years. The alert reader may have noticed that I’ve got a long way through this post before mentioning the E-word. I still think that Elsevier is the publisher that does most damage, and have stuck rigidly to my promise made over six years ago not to submit a paper to them or to do editorial or refereeing work. However, whereas then I thought of Springer as somehow more friendly to mathematics, thanks to its long tradition of publishing important textbooks and monographs, I now feel pretty uncomfortable about all the big four — Elsevier, Springer, Wiley, and Taylor and Francis — with Springer having got a whole lot worse after merging with Nature Macmillan. And in some respects Elsevier is better than Springer: for example, they make all mathematics papers over four years old freely available, while Springer refuses to do so. Admittedly this was basically a sop to mathematicians to keep us quiet, but as sops go it was a pretty good one, and I see now that Elsevier’s open archive, as they call it, includes some serious non-mathematical journals such as Cell. (See their list of participating journals for details.)

I’m also not very comfortable with the society journals and university presses, since although they use their profits to benefit mathematics in various ways, they are fully complicit in the system of big deals, the harm of which outweighs those benefits.

The result is that if I have a paper to submit, I tend to have a lot of trouble finding a suitable home for it, and I end up having to compromise on my principles to some extent (particularly if, as happened recently, I have a young coauthor from a country that uses journal rankings to evaluate academics). An obvious place to submit to would be Discrete Analysis, but I feel uncomfortable about that for a different reason, especially now that I have discovered that the facility that enables all the discussion of a paper to be hidden from selected editors does not allow me, as administrator of the website, to hide a paper from myself. (I won’t have this last problem with Advances in Combinatorics, since the librarians at Queens will have the administrator role on the system.)

So my personal options are somewhat limited, but getting better. If I have willing coauthors, then I would now consider (if I had a suitable paper), Acta Mathematica, Annales Henri Lebesgue, Journal de l’École Polytechnique, Discrete Analysis perhaps (but only if the other editors agreed to process my paper offline), Advances in Combinatorics, the Theory of Computing, Electronic Research Announcements in the Mathematical Sciences, the Electronic Journal of Combinatorics, and the Online Journal of Analytic Combinatorics. I also wouldn’t rule out Forum of Mathematics. A couple of journals to which I have an emotional attachment even if I don’t really approve of their practices are GAFA and Combinatorics, Probability and Computing. (The latter bothers me because it is a hybrid journal — that is, it charges subscriptions but also lets authors pay large APCs to make their articles open access, and I heard recently that if you choose the open access option, CUP retains copyright, so you’re not getting that much for your money. But I think not many authors choose this option. The former is also a hybrid journal, and is published by Springer.) Annals of Mathematics, if I’m lucky enough to have an Annals-worthy paper (though I think now I’d try Acta first), is not too bad — although its articles aren’t open access, their subscription costs are much more reasonable than most journals.

That’s a list off the top of my head: if you think I’ve missed out a good option, then I’d be very happy to hear about it.

As an editor, I have recently made the decision that I want to devote all my energies to promoting journals and “post-journal” systems that I fully approve of. So in order to make time for the work that will be involved in establishing Advances in Combinatorics, I have given notice to Forum of Mathematics and Mathematika, the two journals that took up the most of my time, that I will leave their editorial boards at the end of 2018. I feel quite sad about Forum of Mathematics, since I was involved in it from the start, and I really like the way it runs, with proper discussions amongst all the editors about the decisions we make. Also, I am less hostile (for reasons I’ve given in the past) to its APC model than most mathematicians. However, although I am less hostile, I could never say that I have positively liked it, and I came to the conclusion quite a while ago that, as many others have also said, it simply can’t be made to work satisfactorily: it will lead to just as bad market abuses as there are with the subscription system. In the UK it has been a disaster — government open-access mandates have led to universities paying as much as ever for subscriptions and then a whole lot extra for APCs. And there is a real worry that subscription big deals will be replaced by APC big deals, where a country pays a huge amount up front to a publisher in return for people from that country being able to publish with them. This, for example, is what Germany is pushing for. Fortunately, for the moment (if I understand correctly, though I don’t have good insider information on this) they are asking for the average fee per article to be much lower than Elsevier is prepared to accept: long may that impasse continue.

So my leaving Forum of Mathematics is not a protest against it, but simply a practical step that will allow me to focus my energies where I think they can do the most good. I haven’t yet decided whether I ought to resign in protest from some other editorial boards of journals that don’t ask anything of me. Actually, even the practice of having a long list of names of editors, most of whom have zero involvement in the decisions of the journal, is one that bothers me. I recently heard of an Elsevier journal where almost all the editorial board would be happy to resign en masse and set up an ethical version, but the managing editor is strongly against. “But why don’t the rest of the board resign in that case?” I naively asked, to which the answer was, “Because he’s the one who does all the work!” From what I understood, this is literally true — the managing editor handles all the papers and makes all the decisions — but I’m not 100% sure about that.

Is there any point in starting new journals?

Probably major change, if it happens, will be the result of decisions made by major players such as government agencies, national negotiators, and so on. Compared with big events like the Elsevier negotiations in Germany, founding a new journal is a very small step. And even if all mathematicians gave up using the commercial publishers (not something I expect to see any time soon), that would have almost no direct effect, since mathematics journals are bundled together with journals in other subjects, which would continue with the current system.

However, this is a familiar situation in politics. Big decisions are taken by people in positions of power, but what prompts them to make those decisions is often the result of changes in attitudes and behaviour of voters. And big behavioural changes do happen in academia. For example, as we all know, many people have got into the habit of posting all their work on the arXiv, and this accumulation of individual decisions has had the effect of completely changing the way dissemination works in some subjects, including mathematics, a change that has significantly weakened the hold that journals have — or would have if they weren’t bundled together with other journals. Who would ever subscribe at vast expense to a mathematics journal when almost all its content is available online in preprint form?

So I see Advances in Combinatorics as a small step certainly, but a step that needs to be taken. I hope that it will demonstrate once again that starting a serious new journal is not that hard. I also hope that the current trickle of such journals will turn into a flood, that after the flood it will not be possible for people to argue that they are forced to submit articles to the commercial publishers, and that at some point, someone in a position of power will see what is going on, understand better the absurdities of the current system, and take a decision that benefits us all.

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The covariant power set functor $P : Set \to Set$ can be made into a monad whose multiplication $m_X: P(P(X)) \to P(X)$ turns a subset of the set of subsets of $X$ into a subset of $X$ by taking their union. Algebras of this monad are complete semilattices.

But what about powers of the power set functor? Yesterday Jules Hedges pointed out this paper:

• Bartek Klin and Julian Salamanca, Iterated covariant powerset is not a monad.

The authors prove that $P^n$ cannot be made into a monad for $n \ge 2$.

I’ve mainly looked at their proof for the case $n = 2$. I haven’t completely worked through it, but it focuses on the unit of any purported monad structure for $P^2$, rather than its multiplication. Using a cute Yoneda trick they show there are only four possible units, corresponding to the four elements of $P(P(1))$. Then they show these can’t work. The argument involves sets like this:

As far as I’ve seen, they don’t address the following question:

Question. Does there exist an associative multiplication $m: P^2 P^2 \Rightarrow P^2$? In other words, is there a natural transformation $m: P^2 P^2 \Rightarrow P^2$ such that

$$P^2 P^2 P^2 \stackrel{m P^2 }{\Rightarrow} P^2 P^2 \stackrel{m}{\Rightarrow} P^2$$

equals

$$P^2 P^2 P^2 \stackrel{P^2 m}{\Rightarrow} P^2 P^2 \stackrel{m}{\Rightarrow} P^2 .$$

I’m not very good at these things, so this question might be very easy to answer. But if the answer were “obviously no” then you’d think Klin and Salamanca might have mentioned that. They do prove there is no distributive law $P P \Rightarrow P P$. But they also give examples of monads $T$ for which there’s no distributive law $T T \Rightarrow T T$, yet there’s still a way to make $T^2$ into a monad.

As far as I can tell, my question is fairly useless: does anyone consider “semigroupads”, namely monads without unit? Nonetheless I’m curious.

If there were a positive answer, we’d have a natural way to take a set of sets of sets of sets and turn it into a set of sets in such a way that the two most obvious resulting ways to turn a set of sets of sets of sets of sets of sets into a set of sets agree!

This is a sequel to this previous blog post, in which we discussed the effect of the heat flow evolution

on the zeroes of a time-dependent family of polynomials , with a particular focus on the case when the polynomials had real zeroes. Here (inspired by some discussions I had during a recent conference on the Riemann hypothesis in Bristol) we record the analogous theory in which the polynomials instead have zeroes on a circle , with the heat flow slightly adjusted to compensate for this. As we shall discuss shortly, a key example of this situation arises when is the numerator of the zeta function of a curve.

More precisely, let be a natural number. We will say that a polynomial

of degree (so that ) obeys the functional equation if the are all real and

for all , thus

and

for all non-zero . This means that the zeroes of (counting multiplicity) lie in and are symmetric with respect to complex conjugation and inversion across the circle . We say that this polynomial obeys the Riemann hypothesis if all of its zeroes actually lie on the circle . For instance, in the case, the polynomial obeys the Riemann hypothesis if and only if .

Such polynomials arise in number theory as follows: if is a projective curve of genus over a finite field , then, as famously proven by Weil, the associated local zeta function (as defined for instance in this previous blog post) is known to take the form

where is a degree polynomial obeying both the functional equation and the Riemann hypothesis. In the case that is an elliptic curve, then and takes the form , where is the number of -points of minus . The Riemann hypothesis in this case is a famous result of Hasse.

Another key example of such polynomials arise from rescaled characteristic polynomials

of matrices in the compact symplectic group . These polynomials obey both the functional equation and the Riemann hypothesis. The Sato-Tate conjecture (in higher genus) asserts, roughly speaking, that “typical” polyomials arising from the number theoretic situation above are distributed like the rescaled characteristic polynomials (1), where is drawn uniformly from with Haar measure.

Given a polynomial of degree with coefficients

we can evolve it in time by the formula

thus for . Informally, as one increases , this evolution accentuates the effect of the extreme monomials, particularly, and at the expense of the intermediate monomials such as , and conversely as one decreases . This family of polynomials obeys the heat-type equation

In view of the results of Marcus, Spielman, and Srivastava, it is also very likely that one can interpret this flow in terms of expected characteristic polynomials involving conjugation over the compact symplectic group , and should also be tied to some sort of “” version of Brownian motion on this group, but we have not attempted to work this connection out in detail.

It is clear that if obeys the functional equation, then so does for any other time . Now we investigate the evolution of the zeroes. Suppose at some time that the zeroes of are distinct, then

From the inverse function theorem we see that for times sufficiently close to , the zeroes of continue to be distinct (and vary smoothly in ), with

Differentiating this at any not equal to any of the , we obtain

and

and

Inserting these formulae into (2) (expanding as ) and canceling some terms, we conclude that

for sufficiently close to , and not equal to . Extracting the residue at , we conclude that

which we can rearrange as

If we make the change of variables (noting that one can make depend smoothly on for sufficiently close to ), this becomes

Intuitively, this equation asserts that the phases repel each other if they are real (and attract each other if their difference is imaginary). If obeys the Riemann hypothesis, then the are all real at time , then the Picard uniqueness theorem (applied to and its complex conjugate) then shows that the are also real for sufficiently close to . If we then define the entropy functional

then the above equation becomes a gradient flow

which implies in particular that is non-increasing in time. This shows that as one evolves time forward from , there is a uniform lower bound on the separation between the phases , and hence the equation can be solved indefinitely; in particular, obeys the Riemann hypothesis for all if it does so at time . Our argument here assumed that the zeroes of were simple, but this assumption can be removed by the usual limiting argument.

For any polynomial obeying the functional equation, the rescaled polynomials converge locally uniformly to as . By Rouche’s theorem, we conclude that the zeroes of converge to the equally spaced points on the circle . Together with the symmetry properties of the zeroes, this implies in particular that obeys the Riemann hypothesis for all sufficiently large positive . In the opposite direction, when , the polynomials converge locally uniformly to , so if , of the zeroes converge to the origin and the other converge to infinity. In particular, fails the Riemann hypothesis for sufficiently large negative . Thus (if ), there must exist a real number , which we call the de Bruijn-Newman constant of the original polynomial , such that obeys the Riemann hypothesis for and fails the Riemann hypothesis for . The situation is a bit more complicated if vanishes; if is the first natural number such that (or equivalently, ) does not vanish, then by the above arguments one finds in the limit that of the zeroes go to the origin, go to infinity, and the remaining zeroes converge to the equally spaced points . In this case the de Bruijn-Newman constant remains finite except in the degenerate case , in which case .

For instance, consider the case when and for some real with . Then the quadratic polynomial

has zeroes

and one easily checks that these zeroes lie on the circle when , and are on the real axis otherwise. Thus in this case we have (with if ). Note how as increases to , the zeroes repel each other and eventually converge to , while as decreases to , the zeroes collide and then separate on the real axis, with one zero going to the origin and the other to infinity.

The arguments in my paper with Brad Rodgers (discussed in this previous post) indicate that for a “typical” polynomial of degree that obeys the Riemann hypothesis, the expected time to relaxation to equilibrium (in which the zeroes are equally spaced) should be comparable to , basically because the average spacing is and hence by (3) the typical velocity of the zeroes should be comparable to , and the diameter of the unit circle is comparable to , thus requiring time comparable to to reach equilibrium. Taking contrapositives, this suggests that the de Bruijn-Newman constant should typically take on values comparable to (since typically one would not expect the initial configuration of zeroes to be close to evenly spaced). I have not attempted to formalise or prove this claim, but presumably one could do some numerics (perhaps using some of the examples of given previously) to explore this further.

In particle physics, we almost always use approximations.

Often, we assume the forces we consider are weak. We use a “coupling constant”, some number written or or , and we assume it’s small, so is greater than is greater than . With this assumption, we can start drawing Feynman diagrams, and each “loop” we add to the diagram gives us a higher power of .

If isn’t small, then the trick stops working, the diagrams stop making sense, and we have to do something else.

Except for some times, when everything keeps working fine. This week, along with Simon Caron-Huot, Lance Dixon, Andrew McLeod, and Georgios Papathanasiou, I published what turned out to be a pretty cute example.

We call this fellow . It’s a family of diagrams that we can write down for any number of loops: to get more loops, just extend the “…”, adding more boxes in the middle. Count the number of lines sticking out, and you get six: these are “hexagon functions”, the type of function I’ve used to calculate six-particle scattering in N=4 super Yang-Mills.

The fun thing about is that we don’t have to think about it this way, one loop at a time. We can add up all the loops, times one loop plus times two loops plus times three loops, all the way up to infinity. And we’ve managed to figure out what those loops sum to.

The result ends up beautifully simple. This formula isn’t just true for small coupling constants, it’s true for any number you care to plug in, making the forces as strong as you’d like.

We can do this with because we have equations relating different loops together. Solving those equations with a few educated guesses, we can figure out the full sum. We can also go back, and use those equations to take the s at each loop apart, finding a basis of functions needed to describe them.

That basis is the real reward here. It’s not the full basis of “hexagon functions”: if you wanted to do a full six-particle calculation, you’d need more functions than the ones is made of. What it is, though, is a basis we can describe completely, stating exactly what it’s made of for any number of loops.

We can’t do that with the hexagon functions, at least not yet: we have to build them loop by loop, one at a time before we can find the next ones. The hope, though, is that we won’t have to do this much longer. The basis covers some of the functions we need. Our hope is that other nice families of diagrams can cover the rest. If we can identify more functions like , things that we can sum to any number of loops, then perhaps we won’t have to think loop by loop anymore. If we know the right building blocks, we might be able to guess the whole amplitude, to find a formula that works for any you’d like.

That would be a big deal. N=4 super Yang-Mills isn’t the real world, but it’s complicated in some of the same ways. If we can calculate there without approximations, it should at least give us an idea of what part of the real-world answer can look like. And for a field that almost always uses approximations, that’s some pretty substantial progress.

Important note: As this is not a course in probability, we will try to avoid developing the general theory of stochastic calculus (which includes such concepts as filtrations, martingales, and Ito calculus). This will unfortunately limit what we can actually prove rigorously, and so at some places the arguments will be somewhat informal in nature. A rigorous treatment of many of the topics here can be found for instance in Lawler’s Conformally Invariant Processes in the Plane, from which much of the material here is drawn.

In these notes, random variables will be denoted in boldface.

Definition 1 A real random variable is said to be normally distributed with mean and variance if one has

for all test functions . Similarly, a complex random variable is said to be normally distributed with mean and variance if one has

for all test functions , where is the area element on .

A real Brownian motion with base point is a random, almost surely continuous function (using the locally uniform topology on continuous functions) with the property that (almost surely) , and for any sequence of times , the increments for are independent real random variables that are normally distributed with mean zero and variance . Similarly, a complex Brownian motion with base point is a random, almost surely continuous function with the property that and for any sequence of times , the increments for are independent complex random variables that are normally distributed with mean zero and variance .

Remark 2 Thanks to the central limit theorem, the hypothesis that the increments be normally distributed can be dropped from the definition of a Brownian motion, so long as one retains the independence and the normalisation of the mean and variance (technically one also needs some uniform integrability on the increments beyond the second moment, but we will not detail this here). A similar statement is also true for the complex Brownian motion (where now we need to normalise the variances and covariances of the real and imaginary parts of the increments).

Real and complex Brownian motions exist from any base point or ; see e.g. this previous blog post for a construction. We have the following simple invariances:

Exercise 3

• (i) (Translation invariance) If is a real Brownian motion with base point , and , show that is a real Brownian motion with base point . Similarly, if is a complex Brownian motion with base point , and , show that is a complex Brownian motion with base point .
• (ii) (Dilation invariance) If is a real Brownian motion with base point , and is non-zero, show that is also a real Brownian motion with base point . Similarly, if is a complex Brownian motion with base point , and is non-zero, show that is also a complex Brownian motion with base point .
• (iii) (Real and imaginary parts) If is a complex Brownian motion with base point , show that and are independent real Brownian motions with base point . Conversely, if are independent real Brownian motions of base point , show that is a complex Brownian motion with base point .

The next lemma is a special case of the optional stopping theorem.

Lemma 4 (Optional stopping identities)

• (i) (Real case) Let be a real Brownian motion with base point . Let be a bounded stopping time – a bounded random variable with the property that for any time , the event that is determined by the values of the trajectory for times up to (or more precisely, this event is measurable with respect to the algebra generated by this proprtion of the trajectory). Then

  

  and

  

  and

  

• (ii) (Complex case) Let be a real Brownian motion with base point . Let be a bounded stopping time – a bounded random variable with the property that for any time , the event that is determined by the values of the trajectory for times up to . Then

  

  

  

  

  

By the law of total expectation, we thus have

and

and

where the inner conditional expectations are with respect to the event that attains a particular point in . However, from the independent increment nature of Brownian motion, once one conditions to a fixed point , the random variable becomes a real normally distributed variable with mean and variance . Thus we have

and

and

which give the first two claims, and (after some algebra) the identity

which then also gives the third claim.

Exercise 5 Prove the second part of Lemma 4.

— 1. Conformal invariance of Brownian motion —

Let be an open subset of , and a point in . We can define the complex Brownian motion with base point restricted to to be the restriction of a complex Brownian motion with base point to the first time in which the Brownian motion exits (or if no such time exists). We have a fundamental conformal invariance theorem of Lévy:

Theorem 6 (Lévy’s theorem on conformal invariance of Brownian motion) Let be a conformal map between two open subsets of , and let be a complex Brownian motion with base point restricted to . Define a rescaling by

Note that this is almost surely a continuous strictly monotone increasing function. Set (so that is a homeomorphism from to ), and let be the function defined by the formula

Then is a complex Brownian motion with base point restricted to .

Note that this significantly generalises the translation and dilation invariance of complex Brownian motion.

Proof: (Somewhat informal – to do things properly one should first set up Ito calculus) To avoid technicalities we will assume that is bounded above and below on , so that the map is uniformly bilipschitz; the general case can be obtained from this case by a limiting argument that is not detailed here. With this assumption, we see that almost surely extends continuously to the endpoint time if this time is finite. Once one conditions on the value of and up to this time , we then extend this motion further (if ) by declaring for to be a complex Brownian motion with base point , translated in time by . Now is defined on all of , and it will suffice to show that this is a complex Brownian motion based at . The basing is clear, so it suffices to show for all times , the random variable is normally distributed with mean and variance .

Let be a test function. It will suffice to show that

If we define the field

for and , with , then it will suffice to prove the more general claim

As is well known, is smooth on and solves the backwards heat equation

Next, we obtain the analogous estimate

Taking expectations and applying Lemma 4, (2) and Hölder’s inequality (which can interpolate between the bounds and to conclude ), we obtain the desired claim (3). Subtracting, we now have

The expression in the expectation vanishes unless , hence by the triangle inequality

Iterating this using the fact that vanishes at , and sending to zero (noting that the cumulative error term will go to zero since ), we conclude that for all , giving the claim.

One can use Lévy’s theorem (or variants of this theorem) to prove various results in complex analysis rather efficiently. As a quick example, we sketch a Brownian motion-based proof of Liouville’s theorem (omitting some technical steps). Suppose for contradiction that we have a nonconstant bounded entire function . If is a complex Brownian motion based at , then a variant of Levy’s theorem can be used to show that the image is a time parameterisation of Brownian motion. But it is easy to show that Brownian motion is almost surely unbounded, so the image cannot be bounded.

If is an open subset of whose complement contains an arc, then one can show that for any , the complex Brownian motion based at will hit the boundary of in a finite time . The location where this motion first hits the boundary is then a random variable in ; the law of this variable is called the harmonic measure of with base point , and we will denote it by ; it is a probability measure on . The reason for the terminology “harmonic measure” comes from the following:

Theorem 7 Let be a bounded open subset of , and let be a harmonic (or holomorphic) function that extends continuously to . Then for any , one has the representation formula

 

This theorem can also extend to unbounded domains provided that does not grow too fast at infinity (for instance if is bounded, basically thanks to the neighbourhood recurrent properties of complex Brownian motion); we do not give a precise statement here. Among other things, this theorem gives an immediate proof of the maximum principle for harmonic functions, since if on the boundary then from the triangle inequality one has for all . It also gives an alternate route to Liouville’s theorem: if is entire and bounded, then applying the maximum principle to the complement of a small disk we see that for all distinct .

When the boundary is sufficiently nice (e.g. analytic), the harmonic measure becomes absolutely continuous with respect to one-dimensional Lebesgue measure; however, we will not pay too much attention to these sorts of regularity issues in this set of notes.

From Levy’s theorem on the conformal invariance of Brownian motion we deduce the conformal invariance of harmonic measure, thus for any conformal map that extends continuously to the boundaries and any , the harmonic measure of with base point is the pushforward of the harmonic measure of with base point , thus

for any continuous compactly supported test function , and also

for any (Borel) measurable .

Exercise 8 (Poisson kernel)

• (i) If and , show that the measure on the unit circle is given by

  

  where is arclength measure. In particular, when , then is the uniform measure on the unit circle.

• (ii) If and , show that the measure on the real line is given by

  

  (For this exercise one can assume that harmonic measure is well defined for unbounded domains, and that the representation formula (4) continues to hold for bounded harmonic or holomorphic functions.)

Exercise 9 (Brownian motion description of conformal mapping) Let be the region enclosed by a Jordan curve , and let be three distinct points on in anticlockwise order. Let be three distinct points on the boundary of the unit disk , again traversed in anticlockwise order. Let be the conformal map that takes to for (the existence and uniqueness of this map follows from the Riemann mapping theorem). Let , and for , let be the probability that the terminal point of Brownian motion at with base point lies in the arc between and (here we use the fact that the endpoints are hit with probability zero, or in other words that the harmonic measure is continuous; see Exercise 15 below). Thus are non-negative and sum to . Let be the complex numbers , , . Show the crossratio identity

In principle, this allows one to describe conformal maps purely in terms of Brownian motion.

We remark that the link between Brownian motion and conformal mapping can help gain an intuitive understanding of the Carathéodory kernel theorem (Theorem 12 from Notes 3). Consider for instance the example in Exercise 13 from those notes. It is intuitively clear that a Brownian motion based at the origin will very rarely pass through the slit beween and , instead hitting the right side of the boundary of first. As such, the harmonic measure of the left side of the bounadry should be very small, and in fact one can use this to show that the preimage under of the region to the left of the boundary goes to zero in diameter as , which helps explain why the limiting function does not map to this region at all.

Exercise 10 (Brownian motion description of conformal radius)

• (i) Let and with . Show that the probability that the Brownian motion hits the circle before it hits is equal to . (Hint: is harmonic away from the origin.)
• (ii) Let be a simply connected proper subset of , let be a point in , and let be the conformal radius of around . Show that for small , the probability that a Brownian motion based at a point with will hit the circle before it hits the boundary is equal to , where denotes a quantity that goes to zero as .

Exercise 11 Let be a connected subset of , let be a Brownian motion based at the origin, and let be the first time this motion exits . Show that the probability that hits is at least for some absolute constant . (Hint: one can control the event that makes a “loop” around a point in at radius less than , which is enough to force intersection with , at least if one works some distance away from the boundary of the disk.)

This naive argument does not quite work because a Brownian motion starting at a real number will in fact almost surely cross the real axis an infinite number of times. However it is possible to adapt this argument by redefining the so that after each time , the Brownian motion is forced to move some small distance before one starts looking for the next time it hits the real axis. See the proof of Lemma 6.1 of these notes of Lawler for a complete proof along these lines.

This gives an inequality similar in spirit to the Grötzsch modulus estimate from Notes 2:

Corollary 13 (Beurling projection theorem) Let , and let be a compact connected subset the annulus that intersects both boundary circles of the annulus. Let be a complex Brownian motion based at , and let be the first time this motion hits the outer boundary of the annulus. Then the probability that intersects is greater than or equal to the probability that intersects the interval .

Exercise 14 With the notation as the above corollary, show that the probability that intersects the interval is . (Hint: apply a square root conformal map to the disk with removed, and then compare with the half-plane harmonic measure from Exercise 8(ii).)

The following consequence of the above estimate, giving a sort of Hölder regularity of Brownian measure, is particularly useful in applications.

Exercise 15 (Beurling estimate) Let be an open set not containing , with the property that the connected component of containing intersects the unit circle . Let be such that . Then for any , one has ; that is to say, the probability that a Brownian motion based at exits at a point within from the origin is . (Hint: one can use conformal mapping to show that the probability appearing at the end of Corollary 13 is .) Conclude in particular that harmonic measures are always continuous (they assign zero to any point).

Exercise 16 Let be a region bounded by a Jordan curve, let , let be the Brownian motion based at , and let be the first time this motion exits . Then for any , show that the probability that the curve has diameter at least is at most .

Exercise 17 Let be a conformal map with , and let be a curve with and for . Show that

(Hint: use Exercise 11.)

— 2. Half-plane capacity —

One can use Brownian motion to construct other close relatives of harmonic measure, such Green’s functions, excursion measures. See for instance these lecture notes of Lawler for more details. We will focus on one such use of Brownian motion, to interpret the concept of half-plane capacity; this is a notion that is particularly well adapted to the study of chordal Loewner equations (it plays a role analogous to that of conformal radius for the radial Loewner equation).

Let be the upper half-plane. A subset of the upper half-plane is said to be a compact hull if it is bounded, closed in , and the complement is simply connected. By the Riemann mapping theorem, for any compact hull , there is a unique conformal map which is normalised at infinity in the sense that

Exercise 18 (Examples of half-plane capacity)

• (i) Show the translation invariance and the dilation invariance for any compact hull , any , and any , where and .
• (ii) If is a vertical line segment for some and , show that is given by

  

  (using the standard branch of the square root), so that

  

   

• (iii) If is the semicircle , then is given by

  

  so that

  

   

Proposition 19 Let be a compact hull, with conformal map and half-plane capacity .

• (i) If is complex Brownian motion based at some point , and is the first time this motion exits , then

  

• (ii) We have

  

Proof: (Sketch) Part (i) follows from applying Theorem 7 to the bounded harmonic function . Part (ii) follows from part (i) by setting for a large , rearranging, and sending using (5).

Among other things, this proposition demonstrates that for all , and that the half-plane capacity is always non-negative (in fact it is not hard to show from the above proposition that it is strictly positive as long as is non-empty).

If are two compact hulls with , then will map conformally to the complement of in . Thus is also a convex hull, and by the uniqueness of Riemann maps we have the identity

which on comparing Laurent expansions leads to the further identity

In particular we have the monotonicity , with equality if and only if . One may verify that these claims are consistent with Exercise 18.

Exercise 20 (Submodularity of half-plane capacity) Let be two compact hulls.

• (i) If , show that

  

  (Hint: use Proposition 19, and consider how the times in which a Brownian motion exits , , , and are related.)

• (ii) Show that

  

Exercise 21 Let be a compact hull bounded in a disk . For any , show that

as , where is complex Brownian motion based at and is the first time it exits . Similarly, for any , show that

This formula gives a Brownian motion interpretation for on the portion of the boundary of . It can be used to give useful quantitative estimates for in this region; see Section 3.4 of Lawler’s book.

— 3. The chordal Loewner equation —

We now develop (in a rather informal fashion) the theory of the chordal Loewner equation, which roughly speaking is to conformal maps from the upper half-plane to the complement of complex hulls as the radial Loewner equation is to conformal maps from the unit disk to subsets of the complex plane. A more rigorous treatment can be found in Lawler’s book.

Suppose one has a simple curve such that and . There are important and delicate issues regarding the regularity hypotheses on this curve (which become particularly important in SLE, when the regularity is quite limited), but for this informal discussion we will ignore all of these issues.

For each time , the set forms a compact hull, and so has some half-plane capacity . From the monotonicity of capacity, this half-plane capacity is increasing in . It is traditional to normalise the curve so that

this is analogous to normalising the Loewner chains from Notes 3 to have conformal radius at time . A basic example of such normalised curves would be the curves for some fixed , since the normalisation follows from (6).

Let be the conformal maps associated to these compact hulls. From (8) we will have

for any and , where is the conformal map associated to the compact hull . From (9) this hull has half-plane capacity , thus we have the Laurent expansion

It can be shown (using the Beurling estimate) that extends continuously to the tip of the curve , and attains a real value at that point; furthermore, depends continuously on . See Lemma 4.2 of Lawler’s book. As such, should be a short arc (of length ) starting at . If , it is possible to use a quantitative version of Exercise 21 (again using the Beurling estimate) to obtain an estimate basically of the form

for any fixed . If is non-zero, we instead have

For instance, if , then for all , and from Exercise 18 we have the exact formula

Inserting (12) into (11) and using the chain rule, we obtain

and we then arrive at the (chordal) Loewner equation

for all and . This equation can be justified rigorously for any simple curve : see Proposition 4.4 of Lawler’s book. Note that the imaginary part of is negative, which is consistent with the observation made previously that the imaginary part of is decreasing in .

We have started with a chain of compact hulls associated to a simple curve, and shown that the resulting conformal maps obey the Loewner equation for some continuous driving term . Conversely, suppose one is given a continuous driving term . It follows from Picard existence and uniqueness theorem that for each there is a unique maximal time of existence such that the ODE (13) with initial data can be solved for time , one can show that for each time , is a conformal map from to with the Laurent expansion

hence the complement are an increasing sequence of compact hulls with half-plane capacity . Proving complex differentiability of can be done from first principles, and the Laurent expansion near infinity is also not hard; the main difficulty is to show that the map is surjective, which requires solving (13) backwards in time (and here one can do this indefinitely as now one is moving away from the real axis instead of towards it). See Theorem 4.6 of Lawler’s book for details (in fact a more general theorem is proven, in which the single point is replaced by a probability measure, analogously to how the radial Loewner equation uses Herglotz functions instead of a single driving function when not restricted to slit domains). However, there is a subtlety, in that the hulls are not necessarily the image of simple curves . This is often the case for short times if the driving function does not oscillate too wildly, but it can happen that the curve that one would expect to trace out eventually intersects itself, in which case the region it then encloses must be absorbed into the hull (cf. the “pinching off” phenomenon in the Carathéodory kernel theorem). Nevertheless, it is still possible to have Loewner chains that are “generated” by non-simple paths , in the sense that consists of the unbounded connected component of the complement .

There are some symmetries of the transform from the to the . If one translates by a constant, , then the resulting domains are also translated, , and . Slightly less trivially, for any , if one performs a rescaled dilation , then one can check using (13) that , and the corresponding conformal maps are given by . On the other hand, just performing a scalar multiple on the driving force can transform the behavior of dramatically; the transform from to is very definitely not linear!

— 4. Schramm-Loewner evolution —

In the previous section, we have indicated that every continuous driving function gives rise to a family of conformal maps obeying the Loewner equation (13). The (chordal) Schramm-Loewner evolution () with parameter is the special case in which the driving function takes the form for some real Brownian motion based at the origin. Thus is now a random conformal map from a random domain , defined by solving the Schramm-Loewner equation

with initial condition for , and with defined as the set of all for which the above ODE can be solved up to time taking values in . The parameter cannot be scaled away by simple renormalisations such as scaling, and in fact the behaviour of is rather sensitive to the value of , with special behaviour or significance at various values such as playing particularly special roles; there is also a duality relationship between and which we will not discuss here.

The case is rather boring, in which is deterministic, and is just with the line segment between and removed. The cases are substantially more interesting. It is a non-trivial theorem (particularly at the special value ) that is almost surely generated by some random path ; see Theorem 6.3 of Lawler’s book. The nature of this path is sensitive to the choice of parameter :

• For , the path is almost surely simple and goes to infinity as ; it also avoids the real line (except at time ).
• For ; it also has non-trivial intersection with the real line.
• For , the path is almost surely space-filling (which of course also implies that ), and also hits every point on .

See Section 6.2 of Lawler’s book. The path becomes increasingly fractal as increases: it is a result of Rohde and Schramm and Beffara that the image almost surely has Hausdorff dimension .

We have asserted that defines a random path in that starts at the origin and generally “wanders off” to infinity (though for it keeps recurring back to bounded sets infinitely often). By the Riemann mapping theorem, we can now extend this to other domains. Let be a simply connected open proper subset of whose boundary we will assume for simplicity to be a Jordan curve (this hypothesis can be relaxed). Let be two distinct points on the boundary . By the Riemann mapping theorem and Carathéodory’s theorem (Theorem 20 from Notes 2), there is a conformal map whose continuous extension maps and to and respectively; this map is unique up to rescalings for . One can then define the Schramm-Loewner evolution on from to to be the family of conformal maps for , where is the usual Schramm-Loewner evolution with parameter . The Schramm-Loewner evolution on is well defined up to a time reparameterisation . The Markovian and stationary nature of Brownian motion translates to an analogous Markovian and conformally invariant property of . Roughly speaking, it is the following: if is any reasonable domain with two boundary points , is on this domain from to with associated path , and is any time, then after conditioning on the path up to time , the remainder of the path has the same image as the path on the domain from to . Conversely, under suitable regularity hypotheses, the processes are the only random path processes on domains with this property (much as Brownian motion is the only Markovian stationary process, once one normalises the mean and variance). As a consequence, whenever one now a random path process that is known or suspected to enjoy some conformal invariance properties, it has become natural to conjecture that it obeys the law of (though in some cases it is more natural to work with other flavours of SLE than the chordal SLE discussed here, such as radial SLE or whole-plane SLE). For instance, in the pioneering work of Schramm, this line of reasoning was used to conjecture that the loop-erased random walk in a domain has the law of (radial) ; this conjecture was then established by Lawler, Schramm, and Werner. Many further processes have since been either proven or conjectured to be linked to one of the SLE processes, such as the limiting law of a uniform spanning tree (proven to be ), interfaces of the Ising model (proven to be ), or the scaling limit of self-avoiding random walks (conjectured to be ). Further discussion of these topics is beyond the scope of this course, and we refer the interested reader to Lawler’s book for more details.

A few weeks ago I listened to a bunch of super-enthusiastic high school students share their excitement about astronomy, astrophysics and the space industry. We were at the Royal Astronomical Society of New Zealand’s annual meeting; every year they fund a dozen or so keen students from around the country to attend the event. It’s competitive, with many more kids applying than there are places to take them. 

The students get to share what lit their passion for space, but as they recounted their stories maybe a third of them mentioned a teacher or career counsellor who had done their best to quench that fire. “There’s no jobs in that.” “It’s really hard,” “You can’t do that in New Zealand.” “It won’t help you get into university.” And I wondered who listens to a smart, ambitious youngster's hopes and dreams and tells them to play it safe?

It isn’t all teachers – others told of enthusiastic and supportive mentors – but it was potentially way too many of them, even if the nay-sayers imagine they are protecting kids from disappointment or quietly steering them away from games that are out of their league. Ironically, this is happening at a time when there is a lot of talk about reinventing school – getting kids to code, preparing them for a future where machine learning, AI and 3D printing are big things, building baby entrepreneurs. But none of that will be much use if some teachers are telling students not to chase their dreams.

Ironically, when educators talk about preparing kids for the future, part of their message is usually that the age of linear careers is behind us and today's kids will change jobs more far frequently than their parents did. Which makes it particularly strange for teachers to tell students to aim only at safe and dependable careers as they chart their educational trajectory.

So what should you say to a smart kid with dreams that would take them off the beaten track? Here's my list: 

1. Tell them "Wow that's great. Go for it!"
2. Say to them "That sounds like it's going to be a lot of work."
3. Ask them if they know what they need to do to get to their goal. What skills will they need? 
4. Tell them "That sounds awesome" (whatever it is).
5. Tell them "Go for it. Whatever happens, you will be OK", but help them figure out what they are likely to hit if they miss their actual target. Don't be negative, it just never hurts to look before you leap, to make sure they will be ok.
6. Ask them, "What can you do right now that will get you closer to your dream?" Get them doing things today that will engage their passion. 
7. Ask "How can I help?" 

Item 1 is easy, and it works for anything. Item 2 is honest – no-one sleepwalks to being an astronaut or an All Black – and it is likely part of the fun.

Item 3 is a biggie, and for some goals the answer will be more obvious than others. What should someone who wants to be a professional athlete do, beyond just training for their sport? I'm not the best person to ask about that (really, I'm not) but whatever you want to do, my guess is that some of the things that will get you there won't always be obvious: one of my key skills as a scientist is that I like to write and know how to tell a story. (Seriously: every grant application is a story about what I would do if you gave me money – and it helps if I tell it well.) If a kid at your school wants to get into astronomy (something I do know something about), it could be a long wait before they can actually do astronomy. But astronomers need to know maths, computing, statistics and physics – does the kid you are talking to know this? Set them to finding out what is on their list. 

Item 4 leads into Item 5. If a kid chases their dreams, are they really taking their whole future into the Casino of Life and putting it all on 22? Or are they laying down skills that also lead to all sorts of other opportunities? If I am brutally honest, I will only retire once, and while my field is growing it is certainly not expanding fast enough to accommodate everyone who trains to work within it. Numbers as miserable as a 2 or 3% success rate for making the transition from a PhD to a permanent job in the field are thrown around – the situation is not as dire as that, but in astronomy the success rate may be around 20% for people who actually get PhDs. At some point, it is possible students stand a better chance of success if they don't spend too much time stressing about the odds. But all my former students are doing well. Some have tenure – and others have successful startups or jobs in Silicon Valley where they use knowledge they acquired while they were working with me.

Someone chasing their dreams may be walking a high wire, but they can build a safety net by laying down serious and transferable skills. (And many of the people I know who pulled the cord on their personal Plan B made a conscious decision to do something new; ambitions change.) Help them figure out what those will be for them. The real trick to Item 5 is to help kids take measured risks without hinting you don't think they can do it. And this goes double if you are talking to people who might feel that they won't fit naturally into their chosen field – my friend Chanda Prescod-Weinstein often says that African American students hear well-meaning talk about a personal Plan B as "I don' think this is for you"; the same likely goes for Māori and Pasifika kids in New Zealand, or girls heading into fields where women are scarce. 

Item 6 and 7: whatever it is they want to do, there will always be something they can start on right away. There's no time like the present. And be there for them. Check in. Ask how they are getting on. And wish them well.

And one last question: if you're a teacher, what's your ambition? It's got to be better to watch your students chase their dreams than, when they come to tell the story of their success, be remembered as the person who said they couldn't do it.

CODA: It is also true that if we are talking about academia, it is not just the student's job to be sure they have a Plan B, it is up to their teachers to recognise that many of them will need one, make sure that they get the chance to maximise their transferable skills and know how to demonstrate them to potential employers. But that's a story for another day.  

IMAGE CREDIT: The image is from NASA.  

We now approach conformal maps from yet another perspective. Given an open subset of the complex numbers , define a univalent function on to be a holomorphic function that is also injective. We will primarily be studying this concept in the case when is the unit disk .

Clearly, a univalent function on the unit disk is a conformal map from to the image ; in particular, is simply connected, and not all of (since otherwise the inverse map would violate Liouville’s theorem). In the converse direction, the Riemann mapping theorem tells us that every open simply connected proper subset of the complex numbers is the image of a univalent function on . Furthermore, if contains the origin, then the univalent function with this image becomes unique once we normalise and . Thus the Riemann mapping theorem provides a one-to-one correspondence between open simply connected proper subsets of the complex plane containing the origin, and univalent functions with and . We will focus particular attention on the univalent functions with the normalisation and ; such functions will be called schlicht functions.

One basic example of a univalent function on is the Cayley transform , which is a Möbius transformation from to the right half-plane . (The slight variant is also referred to as the Cayley transform, as is the closely related map , which maps to the upper half-plane.) One can square this map to obtain a further univalent function , which now maps to the complex numbers with the negative real axis removed. One can normalise this function to be schlicht to obtain the Koebe function

which now maps to the complex numbers with the half-line removed. A little more generally, for any we have the rotated Koebe function

that is a schlicht function that maps to the complex numbers with the half-line removed.

Intuitively, the Koebe function and its rotations should be the “largest” schlicht functions available. This is formalised by the famous Bieberbach conjecture, which asserts that for any schlicht function, the coefficients should obey the bound for all . After a large number of partial results, this conjecture was eventually solved by de Branges; see for instance this survey of Korevaar or this survey of Koepf for a history.

It turns out that to resolve these sorts of questions, it is convenient to restrict attention to schlicht functions that are odd, thus for all , and the Taylor expansion now reads

for some complex coefficients with . One can transform a general schlicht function to an odd schlicht function by observing that the function , after removing the singularity at zero, is a non-zero function that equals at the origin, and thus (as is simply connected) has a unique holomorphic square root that also equals at the origin. If one then sets

it is not difficult to verify that is an odd schlicht function which additionally obeys the equation

Conversely, given an odd schlicht function , the formula (4) uniquely determines a schlicht function .

For instance, if is the Koebe function (1), becomes

which maps to the complex numbers with two slits removed, and if is the rotated Koebe function (2), becomes

De Branges established the Bieberbach conjecture by first proving an analogous conjecture for odd schlicht functions known as Robertson’s conjecture. More precisely, we have

Theorem 1 (de Branges’ theorem) Let be a natural number.

• (i) (Robertson conjecture) If is an odd schlicht function, then

  

• (ii) (Bieberbach conjecture) If is a schlicht function, then

  

It is easy to see that the Robertson conjecture for a given value of implies the Bieberbach conjecture for the same value of . Indeed, if is schlicht, and is the odd schlicht function given by (3), then from extracting the coefficient of (4) we obtain a formula

for the coefficients of in terms of the coefficients of . Applying the Cauchy-Schwarz inequality, we derive the Bieberbach conjecture for this value of from the Robertson conjecture for the same value of . We remark that Littlewood and Paley had conjectured a stronger form of Robertson’s conjecture, but this was disproved for by Fekete and Szegö.

To prove the Robertson and Bieberbach conjectures, one first takes a logarithm and deduces both conjectures from a similar conjecture about the Taylor coefficients of , known as the Milin conjecture. Next, one continuously enlarges the image of the schlicht function to cover all of ; done properly, this places the schlicht function as the initial function in a sequence of univalent maps known as a Loewner chain. The functions obey a useful differential equation known as the Loewner equation, that involves an unspecified forcing term (or , in the case that the image is a slit domain) coming from the boundary; this in turn gives useful differential equations for the Taylor coefficients of , , or . After some elementary calculus manipulations to “integrate” this equations, the Bieberbach, Robertson, and Milin conjectures are then reduced to establishing the non-negativity of a certain explicit hypergeometric function, which is non-trivial to prove (and will not be done here, except for small values of ) but for which several proofs exist in the literature.

The theory of Loewner chains subsequently became fundamental to a more recent topic in complex analysis, that of the Schramm-Loewner equation (SLE), which is the focus of the next and final set of notes.

— 1. The area theorem and its consequences —

We begin with the area theorem of Grönwall.

Theorem 2 (Grönwall area theorem) Let be a univalent function with a convergent Laurent expansion

Then

Corollary 4 (Bieberbach inequality)

• (i) If is an odd schlicht function, then .
• (ii) If is a schlicht function, then .

Proof: For (i), we apply Theorem 2 to the univalent function defined by , which has a Laurent expansion , to give the claim. For (ii), apply part (i) to the square root of with first term .

Exercise 5 Show that equality occurs in Corollary 4(i) if and only if takes the form for some , and in Corollary 4(ii) if and only if takes the form of a rotated Koebe function for some .

The Bieberbach inequality can be rescaled to bound the second coefficient of univalent functions:

Exercise 6 (Rescaled Bieberbach inequality) If is a univalent function, show that

When does equality hold?

The Bieberbach inequality gives a useful lower bound for the image of a univalent function, known as the Koebe quarter theorem:

Corollary 7 (Koebe quarter theorem) Let be a univalent function. Then contains the disk .

Applying Exercise 6 we then have

while from the Bieberbach inequality one also has . Hence by the triangle inequality , which is incompatible with the hypothesis .

Exercise 8 Show that the radius is best possible in Corollary 7 (thus, does not contain any disk with ) if and only if takes the form for some complex numbers and real .

Remark 9 The univalence hypothesis is crucial in the Koebe quarter theorem. Consider for instance the functions defined by . These are locally univalent functions (since is holomorphic with non-zero derivative) and , , but avoids the point .

Exercise 10 (Koebe distortion theorem) Let be a schlicht function, and let have magnitude .

• (i) Show that

  

  (Hint: compose on the right with a Möbius automorphism of that sends to and then apply the rescaled Bieberbach inequality.)

• (ii) Show that

  

  (Hint: use (i) to control the radial derivative of .)

• (iii) Show that

  

• (iv) Show that

  

  (This cannot be directly derived from (ii) and (iii). Instead, compose on the right with a Mobius automorphism that sends to and to , rescale it to be schlicht, and apply (iii) to this function at .)

• (v) Show that the space of schlicht functions is a normal family. In other words, if is any sequence of schlicht functions, then there is a subsequence that converges locally uniformly on compact sets.
• (vi) (Qualitative Bieberbach conjecture) Show that for each natural number there is a constant such that whenever is a schlicht function with Taylor expansion

  

Exercise 11 (Conformal radius) If is a non-empty simply connected open subset of that is not all of , and is a point in , define the conformal radius of at to be the quantity , where is any conformal map from to that maps to (the existence and uniqueness of this radius follows from the Riemann mapping theorem). Thus for instance the disk has conformal radius around .

• (i) Show that the conformal radius is strictly monotone in : if are non-empty simply connected open subsets of , and , then the conformal radius of around is strictly greater than that of .
• (ii) Show that the conformal radius of a disk around an element of the disk is given by the formula .
• (iii) Show that the conformal radius of around lies between and , where is the radius of the maximal disk that is contained in .
• (iv) If the conformal radius of around is equal to , show that for all sufficiently small , the ring domain has modulus , where denotes a quantity that goes to zero as , and the modulus of a ring domain was defined in Notes 2.

We can use the distortion theorem to obtain a nice criterion for when univalent maps converge to a given limit, known as the Carathéodory kernel theorem.

Theorem 12 (Carathéodory kernel theorem) Let be a sequence of simply connected open proper subsets of containing the origin, and let be a further simply connected open proper subset of containing . Let and be the conformal maps with and (the existence and uniqueness of these maps are given by the Riemann mapping theorem). Then the following are equivalent:

• (i) converges locally uniformly on compact sets to .
• (ii) For every subsequence of the , is the set of all such that there is an open connected set containing and that is contained in for all sufficiently large .

Proof: Suppose first that converges locally uniformly on compact sets to . If , then for some . If , then the holomorphic functions converge uniformly on to the function , which is not identically zero but has a zero in . By Hurwitz’s theorem we conclude that also has a zero in for all sufficiently large ; indeed the same argument shows that one can replace by any element of a small neighbourhood of to obtain the same conclusion, uniformly in . From compactness we conclude that for sufficiently large , has a zero in for all , thus for sufficiently large . Since is open connected and contains and , we see that is contained in the set described in (ii).

Conversely, suppose that is a subsequence of the and is such that there is an open connected set containing and that is contained in for sufficiently large . The inverse maps are holomorphic and bounded, hence form a normal family by Montel’s theorem. By refining the subsequence we may thus assume that the converge locally uniformly to a holomorphic limit . The function takes values in , but by the open mapping theorem it must in fact map to . In particular, . Since converges to , and converges locally uniformly to , we conclude that converges to , thus and hence . This establishes the derivation of (ii) from (i).

Now suppose that (ii) holds. It suffices to show that every subsequence of has a further subsequence that converges locally uniformly on compact sets to (this is an instance of the Urysohn subsequence principle). Then (as contains ) in particular there is a disk that is contained in the for all sufficiently large ; on the other hand, as is not all of , there is also a disk which is not contained in the for all sufficiently large . By Exercise 11, this implies that the conformal radii of the around zero is bounded above and below, thus is bounded above and below.

By Exercise 10(v), and rescaling, the functions then form a normal family, thus there is a subsequence of the that converges locally uniformly on compact sets to some limit . Since is positive and bounded away from zero, is also positive, so is non-constant. By Hurwitz’s theorem, is therefore also univalent, and thus maps to some region . By the implication of (ii) from (i) (with replaced by ) we conclude that is the set of all such that there is an open connected set containing and that is contained in for all sufficiently large ; but by hypothesis, this set is also . Thus , and then by the uniqueness part of the Riemann mapping theorem, as desired.

The condition in Theorem 12(ii) indicates that “converges” to in a rather complicated sense, in which large parts of are allowed to be “pinched off” from and disappear in the limit. This is illustrated in the following explicit example:

Exercise 13 (Explicit example of kernel convergence) Let be the function from (5), thus is a univalent function from to with the two vertical rays from to , and from to , removed. For any natural number , let and let , and define the transformed functions .

• (i) Show that is a univalent function from to with the two vertical rays from to , and from to , removed, and that and .
• (ii) Show that converges locally uniformly to the function , and that this latter map is a univalent map from to the half-plane . (Hint: one does not need to compute everything exactly; for instance, any terms of the form can be written using the notation instead of expanded explicitly.)
• (iii) Explain why these facts are consistent with the Carathéodory kernel theorem.

As another illustration of the theorem, let be two distinct convex open proper subsets of containing the origin, and let be the associated conformal maps from to respectively with and . Then the alternating sequence does not converge locally uniformly to any limit. The set is the set of all points that lie in a connected open set containing the origin that eventually is contained in the sequence ; but if one passes to the subsequence , this set of points enlarges to , and so the sequence does not in fact have a kernel.

However, the kernel theorem simplifies significantly when the are monotone increasing, which is already an important special case:

Corollary 14 (Monotone increasing case of kernel theorem) Let the notation and assumptions be as in Theorem 12. Assume furthermore that

and that . Then converges locally uniformly on compact sets to .

Loewner observed that the kernel theorem can be used to approximate univalent functions by functions mapping into slit domains. More precisely, define a slit domain to be an open simply connected subset of formed by deleting a half-infinite Jordan curve connecting some finite point to infinity; for instance, the image of the Koebe function is a slit domain.

Theorem 15 (Loewner approximation theorem) Let be a univalent function. Then there exists a sequence of univalent functions whose images are slit domains, and which converge locally uniformly on compact subsets to .

The material in this section is based on these lecture notes of Contreras.

An important tool in analysing univalent functions is to study one-parameter families of univalent functions, parameterised by a time parameter , in which the images are increasing in ; roughly speaking, these families allow one to study an arbitrary univalent function by “integrating” along such a family from back to . Traditionally, we normalise these families into (radial) Loewner chains, which we now define:

Definition 16 (Loewner chain) A (radial) Loewner chain is a family of univalent maps with and (so in particular is schlicht), such that for all . (In these notes we use the prime notation exclusively for differentiation in the variable; we will use later for differentiation in the variable.)

A key example of a Loewner chain is the family

of dilated Koebe functions; note that the image of each is the slit domain , which is clearly monotone increasing in . More generally, we have the rotated Koebe chains

for any real .

Whenever one has a family of simply connected proper open subsets of containing with for , and . By definition, is then the conformal radius of around , which is a strictly increasing function of by Exercise 11. If this conformal radius is equal to at and increases continuously to infinity as , then one can reparameterise the variable so that , at which point one obtains a Loewner chain.

From the Koebe quarter theorem we see that each image in a Loewner chain contains the disk . In particular the increase to fill out all of : .

Let be a Loewner chain, Let . The relation is sometimes expressed as the assertion that is subordinate to . It has the consequence that one has a composition law of the form

for a univalent function , uniquely defined as , noting taht is well-defined on . By construction, we have and

as well as the composition laws

for . We will refer to the as transition functions.

From the Schwarz lemma, we have

for , with strict inequality when . In particular, if we introduce the function

for and , then (after removing the singularity at infinity and using (10)) we see that is a holomorphic map to the right half-plane , normalised so that

Define a Herglotz function to be a holomorphic function , thus is a Herglotz function for all . A key family of examples of a Herglotz function are the Möbius transforms for . In fact, all other Herglotz functions are basically just averages of this one:

Exercise 17 (Herglotz representation theorem) Let be a Herglotz function, normalised so that .

• (i) For any , show that

  

  for . (Hint: The real part of is harmonic, and so has a Poisson kernel representation. Alternatively, one can use a Taylor expansion of .)

• (ii) Show that there exists a (Radon) probability measure on such that

  

  for all . (One will need a measure-theoretic tool such as Prokhorov’s theorem, the Riesz representation theorem, or the Helly selection principle.) Conversely, show that every probability measure on generates a Herglotz function with by the above formula.

• (iii) Show that the measure constructed on (ii) is unique.

This has a useful corollary, namely a version of the Harnack inequality:

Exercise 18 (Harnack inequality) Let be a Herglotz function, normalised so that . Show that

for all .

This gives some useful Lipschitz regularity properties of the transition functions and univalent functions in the variable:

Lemma 19 (Lipschitz regularity) Let be a compact subset of , and let . Use to denote a quantity bounded in magnitude by , where depends only on .

• (i) For any and , one has

  

• (ii) For any and , one has

  

One can make the bounds much more explicit if desired (see e.g. Lemma 2.3 of these notes of Contreras), but for our purposes any Lipschitz bound will suffice.

Proof: To prove (i), it suffices from (11) and the Schwarz-Pick lemma (Exercise 13 from Notes 2) to establish this claim when . We can also assume that since the claim is trivial when . From the Harnack inequality one has

for , which by (12) and some computation gives

giving the claim.

Now we prove (ii). We may assume without loss of generality that is convex. From Exercise 10 (normalising to be schlicht) we see that for , and hence has a Lipschitz constant of on . Since , the claim now follows from (13).

As a first application of this we show that every schlicht function starts a Loewner chain.

Lemma 20 Let be schlicht. Then there exists a Loewner chain with .

Proof: This will be similar to the proof of Theorem 15. First suppose that extends to be univalent on for some , then is a Jordan curve. Then by Carathéodory’s theorem (Theorem 20 of Notes 2) (and the Möbius inversion ) one can find a conformal map from the exterior of to the exterior of that sends infinity to infinity. If we define for to be the region enclosed by the Jordan curve , then the are increasing in with conformal radius going to infinity as . If one sets to be the conformal maps with and , then (by the uniqueness of Riemann mapping) and by the Carathéodory kernel theorem, converges locally uniformly to as . In particular, the conformal radii are continuous in . Reparameterising in one can then obtain the required Loewner chain.

Now suppose is only univalent of . As in the proof of Theorem 15, one can express as the locally uniform limit of schlicht functions , each of which extends univalently to some larger disk . By the preceding discussion, each of the extends to a Loewner chain . From the Lipschitz bounds (and the Koebe distortion theorem) one sees that these chains are locally uniformly equicontinuous in and , uniformly in , and hence by Arzela-Ascoli we can pass to a subsequence that converges locally uniformly in to a limit ; one can also assume that the transition functions converge locally uniformly to limits . It is then not difficult by Hurwitz theorem to verify the limiting relations (9), (11), and that is a Loewner chain with as desired.

Suppose that are close to each other: . Then one heuristically has the approximations

and hence by (12) and some rearranging

and hence on applying , (9), and the Newton approximation

This suggests that the should obey the Loewner equation

for some Herglotz function . This is essentially the case:

Theorem 21 (Loewner equation) Let be a Loewner chain. Then, for outside of an exceptional set of Lebesgue measure zero, the functions are differentiable in time for each , and obey the equation (14) for all and , and some Herglotz function for each with . Furthermore, the maps are measurable for every .

Proof: Let be a countable dense subset of . From Lemma 19, the function is Lipschitz continuous, and thus differentiable almost everywhere, for each . Thus there exists a Lebesgue measure zero set such that is differentiable in outside of for each . From the Koebe distortion theorem is also locally Lipschitz (hence locally uniformly equicontinuous) in the variable, so in fact is differentiable in outside of for all . Without loss of generality we may assume contains zero.

Let , and let . Then as approaches from below, we have

uniformly; from (9) and Newton approximation we thus have

which implies that

Also we have

and hence by (12)

Taking limits, we see that the function is Herglotz with , giving the claim. It is also easy to verify the measurability (because derivatives of Lipschitz functions are measurable)

Example 22 The Loewner chain (7) solves the Loewner equation with the Herglotz function . With the rotated Koebe chains (8), we instead have .

Although we will not need it in this set of notes, there is also a converse implication that for every family of Herglotz functions depending measurably on , one can associate a Loewner chain.

Let us now Taylor expand a Loewner chain at each time as

as , we have . As is differentiable in almost every for each , and is locally uniformly continuous in , we see from the Cauchy integral formulae that the are also differentiable almost everywhere in . If we similarly write

for all outside of , then , and we obtain the equations

for every , thus

and so forth. For instance, for the Loewner chain (7) one can verify that and for solve these equations. For (8) one instead has and .

We have the following bounds on the first few coefficients of :

Exercise 23 Let be a Herglotz function with . Let be the measure coming from the Herglotz representation theorem.

• (i) Show that for all . In particular, for all . Use this to give an alternate proof of the upper bound in the Harnack inequality.
• (ii) Show that .

Theorem 24 ( cases of Bieberbach) If is schlicht, then and .

The bound is not new, and indeed was implicitly used many times in the above arguments, but we include it to illustrate the use of the equations (15), (16).

Proof: By Lemma 20, we can write (and ) for some Loewner chain .

We can write (15) as . On the other hand, from the Koebe distortion theorem applied to the schlicht functions , we have , so in particular goes to zero at infinity. We can integrate from to infinity to obtain

From Harnack’s inequality we have , giving the required bound .

In a similar vein, writing (16) as

we obtain

As , we may integrate from to infinity to obtain the identity

Taking real parts using Exercise 23(ii) and (17), we have

Since , we thus have

where . By Cauchy-Schwarz, we have , and from the bound , we thus have

Replacing by the schlicht function (which rotates by ) and optimising in , we obtain the claim .

Exercise 25 Show that equality in the above bound is only attained when is a rotated Koebe function.

The Loewner equation (14) takes a special form in the case of slit domains. Indeed, let be a slit domain not containing the origin, with conformal radius around , and let be the Loewner chain with . We can parameterise so that the sets have conformal radius around for every , in which case we see that must be the unique conformal map from to with and . For instance, for the chain (7) we would have .

Theorem 26 (Loewner equation for slit domains) In the above situation, we have the Loewner equation holding with

for almost all and some measurable .

Proof: Let be a time where the Loewner equation holds. For , the function extends continuously to the boundary, and is two-to-one on the split , except at the tip where there is a single preimage on the unit circle; this can be seen by taking a holomorphic square root of , using a Möbius transformation to map the resulting image to a set bounded by a Jordan curve, and applying Carathéodory's theorem (Theorem 20 from Notes 2) to the resulting conformal map. The image is then with a Jordan arc removed, where is a point on the boundary of the sphere. Applying Carathéodory’s theorem to a holomorphic square root of , we see that extends continuously to be a map from to , with an arc on the boundary mapping (in two-to-one fashion) to the arc , and the endpoints of this arc mapping to . From this and (12), we see that converges to zero outside of the arc , which by the Herglotz representation theorem implies that the measure associated to is supported on the arc . An inspection of the proof of Carathéodory’s theorem also reveals that the are equicontinuous on as , and thus converge uniformly to (which is the identity function) as . This implies that must converge to the point as approaches , and so converges vaguely to the Dirac mass at . Since converges locally uniformly to , we conclude the formula (18). As depends measurably in , we conclude that does also.

In fact one can show that extends to a continuous function , and that the Loewner equation holds for all , but this is a bit trickier to show (it requires some further distortion estimates on conformal maps, related to the arguments used to prove Carathéodory’s theorem in the previous notes) and will not be done here. One can think of the function as “driving force” that incrementally enlarges the slit via the Loewner equation; this perspective is often used when studying the Schramm-Loewner evolution, which is the topic of the next (and final) set of notes.

— 3. The Bieberbach conjecture —

We now turn to the resolution of the Bieberbach (and Robertson) conjectures. We follow the simplified treatment of de Branges’ original proof, due to FitzGerald and Pommerenke, though we omit the proof of one key ingredient, namely the non-negativity of a certain hypergeometric function.

The first step is to work not with the Taylor coefficients of a schlicht function or with an odd schlicht function , but rather with the (normalised) logarithm of a schlicht function , as the coefficients end up obeying more tractable equations. To transfer to this setting we need the following elementary inequalities relating the coefficients of a power series with the coefficients of its exponential.

Lemma 27 (Second Lebedev-Milin inequality) Let be a formal power series with complex coefficients and no constant term, and let be its formal exponential, thus

where is the formal series . Then for any , one has

Proof: If we formally differentiate (19) in , we obtain the identity

extracting the coefficient for any , we obtain the formula

By Cauchy-Schwarz, we thus have

which we can rearrange as

Exercise 28 Show that equality holds in (20) for a given if and only if there is such that for all .

Exercise 29 (First Lebedev-Milin inequality) With the notation as in the above lemma, and under the additional assumption , prove that

(Hint: using the Cauchy-Schwarz inequality as above, first show that the power series is bounded term-by-term by the power series of .) When does equality occur?

Exercise 30 (Third Lebedev-Milin inequality) With the notation as in the above lemma, show that

(Hint: use the second Lebedev-Milin inequality and (21), together with the calculus inequality for all .) When does equality occur?

Using these inequalities, one can reduce the Robertson and Bieberbach conjectures to the following conjecture of Milin, also proven by de Branges:

Theorem 31 (Milin conjecture) Let be a schlicht function. Let be the branch of the logarithm of that equals at the origin, thus one has

for some complex coefficients . Then one has

for all .

Indeed, if

is an odd schlicht function, let be the schlicht function given by (4), then

Applying Lemma 27 with , we obtain the Robertson conjecture, and the Bieberbach conjecture follows.

Example 32 If is the Koebe function (1), then

so in this case and . Similarly, for the rotated Koebe function (2) one has and again . If one works instead with the dilated Koebe function , we have , thus the time parameter only affects the constant term in . This is already a hint that the coefficients of could be worth studying further in this problem.

To prove the Milin conjecture, we use the Loewner chain method. It suffices by Theorem 15 and a limiting argument to do so in the case that is a slit domain. Then, by Theorem 26, is the initial function of a Loewner chain that solves the Loewner equation

for all and almost every , and some function .

We can transform this into an equation for . Indeed, for non-zero we may divide by to obtain

(for any local branch of the logarithm) and hence

Since , is equal to at the origin (for an appropriate branch of the logarithm). Thus we can write

The are locally Lipschitz in (basically thanks to Lemma 19) and for almost every we have the Taylor expansions

and

Comparing coefficients, we arrive at the system of ordinary differential equations

for every .

Fix (we will not need to use any induction on here). We would like to use the system (22) to show that

The most naive attempt to do this would be to show that one has a monotonicity formula

for all , and that the expression goes to zero as , as the claim would then follow from the fundamental theorem of calculus. This turns out to not quite work; however it turns out that a slight modification of this idea does work. Namely, we introduce the quantities

where for each , is a continuously differentiable function to be chosen later. If we have the initial condition

for all , then the Milin conjecture is equivalent to asking that . On the other hand, if we impose a boundary condition

for , then we also have as , since is schlicht and hence is a normal family, implying that the are bounded in for each . Thus, to solve the Milin, Robertson, and Bieberbach conjectures, it suffices to find a choice of weights obeying the initial and boundary conditions (23), (24), and such that

for almost every (note that will be Lipschitz, so the fundamental theorem of calculus applies).

Let us now try to establish (25) using (22). We first write , and drop the explicit dependence on , thus

for . To simplify this equation, we make a further transformation, introducing the functions

(with the convention ); then we can write the above equation as

We can recover the from the by the formula

It may be worth recalling at this point that in the example of the rotated Koebe Loewner chain (2) one has , , and , for some real constant . Observe that has a simpler form than in this example, suggesting again that the decision to transform the problem to one about the rather than the is on the right track.

We now calculate

Conveniently, the unknown function no longer appears explicitly! Some simple algebra shows that

and hence by summation by parts

with the convention .

In the example of the rotated Koebe function, with , the factors and both vanish, which is consistent with the fact that vanishes in this case regardless of the choice of weights . So these two factors look to be related to each other. On the other hand, for more general choices of , these two expressions do not have any definite sign. For comparison, the quantity also vanishes when , and has a definite sign. So it is natural to see of these three factors are related to each other. After a little bit of experimentation, one eventually discovers the following elementary identity giving such a connection:

Inserting this identity into the above equation, we obtain

which can be rearranged as

We can kill the first summation by fiat, by imposing the requirement that the obey the system of differential equations

for ; then we just have

Hence if we also have the non-negativity condition

for all and , we will have obtained the desired monotonicity (25).

To summarise, in order to prove the Milin conjecture for a fixed value of , we need to find functions obeying the initial condition (23), the boundary condition (24), the differential equation (26), and the nonnegativity condition (27), with the convention . This is a significant reduction to the problem, as one just has to write down an explicit formula for such functions and verify all the properties.

Let us work out some simple cases. First consider the case . Now our task is to solve the system

for all . This is easy: we just take (indeed this is the unique choice). This gives the case of the Milin conjecture (which corresponds to the case of Bieberbach).

Next consider the case . The system is now

Again, a routine computation shows that there is a unique solution here, namely and . This gives the case of the Milin conjecture (which corresponds to the case of Bieberbach). One should compare this argument to that in Theorem 24, in particular one should see very similar weight functions emerging.

Let us now move on to . The system is now

A slightly lengthier calculation gives the unique explicit solution

to the above conditions.

These simple cases already indicate that there is basically only one candidate for the weights that will work. A calculation can give the explicit formula:

Exercise 33 Let .

• (i) Show there is a unique choice of continuously differentiable functions that solve the differential equations (26) with initial condition (23), with the convention . (Use the Picard existence theorem.)
• (ii) For any , show that the expression

  

  is equal to when is even and when is odd.

• (iii) Show that the functions

  

  for obey the properties (23), (26), (24). (Hint: for (23), first use (ii) to show that is equal to when is even and when is odd, then use (26).)

The Bieberbach conjecture is then reduced to the claim that

for any and . This inequality can be directly verified for any fixed ; for general it follows from general inequalities on Jacobi polynomials by Askey and Gasper, with an alternate proof given subsequently by Gasper. A further proof of (28), based on a variant of the above argument due to Weinstein that avoids explicit use of (28), appears in this article of Koepf. We will not detail these arguments here.

My course on applied category theory is continuing! After a two-week break where the students did exercises, I went back to lecturing about Fong and Spivak’s book Seven Sketches. The second chapter is about ‘resource theories’.

Resource theories help us answer questions like this:

1. Given what I have, is it possible to get what I want?
2. Given what I have, how much will it cost to get what I want?
3. Given what I have, how long will it take to get what I want?
4. Given what I have, what is the set of ways to get what I want?

Resource theories in their modern form were arguably born in these papers:

• Bob Coecke, Tobias Fritz and Robert W. Spekkens, A mathematical theory of resources.

• Tobias Fritz, Resource convertibility and ordered commutative monoids.

We are lucky to have Tobias in our course, helping the discussions along! He’s already posted some articles on resource theory on the Azimuth blog:

• Tobias Fritz, Resource convertibility (part 1), Azimuth, 7 April 2015.

• Tobias Fritz, Resource convertibility (part 2), Azimuth, 10 April 2015.

• Tobias Fritz, Resource convertibility (part 3), Azimuth, 13 April 2015.

In the course, we had fun bouncing between the relatively abstract world of monoidal preorders and their very concrete real-world applications to chemistry, scheduling, manufacturing and other topics. Here are the lectures:

• Lecture 18 - Chapter 2: Resource Theories
• Lecture 19 - Chapter 2: Chemistry and Scheduling
• Lecture 20 - Chapter 2: Manufacturing
• Lecture 21 - Chapter 2: Monoidal Preorders
• Lecture 22 - Chapter 2: Symmetric Monoidal Preorders
• Lecture 23 - Chapter 2: Commutative Monoidal Posets
• Lecture 24 - Chapter 2: Pricing Resources
• Lecture 25 - Chapter 2: Reaction Networks
• Lecture 26 - Chapter 2: Monoidal Monotones
• Lecture 27 - Chapter 2: Adjoints of Monoidal Monotones
• Lecture 28 - Chapter 2: Ignoring Externalities
• Lecture 29 - Chapter 2: Enriched Categories
• Lecture 30 - Chapter 2: Preorders as Enriched Categories
• Lecture 31 - Chapter 2: Lawvere Metric Spaces
• Lecture 32 - Chapter 2: Enriched Functors
• Lecture 33 - Chapter 2: Tying Up Loose Ends

I’d completed about half my candidacy exam.

Four Caltech faculty members sat in front of me, in a bare seminar room. I stood beside a projector screen, explaining research I’d undertaken. The candidacy exam functions as a milepost in year three of our PhD program. The committee confirms that the student has accomplished research and should continue.

I was explaining a quantum-thermodynamics problem. I reviewed the problem’s classical doppelgänger and a strategy for solving the doppelgänger. Could you apply the classical strategy in the quantum problem? Up to a point. Beyond it, you’d need

“Does anyone here like the Beatles?” I asked the committee. Three professors had never participated in an exam committee before. The question from the examinee appeared to startle them.

One committee member had participated in cartloads of committees. He recovered first, raising a hand.

The committee member—John Preskill—then began singing the Beatles song.

In the middle of my candidacy exam.

The moment remains one of the highlights of my career.

Throughout my PhD career, I’ve reported to John. I’ve emailed an update every week and requested a meeting about once a month. I sketch the work that’s firing me, relate my plans, and request feedback.

Much of the feedback, I’ve discerned over the years, condenses into aphorisms buried in our conversations. I doubt whether John has noticed his aphorisms. But they’ve etched themselves in me, and I hope they remain there.

“Think big.” What would impact science? Don’t buff a teapot if you could be silversmithing.

Education serves as “money in the bank.” Invest in yourself, and draw on the interest throughout your career.

“Stay broad.” (A stretching outward of both arms accompanies this aphorism.) Embrace connections with diverse fields. Breadth affords opportunities to think big.

“Keep it simple,” but “do something technical.” A teapot cluttered with filigree, spouts, and eighteen layers of gold leaf doesn’t merit a spot at the table. A Paul Revere does.

“Do what’s best for Nicole.” I don’t know how many requests to speak, to participate on committees, to explain portions of his lecture notes, to meet, to contribute to reports, and more John receives per week. The requests I receive must look, in comparison, like a mouse to a mammoth. But John exhorts me to to guard my time for research—perhaps, partially, because he gives so much time, including to students.

“Move on.” If you discover an opportunity, study background information for a few months, seize the opportunity, wrap up the project, and seek the next window.

John has never requested my updates, but he’s grown used to them. I’ve grown used to how meetings end. Having brought him questions, I invite him to ask questions of me.

“Are you having fun?” he says.

I tell the Beatles story when presenting that quantum-thermodynamics problem in seminars.

“I have to digress,” I say when the “Help!” image appears. “I presented this slide at a talk at Caltech, where John Preskill was in the audience. Some of you know John.” People nod. “He’s a…mature gentleman.”

I borrowed the term from the apparel industry. “Mature gentleman” means “at a distinguished stage by which one deserves to have celebrated a birthday of his with a symposium.”

Many physicists lack fluency in apparel-industry lingo. My audience members take “mature” at face value.

Some audience members grin. Some titter. Some tilt their heads from side to side, as though thinking, “Eh…”

John has impact. He’s logged boatloads of technical achievements. He has the scientific muscle of a scientific rhinoceros.

And John has fun. He doesn’t mind my posting an article about audience members giggling about him.

Friends ask me whether professors continue doing science after meriting birthday symposia, winning Nobel Prizes, and joining the National Academy of Sciences. I point to the number of papers with which John has, with coauthors, electrified physics over the past 20 years. Has coauthored because science is fun. It merits singing about during candidacy exams. Satisfying as passing the exam felt two years ago, I feel more honored when John teases me about my enthusiasm for science.

A year ago, I ate lunch with an alumnus who’d just graduated from our group. Students, he reported, have a tradition of gifting John a piece of art upon graduating. I relayed the report to another recent alumnus.

“Really?” the second alumnus said. “Maybe someone gave John a piece of art and then John invented the tradition.”

Regardless of its origin, the tradition appealed to me. John has encouraged me to blog as he’s encouraged me to do theoretical physics. Writing functions as art. And writing resembles theoretical physics: Each requires little more than a pencil, paper, and thought. Each requires creativity, aesthetics, diligence, and style. Each consists of ideas, of abstractions; each lacks substance but can outlive its creator. Let this article serve as a finger painting for John Preskill.

Thanks for five fun years.

With my PhD-thesis committee, after my thesis defense. Photo credit to Nick Hutzler, who cracked the joke that accounts for everyone’s laughing. (Left to right: Xie Chen, Fernando Brandão, John Preskill, Nicole Yunger Halpern, Manuel Endres.)