## Applied Algebraic Topology 2017

In the comments on this blog post I’m taking some notes on this conference:

Applied Algebraic Topology 2017, August 8-12, 2017, Hokkaido University, Sapporo, Japan.

Unfortunately these notes will not give you a good summary of the talks—and almost nothing about applications of algebraic topology. Instead, I seem to be jotting down random cool math facts that I’m learning and don’t want to forget.

### 31 Responses to Applied Algebraic Topology 2017

1. John Baez says:

A young mathematician came up after my talk and asked me: “What did Gromov mean when he said the brain is a 2-category?”

She was not able to offer any clues, so I said I had no idea. I said I might ask him. If anyone knows where he said this, that would help a lot.

• retlav46 says:

I found this: “Probably, the full concept of ”ergo-brain“ can be expressed in a language similar to that of n-categories” (p. 53 at https://www.ihes.fr/~gromov/PDF/ergobrain.pdf)

• John Baez says:

Thanks! That’s a quite interesting screed by Gromov—I hadn’t seen it. It has some nice remarks on category theory, but I haven’t seen any real detail on n-categories. The full sentence is:

Probably, the full concept of “ergo-brain” can be expressed in a language similar to that of n-categories (see ???) which make an even more refined (than categories/functors) tool for dealing with “arbitrariness and ambiguity”.

where he hasn’t put in the reference for ???.

2. John Baez says:

I’m listening to a talk about biochemical networks, how to simplify them, how to extract dynamical systems from them, and how to test these models:

Konstantin Mischaikov, Towards a computational, discrete geometric foundation for nonlinear dynamics.

Abstract. The current paradigm for nonlinear dynamics focuses on the existence and structure of invariant sets. As over a century of work shows this is an incredibly rich subject and perhaps, from the perspective of modern applications, too rich. For example, in general, invariant sets are not computable and structurally stable systems are not generic. As consequence it appears difficult to develop a natural methodology for analyzing dynamics in the context of scientific challenges driven by data as opposed to mathematical models.

In this talk I will describe an alternative approach to dynamics based on using order structures to identify gradient versus recurrent-like behavior and algebraic topology to characterize local and global structures of the dynamics. Using regulatory networks arising from systems biology as motivation I will demonstrate some advantages of this approach: it allows for finite queryable descriptions of global dynamics; it leads to natural decompositions of parameter space; it lends itself to efficient computations; and the results lead to mathematically rigorous statements about the possible dynamics of more traditional models based on differential equations.

3. John Baez says:

If there’s a binomial identity like

$\binom{6}{0} + \binom{6}{3} = \binom{6}{1} + \binom{6}{2}$

where all the numbers on the bottom are distinct (in this case 0, 3, 1 and 2), then you can write a protocol for ‘weak symmetry breaking’ that uses just 3 communication steps: that is, a way to get $n$ people to collectively make a binary choice under certain constraints, including that they can send messages just 3 times:

• D. N. Kozlov, All binomial identities are orderable.

Such identities exist for infinitely many $n,$ but they also fail to exist for infinitely many $n.$

When we have an identity of this sort, it means there’s a bijection between some collection $A$ of subsets of an $n$-element and some other collection $B.$ In the above example, there’s a bijection between the collection of 0- and 3-element subsets of a 6-element set, and the collection of 1- and 2-element subsets.

That much is obvious. Less obviously, Kozylov proved that when we have such an identity we can choose a bijection, say

$\Phi : A \to B$

such for any $S \in A$ we either have $\Phi(S) \subseteq S$ or $S \subseteq \Phi(S).$ That’s pretty cool!

4. John Baez says:

Here’s a juicy little fact from this talk:

• Yasuaki Hiraoka, Materials topological data analysis and random/statistical topology.

Suppose you take the complete graph on $n$ vertices and randomly assign a number between 0 and 1 to each edge. Call these numbers lengths, and let them be independent and uniformly distributed random variables.

What is the total length of the minimal (= shortest possible) spanning tree?

Of course, this is a random variable. But Frieze showed that as $n \to \infty,$ it converges in probability to

$\zeta(3) = \frac{1}{1^3} + \frac{1}{2^3} + \frac{1}{3^3} + \cdots$

In other words, for any $\epsilon > 0,$ the probability that this total length differs from $\zeta(3)$ by more than $\epsilon$ approaches zero as $n \to \infty.$

I find this to be a wonderfully fundamental appearance of Apery’s constant $\zeta(3).$ It was proved by Frieze in 1983, and later published here:

• Alan M. Frieze, On the value of a random minimum spanning tree problem, Discrete Applied Mathematics 10 (1985), 47–56.

He actually proved a more general result that applies to non-uniform probability distributions on $[0,L]$ obeying some conditions, and this was generalized further here:

• J. Michael Steele, On Frieze’s zeta(3) limit for lengths of minimal spanning trees, Discrete Applied Mathematics 18 (1987), 99–103.

One gets a simple explicit formula involving $\zeta(3)$ that only depends on the probability distribution for very small lengths. The last part makes perfect sense, since small lengths are the most important for finding the shortest spanning tree! But the appearance of $\zeta(3)$ is still mysterious to me.

• John Baez says:

“Here’s another really nice way that Apery’s constant shows up. It’s the reciprocal of the probability that 3 positive integers chosen at random are relatively prime!”

Akira Bergman wrote:

Apparently this generalizes to arbitrary positive integer n, as in ζ(n). Is there a graph interpretation for each n, or just for n=3?

I wrote:

I don’t know a graph interpretation of the other ζ(n), and this seems hard because the number “3” doesn’t appear in the puzzle “what’s the total length of the minimal spanning tree of a large complete graph with each edge weighted by a random number between 0 and 1?” At least not in any obvious way!

But I’m not an expert on this stuff.

Matt McIrvin wrote:

Appearances that don’t obviously involve the number 3 seem far more remarkable. Perhaps if we proceed from graphs to creatures with higher-dimensional pieces, like polytopes and honeycombs, we can get analogues with other integers in them?

I wrote:

That’s a fun thought. I don’t know what the analogue of a spanning tree is, in these higher-dimensional ‘cell complexes’. However, I know that Kirchhoff’s matrix tree theorem has an analogue in higher dimensions – I just don’t know what this analogue is.

Kirchhoff’s matrix tree theorem says that if you take the graph Laplacian (the obvious discretization of the Laplace operator) on a connected graph, and take the product of its nonzero eigenvalues, this is the number of spanning trees in the graph times the number of vertices.

More generally you can compute various things about circuits made of resistors by summing over spanning trees—presumably that’s how Kirchhoff got into this game. This has been on my mind a lot lately.

You can generalize the graph Laplacian to higher-dimensional cell complexes, so presumably the generalization of Kirchhoff’s matrix tree theorem to higher dimensions involves higher-dimensional analogues of spanning trees.

Okay, now I’ve psyched myself up for reading this paper!

• Art M. Duval, Caroline J. Klivans and Jeremy L. Martin, Simplicial and cellular trees.

This chapter is about the more recent theory of spanning trees in cell complexes, which are natural higher-dimensional analogues of graphs. The theory for cell complexes parallels the graph-theoretical version in many ways, including the connection to matroid theory. However, higher-dimensional spaces can have much richer topology, which complicates the algebraic and enumerative parts of the story in a very concrete way. It turns out that the simple number of spanning trees of a cell complex is not a well-behaved invariant, and it is better to account for topological complexity in the enumeration. On the other hand, many essential algebraic tools for working with spanning trees of a graph do extend well to arbitrary dimension.

• The link to Simplicial and Cellular Trees appears to be broken.

• John Baez says:

Thanks—I’ve fixed it. It had an unnecessary period at the end.

• John Baez says:

So then I read that paper….

It turns out Gil Kalai did some crucial work in figuring out the generalization of spanning trees to higher dimensions, and also Kirchhoff’s matrix tree theorem, and also the right generalization of a complete graph, and the generalization of Cayley’s formula counting the spanning trees in a complete graph.

The d-dimensional generalization of a complete graph on n vertices is pretty simple: take all the d-dimensional faces of an (n-1)-simplex. The generalization of a spanning tree is not so simple: a beautiful generalization of Cayley’s formula says there should be

$n^{\binom{n - 2}{d}}$

of them, but with a naive definition of this formula fails with n = 6, d = 2: we have

$n^{\binom{n - 2}{d}} = 6^6 = 46656$

while there are only 46620 ‘spanning hypertrees’: ways of choosing triangles in the 5-simplex such that every edge lies in some triangle and the integral 2nd homology of the resulting space vanishes.

Thus, Kalai had to invent a subtler definition to save the formula. In a blog article on this topic he writes:

With such a nice conjecture we should not take no for an answer.

The answer is that we should call a collection of d-dimensional faces of a simplex a spanning hypertree if:

1) every (d-1)-dimensional face lies on one of these d-dimensional faces (that’s the spanning part), and

2) the dth rational homology of the space formed by these d-dimensional faces vanishes.

The subtlety is that we use rational homology: homology with rational coefficients (or if you prefer, real coefficients). The homology with integer coefficients may still be nonzero! If the dth rational homology vanishes, the dth integral homology is a finite group G, and we should count each spanning hypertree |G|2 times if we want the total number of spanning hypertrees of an (n-1)-simplex to come out to

$n^{\binom{n-2}{d}}$

This is Kalai’s result.

So maybe someone should try this:

Take an (n-1)-simplex and label all its d-dimensional faces with numbers in [0,1], randomly chosen from independent uniform distributions. Define the volume of a spanning hypertree to be the sum of these numbers over all d-dimensional faces of the hypertree. Define a minimal spanning hypertree to be a spanning hypertree of minimal volume.

Calculate the expected value of the volume of a minimal spanning hypertree. Take the limit as n → ∞.

• John Baez says:

Here’s how $\zeta(3)$ shows up as the expected length of the minimal spanning tree:

$\displaystyle{ 2 \sum_{k = 1}^\infty \frac{k^{k-2}}{k!} \int_0^\infty (2a)^{k-1} e^{-2ak} \, da = \sum_{k = 1}^\infty \frac{1}{k^3}}$

I don’t really understand how the left hand side shows up in this problem, but Cayley’s formula says that $k^{k-2}$ is the number of spanning trees with $k$ vertices.

• arch1 says:

Thanks John. I’m pretty sure the math behind the spanning tree result is beyond me but have some elementary observations which I hope aren’t all wet.
1) Given one of these complete length-assigned n-graphs, if we try to choose a spanning subset of arcs by picking the n vertices in random order, choosing for each vertex the shortest arc from that vertex to any other vertex, then the expected total length of the chosen arcs will be n/(n+1) (right?), which approaches 1 for large n. Alas in general the chosen arcs won’t all be connected, so we’re not guaranteed a spanning tree.
2) If we modify the above algorithm by replacing “any other” with “already-picked”, we do get a spanning tree, but now the chosen arcs have an expected total length = the sum over k=1 to n of 1/(k+1) , which diverges as log n. Of course there’s no reason to expect this ‘greedy’ spanning tree algorithm to be optimal, so no conflict with the result you reported.
3) What is surprising to me is that the optimal spanning tree (whose expected length obviously lies between the above two values, 1 and log n) comes so close to the former: The constraint of having to span all n vertices has only a fixed cost relative to the unconstrained case #1 (and a modest one at that, only 20%).

• John Baez says:

I’m sorry, I don’t have much intuition for this stuff, so I don’t have anything interesting to say! Maybe someone else does?

• Graham Jones says:

I think you can get a rough estimate using the theory about giant components in random graphs. See Theorem 16.1 from https://people.eecs.berkeley.edu/~sinclair/cs271/n16.pdf. (In statement 2, that should be (0,1] not [0,1] because b=0 is always a solution.)

First use all edges smaller than 2c/n. There will be about cn of them, so for c>1, there will be a giant component C of size bn where b + exp(-bc) – 1 = 0. Delete edges from C until it is a tree T. The sum of weights in T is about bn(c/n) = bc.

Next, use edges with weights in [2c/n,1] to join G-T to T, using the edge with smallest weight for each vertex in G-T. There are about bn choices for each one, so these weights add up to about (1-b)n (2c/n + 1/(bn)) = (1-b)(2c_1/b). The total is bc + 2c – 2cb + 1/b – 1 = 2c – 1 – bc – 1/b. For example, if c=2, b is about .8 and this is 3-1.6+1.25=2.65.

• John Baez says:

Thanks, Graham! That gives a nice intuition for why the answer works out to be finite. Now that I reread Frieze’s paper I see some of this intuition lurking between the lines:

• Alan M. Frieze, On the value of a random minimum spanning tree problem, Discrete Applied Mathematics 10 (1985), 47–56.

Perhaps this was before people knew about, or talked about, the ‘giant component’ of a random graph.

• Graham Jones says:

I thought I’d heard about the ‘phase transition’ at c=1 as edges are added to an empty graph ages ago, possibly via you! So I did a little digging. Erdős and Rényi (1960) mention ‘giant’ components. https://www.renyi.hu/~p_erdos/1961-15.pdf

• arch1 says:

Thank you Graham and John. Graham you seem to have lowered the comprehension bar sufficiently that on a good day I may be able to clear it. Maybe this Saturday:-)

5. John Baez says:

• Satohiro Tajima, Toru Yanagawa, Naotaka Fujii and Taro Toyoizumi, Untangling brain-wide dynamics in consciousness by cross-embedding, PLOS Computational Biology.

Abstract. Brain-wide interactions generating complex neural dynamics are considered crucial for emergent cognitive functions. However, the irreducible nature of nonlinear and high-dimensional dynamical interactions challenges conventional reductionist approaches. We introduce a model-free method, based on embedding theorems in nonlinear state-space reconstruction, that permits a simultaneous characterization of complexity in local dynamics, directed interactions between brain areas, and how the complexity is produced by the interactions. We demonstrate this method in large-scale electrophysiological recordings from awake and anesthetized monkeys. The cross-embedding method captures structured interaction underlying cortex-wide dynamics that may be missed by conventional correlation-based analysis, demonstrating a critical role of time-series analysis in characterizing brain state. The method reveals a consciousness-related hierarchy of cortical areas, where dynamical complexity increases along with cross-area information flow. These findings demonstrate the advantages of the cross-embedding method in deciphering large-scale and heterogeneous neuronal systems, suggesting a crucial contribution by sensory-frontoparietal interactions to the emergence of complex brain dynamics during consciousness.

6. Blake Stacey says:

This thread is full of fun and interesting things!

• John Baez says:

Thanks! I just added an apology at the top, saying that I’m not really summarizing the talks—just taking notes on random cool facts that I’m learning here and don’t want to forget. Glad you like ’em.

7. Ilyas says:

Dear John, do you know if there is likely to be a video link for the talks? I know it’s not always the case in Japan, but if you happen to know I would appreciate it. I cannot tell from the workshop website.

• John Baez says:

Hi! No, unfortunately they are not making videos of the talks or even collecting the speakers’ slides on a website.

• John Baez says:

I got an email today from the conference organizers asking for my slides. So, many speakers’ slides will eventually appear on the conference website.

8. John Baez says:

Nice work on network motifs in neuroscience, good for me to learn so I can start thinking about the network dynamics of neurons:

• Sen Song, Per Jesper Sjöström, Markus Reigl, Sacha Nelson, and Dmitri B Chklovskii, Highly nonrandom features of synaptic connectivity in local cortical circuits, PLOS Biology (2005).

• Michael W. Reimann, Max Nolte, Martina Scolamiero, Katharine Turner, Rodrigo Perin, Giuseppe Chindemi, Paweł Dłotko, Ran Levi, Kathryn Hess and Henry Markram, Cliques of neurons bound into cavities provide a missing link between structure and function, Frontiers in Computational Neuroscience 11 (2017), 48.

One coauthor on the second one is my friend the homotopy theorist Kathryn Hess, who invited me to this conference!

• John Baez says:

I heard about the above papers in this great talk:

Carina Curto, Topological organization of neural networks.

Abstract. Neural networks serve as data summaries and dynamic models of neural activity in the brain. Recently, methods from topological data analysis have been used to gain insights into the structure of such networks, using various types neural activity data. In this talk, we will turn our attention to how network structures uncovered by topological methods might shape dynamics. We will illustrate these ideas in the context of threshold-linear networks of simple neurons, whose dynamics are controlled purely by the pattern of connectivity as defined by a directed graph. This enables us to study directly the role of connectivity in shaping network dynamics, without worrying about effects stemming from the intrinsic properties of neurons. Here we find some interesting connections between topologically-relevant features of the network structure and its dynamic attractors. We also identify some aspects of the connectivity that are not picked up by standard tools, but may be amenable to new types of topological analyses.

It was a great talk because she gave a simple recipe to get ODEs from directed graphs, based on ideas of how neurons affect the firing of their neighbors, and has figured out how to read off qualitative properties of the ODE from the graph! She calls these ODEs, or maybe the graphs they come from, combinatorial threshold-linear networks.

Theorem. If the graph has no sinks, then the ODE has no stable fixed points.

Theorem. A clique is the support of a stable fixed point iff it is a clique with no target (i.e. a vertex outside the clique such that every vertex in the clique has an edge going to that vertex).

Her work is here:

• Katherine Morrison, Anda Degeratu, Vladimir Itskov and Carina Curto, Diversity of emergent dynamics in competitive threshold-linear networks: a preliminary report.

and here’s a poster summarizing it:

• Katherine Morrison, Anda Degeratu, Vladimir Itskov and Carina Curto, Emergent dynamics from network connectivity: a minimal model.

• John Baez says:

Carina Curto also has a link to tunes made by neural networks.

• Blake Stacey says:

I worked on motifs in (undirected) graphs, ages ago. The best idea I had was the following:

Let $p$ be the probability that two vertices chosen at random from a graph are connected by an edge, and let $C$ be the clustering coefficient, i.e., the probability that two vertices which have a common neighbor are themselves directly linked by an edge. Then, given a pattern of connections among a set of four vertices, one can make an educated guess about how many times that pattern will occur in the graph overall, by multiplying together some combination of $p$ and $C$ and $(1-p)$ and $(1-C)$, with some symmetry factor. I think I got the formulae right in the end, because they made pretty good predictions for motif abundances in a certain class of graphs. Specifically, this worked well for random spatial graphs, made by sprinkling vertices randomly throughout a 2- or 3-dimensional box and connecting them when they fall within a threshold distance of each other. I never made a satisfactory write-up of this, mostly because I couldn’t find where to go with the idea next, but I did give a couple talks about it. In fact, back in 2013 I gave an hour-long lecture on “Mesoscale Structure in Complex Networks”, in which I described several “solutions looking for a problem”: ways of characterizing graphs that sounded interesting and seemed well-motivated at the time, but for which I’d never found a truly incisive application. And the general idea of a system having “structure at a scale of two components or three components, but no further structure at scale four” did feed into stuff that I worked on later.

• John Baez says:

You might be interested in this:

• John Baez, Networks and population biology (Part 4), 6 May 2011.

It’s about an attempt by Persi Diaconis to define random graphs where certain features show up. Here’s a teaser, which leaves out the shocking conclusion:

People have studied the Erdős–Rényi random graphs very intensively, so now people are eager to study random graphs with more interesting correlations. For example, consider the graph where we draw an edge between any two people who are friends. If you’re my friend and I’m friends with someone else, that improves the chances that you’re friends with them! In other words, friends tend to form ‘triangles’. But in an Erdős–Rényi random graph there’s no effect like that.

‘Exponential families’ of random graphs seem like a way around this problem. The idea here is to pick a specific collection of graphs $H_1, \dots, H_k$ and say how commonly we want these to appear in our random graph. If we only use one graph $H_1,$ and we take this to be two vertices connected by an edge, we’ll get an Erdős–Rényi random graph. But, if we also want our graph to contain a lot of triangles, we can pick $H_2$ to be a triangle.

His approach seems very well-motivated by statistical mechanics. However, it doesn’t work! But the way it fails is itself interesting.

• Blake Stacey says:

Interesting failures can be quite instructive!

• Blake Stacey says:

And, funnily enough, the two papers I linked at the end there bring me around to the subject of Tom Leinster’s talk.

While I was working on the “multiscale information theory” effort described in the aforelinked papers, I tried to come up with a simpler setting for the basic ideas, one that would illustrate the basics and wouldn’t require taking logarithms of probabilities all over the place. Take a set $Q$ of questions, all of which pertain to some physical system—to keep it simple, make them all binary questions. We suppose that each question $q \in Q$ carries one unit of information; or, to say it another way, answering any $q \in Q$ removes one unit of uncertainty. Furthermore, we suppose that the questions are independent, in that answering any $q \in Q$ does not help us to answer any other $q' \in Q$.

Our physical system may contain multiple pieces, or components; call them $a$, $b$, $c$ and so forth. Some of the questions in $Q$ may pertain to part $a$, but not all of them have to, so let’s call $Q_a$ the subset of $Q$ consisting of questions that each pertain to component $a$. Likewise for $Q_b$ and $Q_c$. So, the total information content of component $a$ is $H(a) = |Q_a|$: This is the total number of questions which we must get the answers to in order to remove our uncertainty about $a$. Then, the mutual information between components $a$ and $b$ is

$I(a;b) = |Q_a \cap Q_b| = H(a) + H(b) - H(a,b).$

We can go further and define a tertiary mutual information $I(a;b;c)$ in the same manner, and we can do even fancier things as the number of components in the system grows larger. This provides a kind of “toy version” of information theory: It doesn’t have the richness of Shannon theory, but it does furnish functions that satisfy some of the important properties of Shannon information.

So, keeping the familiar concepts of Shannon theory firmly set aside for the moment, in this more abstract setting, a natural puzzle arises. What if the questions in the set $Q$ are informative about each other? Maybe we want to keep probability theory at a distance for now, because it doesn’t feel natural to apply. Perhaps our notion of “similarity” between questions is something like, “Given the answer to question $q \in Q$, we could conceivably compute the answer to question $q'$, given sufficient computer time and memory.” So, the first thing we try is to quantify this with a notion of “distance” $d(q,q')$. Why should this work like a distance? Well, if $d(q,q')$ indicates how hard it is to calculate the answer to $q'$ given an answer to $q$, and $d(q',q'')$ quantifies the difficulty of computing an answer to $q''$ given one for $q'$, then the worst-case scenario for calculating an answer for $q''$ starting with $q$ feels like it should be the sum total difficulty for the two-step path, $d(q,q') + d(q',q'')$.

Thus, we put a metric on $Q$, and we can talk about the magnitude of $Q$, as well as of the subspaces within it. All our measures of information pick up a scale parameter $t$, which indicates the cost of computation.

Fun to think about, maybe, as a setting where various abstract ideas can be applied.

This site uses Akismet to reduce spam. Learn how your comment data is processed.