Back from San Diego, recovering from the redeye. It was a terrific Joint Math Meetings this year; I saw lots of old friends and great talks, but had to miss a lot of both, too.

A couple of people asked me for the slides of my talk, “How To Count with Topology.” Here they are, in .pdf:

How To Count With Topology

If you find this stuff interesting, these blog posts give a somewhat more detailed sketch of the papers with Venkatesh and Westerland I talked about.

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Nice talk. The situation as summed up on your last slide (“Topology as conjecture machine”) reminds me strongly of the philosophy of random matrix theory as a conjecture machine in analytic number theory.

Actually, there’s a very close connection. There’s a parallel tradition, starting with Friedman and Washington, of developing heuristics for conjectures of this kind via the statistics of l-adic random matrices. Part of the story, which I didn’t have time to put in the talk, is that in situations where both machines make a prediction, they are typically the same — so the topological story serves to validate random matrix heuristics. But there are some situations (e.g. those treated in my paper with Jain and Venkatesh about lambda-invariants, or in my paper with Cais and Zureick-Brown about random Dieudonne modules) where I don’t know how to tell a topology story, only a random matrix story, and some (like Batyrev-Manin) where I don’t know how to tell a random matrix story, only a topology story. Maybe I’ll make another post about this at some point!

I agree, these slides are very nice! I’d have been happy to attend the talk…

A minor addendum to the slides : explicit error terms in the Davenport-Heilbronn theorem were known before the papers which are mentioned (the first due to Belabas, Bhargava and Pomerance, but also, already around 1995, Belabas and Fouvry add versions on average over arithmetic progressions to perform sieve for quadratic number fields with almost prime discriminant and small 3-rank.)

Concerning the previous comment, there are definitely connections between these types of conjectures and those involving random matrices, but one may note that whereas the Cohen-Lenstra-type conjectures typically involve the “behavior at s=1” of Dirichlet series (in particular, location and order of poles, or in geometric terms, number of irreducible components), the RMT conjectures for, e.g., moments of L-functions, involve the behavior on the critical line (i.e., they concern zeros of L-functions).