The n-Category Café

Algebraic Topology lets robots lift things they haven’t seen before, discovers when sensors completely cover an area, discovers new subtypes of breast cancer, classifies basketball players and discovers new methods for fraud detection.

One of the currently very hot topics is compressed sensing, which began with ideas from probability and Banach space geometry and is resulting in faster MRIs, digital cameras which take pictures that can be refocused after being taken, and other such exciting things. (Though this field is moving so fast that now — less than ten years after its birth — it’s already making the transition from being “mathematics” to being “engineering”.) There are millions of links (well, maybe just hundreds of thousands) and it’s not easy to find much written for an audience with no background in signal processing, but here’s a link that might interest you.

My personal favorite unexpected mathematical connection is the (still unexplained, but nevertheless extremely fruitful) connection between eigenvalues of random matrices and zeros of the Riemann zeta function. This may be old news now — certainly everyone working in either random matrix theory or in number theory knows it well, even if they don’t work on this aspect of things — but nobody saw it coming ahead of time. Try googling “Dyson Montgomery Riemann” for many retellings of how that story got started.

Just repeat something by Atiyah ;)

How about The Unity of Mathematics?

Well, rather close to home there is something you might know about: magnitude of metric spaces. That connects category theory and combinatorics (the Euler characteristic of a poset) with metric spaces, biological diversity, entropy, differential geometry, integral geometry, analysis, and probably some other stuff.

If you can count the theoretical side of computing science as maths… One thing that struck me as being remarkable, at least until I understood it (thanks) is how Kleisli categories are exactly what’s needed to handle computation and effects in functional programming. It seems “obvious” that simple equational reasoning can’t be used when you have mutable state (x used to be 3, and now it is 4), functions that can fail, input and output and so on. It’s a very pleasant surprise to find out that with the help of a little bit of category theory you can go right back to using everything you know from high school algebra to prove properties of programs.

Here are a couple of relevant MathOverflow threads that I’d forgotten about:

Real-world applications of mathematics, by arxiv subject area

Your favorite surprising connections in Mathematics

I’m also coming more from the computer science side of things, but we do have some doozies.

One of my favorites is what Russell O’Connor dubs the Gauss-Jordan-Floyd-Warshall-McNaughton-Yamada algorithm, though names like Roy, Warshall, Kleene, and Thompson could be included as well. That is, the Gauss–Jordan (1887) algorithm for solving linear algebra problems, the Roy–Floyd-Warshall (1959; 1962a–c) algorithm for computing all-pairs shortest path on weighted graphs (also for taking transitive closures of graphs/relations), and the McNaughton-Yamada (1960) algorithm for constructing finite automata from their outputs, are all in fact the exact same algorithm just with different choices of *-semiring (i.e. closed semiring, not semiring with involution). As far as reinventing the wheel, this one is up there with the constant rediscovery of calculus (with variants of Gaussian elimination going back to 150BCE–179CE as well as being rediscovered by Newton in 1670). And yet, unlike calculus, the connections here are often missed by modern audiences.

Similarly, it took a terrifyingly long time for for the A*-search algorithm (1968) popular in artificial intelligence/machine learning to finally cross the hall (A*-parsing, 2003) into natural language processing, despite the fact that the connection is glaringly obvious— the parsing problem in NLP requires the exact same kind of search as path-finding in ML. It may seem passe, since these days we recognize the huge overlap between NLP and ML, but evidently this was not always so obvious.

The connection between category theory and type theory is very fruitful these days— the connection between co/monads and effects being one of the more popular unexpected connections. The Curry–Howard correspondence is another big one; although it’s now widely known, it’s still a fruitful area for drawing connections between logic and programming. And the connection between domain theory and semantics is still alive; there’s the classic work by Dana Scott on handling partially-defined terms, but there’s also more recent work on things like deterministic parallelism (LVars/LVish).

How about Sturmfels at al.’s Algebraic Statistics? There should be some good applications in Algebraic Statistics for Computational Biology.

How about this Real Life Story${}^{\text{TM}}$? When I was a PhD student at Edinburgh doing knots and topology, I went to an Edinburgh Mathematical Society talk in St Andrews on Solar Physics given by Mitchell Berger. He was trying to calculate braiding in magnetic field lines of solar flares by defining combinatorial ‘higher order winding numbers’. I realized that the rational homotopy theory and finite type invariants of braids that I’d be thinking about could be used to tackle this kind of problem. The resulting ideas ended up forming half of my thesis.

Here is a pair of separate applied problems that surprisingly have the same underlying combinatorial principles organizing their solutions. One is the problem of classifying colliding waves in a shallow puddle, the other of organizing the likely genetic histories of a collection of related species. Compare the pictures for a quick impression: Figure 1 of Levy and Pachter with Figure 34 of Kodama and Williams.

It was fun to hear the two papers presented in succession at a conference a couple of years ago: Pachter joked that he could simply recycle the graphics from the title slide from Williams’s talk.

Bonus: the colliding wave picture has something very much in common with Postnikov’s amplituhedron , but I am in no position to comment more precisely!

Thanks very much to everyone for your suggestions, including the ones I didn’t use (which was mostly out of ignorance — in order to stand up and talk about something in front of several dozen people, I need to have at least a tiny sense of knowing what I’m talking about).

In the end, I decided to make these my main topics:

• algebraic topology of data sets (persistent homology etc.)
• topology applied to DNA (un)knotting
• random matrices and Riemann’s $\zeta$
• statistical inference as a branch of logic
• homotopy type theory.

You can see my slides here, though I suspect they won’t entirely make sense without the extensive waffle that accompanied them.

A very fruiful source of unexpected connections is the subject of expander graphs. To someone who loves self-referentiality, it will be even more intruiging to know that expander graphs are themselves defined through their high connectivity!

My own understanding is way too shallow to explain anything in detail, so I can only quote from an expository paper of Lubotzky,

Various deep mathematical theories have been used to give explicit constructions, e.g., the Kazhdan property (T) from representation theory of semisimple Lie groups and their discrete subgroups, the Ramanujan Conjecture (proved by Deligne) from the theory of automorphic forms, and more. [..]

We now start to see applications in the opposite direction: from expander graphs to number theory. The most notable one is the development of the affine sieve method. [..]

We will describe several results about groups whose formulations do not mention expanders but expanders come out substantially in the proofs. [..]

several ways in which expanders have appeared, somewhat unexpectedly, in geometry. Most of these applications are for hyperbolic manifolds.

and Wikipedia,

The original motivation for expanders is to build economical robust networks (phone or computer): an expander with bounded valence is precisely an asymptotic robust graph with number of edges growing linearly with size (number of vertices), for all subsets.

Expander graphs have found extensive applications in computer science, in designing algorithms, error correcting codes, extractors, pseudorandom generators, sorting networks and robust computer networks. They have also been used in proofs of many important results in computational complexity theory, such as SL=L and the PCP theorem. In cryptography, expander graphs are used to construct hash functions.

Lubotzky’s book also explains connections to the Banach-Tarski paradox via the Ruziewicz problem.

Very interesting thread, and nice slides.

Each of all these interesting connections looks
like worth a lifetime study. But there seem to
be thousands of such interesting connections!
Perhaps we should ask ourselves what are the
connections between all these unexpected
connections?

Cheers,
Joachim.

PS: I like Mike’s remark about infinity-groupoids.

PPS: Re 404: this means that the server did not
expect the connection.

The day after I gave my talk, I got an email from science writer Monica Kortsha at the University of Texas, pointing me to a story she’d written on the interplay between numerical analysis and engineering — which contains a surprise.

Category theory meets music meets materials science in the newscientist: www.newscientist.com/article/mg22129540.200-stuff-symphony-beautiful-music-makes-better-materials.html

(There’s also a book published by springer called “Biomateriomics” which goes into much more depth)