# 7ECM - Some personal thoughts

So I’m currently on my way back from the 7th European Congress of Mathematics, which took place in Berlin over the course of the last week. While everything is still fresh(ish) in my memory I wanted to record some of the bits that I (personally) found most interesting so that I’ll be able to look back when I invariably do forget what I did for a whole week. If you like similar things to me then maybe you’ll find something interesting here too (if you don’t like similar things to me then I question your choice to read this blog). Some of these topics I’d like to revisit in more detail (possibly in a future post?!) but for now these short snippets will have to do. So in approximately no particular order and undoubtedly with some gaps.

## Peter Scholze

This is the most obvious one for me, Scholze talked about Perfectoid spaces, which since he introduced them in his thesis have become a hot topic in number theory. Scholze himself has been awarded various prizes and honours for developing this theory (including another at the ECM).

The motivation is to transfer problems involving the $p$-adics (which can be problematic as they are mixed characteristic) over to the similar looking ring $\mathbf{F}_p((t))$. Scholze said he wants to fight for the freedom of $p$. The way he does this is via a technique called tilting, which was originally used by Fontaine and Wintenberger to prove a result relating the Galois theories of these fields. Scholze (and independently Kedlaya-Liu) takes this result and distils out the definition of a field where this works and calls it a perfectoid field. He then does some natural looking things (which I’m sure are actually very complicated to do properly) and constructs perfectoid algebras and perfectoid spaces. He shows that various things work out nicely and that one can tilt the spaces too in a way that matches what you’d hope for simple spaces like projective space.

In the last part of the talk Scholze got increasingly high level so rather than embarrass myself trying to replicate it I’ll just say that it definitely looks like he has more impressive things forthcoming.

One thing that was especially interesting for me in this talk was mention of work by Jared Weinstein who works at Boston University (where I will also be from September). I was aware that Weinstein was doing work in this area so it was really interesting to have a bit of it put into context.

## Karin Vogtmann

Vogtmann first gave a nice introduction to outer automorphisms of the free group and why one might care, and got me interested with the observation that abelianising gives automorphisms of $\mathbf{ZZ}^d$. She also described various related “outer space”s which are spaces which the outer automorphisms of free groups act on. The construction is really cool and involves deforming certain metrized graphs using the interpretation of $\operatorname{Out}(F_n)$ as the automorphisms of the rose graph. (One thing she mentioned in passing here that I’d like to learn more about is Gromov’s topology on the space of all metric spaces, so meta.) The gist was that the various slightly different outer spaces were all very related to the main one. The pictures were very good and extremely helpful and for me this talk definitely wins the Oscar for best visual effects.

## Don Zagier

I found Don’s style of presenting (rapidly moving through hand written slides while barely breathing) engaging and super impressive, multiple times he switched to a slide and corrected something before I’d even had time to read the first sentence. The content itself was also really fascinating, he started with a fairly random looking recurrently defined sequence that Apery used to prove that $\zeta(3)$ is irrational and gave multiple interpretations of where such a sequence comes from (though in fact there is still something very special going on). He ended up giving some motivation for the concept motives (pun not intended) and presented us with a few results concerning various special values that were conjectured by a coauthor of his based on pure thought on the level of motives and proved by Zagier and others on the concrete level.

Unfortunately I can’t remember quite as much of what went on as I’d like due to the above-mentioned speed and Don’s recommendation at the start that we “don’t even bother trying to take notes”. I do hope a video of the lecture will appear at some point as out of all the talks I saw I think this is the one I’d most benefit from rewatching in its entirety.

Another fun coincidence for me personally was that he mentioned some work of Irene Bouw, who I met only days beforehand in Ulm.

## Tommy Hofman

Tommy talked about an algorithm he and Claus Fieker developed to compute Hermite normal forms over Dedekind domains. This was another exceptional example of the law of large numbers (or whichever law it is that explains funny coincidences) as previously I worked on implementing algorithms for computing HNFs and doing this work was in some way the original cause of me coming to Kaiserslautern, which is where Tommy works! The main takeaway from this talk (other than a nice new algorithm) is that quotients of rings of integers by powers of prime ideals are always Euclidean rings.

## Other nice talks

• Fedor Pakovich found a relationship between Davenport-Zannier pairs (generalisations of pairs of polynomials $(f,g)$ such that $f^3-g^2$ is of minimal possible degree, excluding trivial things) and Dessins d’Enfants. He used this to obtain a classification of these pairs (roughly 10 famillies and 11 sporadic pairs) by couting the corresponding graphs (which ended up being interpretable as certain weighted trees or something like this). One of the reasons I found this so interesting was that it’s the first time I think I’ve seen Dessins used in anger, I started reading Girondo and Gonzailez-Diaz’s book on compact Riemann surfaces and Dessins about year ago but got distracted before I got to the juicy bit.

• Thomas Willwacher talked about cohomology of graph complexes which it turns out is really hard to compute but relates many other areas of maths (including automorphisms of free groups!).

• Maryna Viazovska gave a nice talk about her (and others) recent proof of sphere packing in dimensions 8 and 24, which got a lot of popular attention at the time so was nice to hear about.

• Meinolf Geck spoke about his work computing with groups, remarked that finite groups of Lie type $G(\mathbf{F}_q)$ make up a large portion of the classification of finite simple groups. He wants to construct generic character tables as $q$ varies for different algebraic groups and possibly different primes - and this actually looks to work! That this is even possible totally blew my mind.

• Uli Wagner spoke about the topological Tverberg conjecture, Gunter Ziegler gave a talk that touched on this in Warwick in 2014 which I really enjoyed, so it was cool to hear about recent progress regarding it.

• Jeremy Grey spoke about Poincare and Weyl and it was a lot more philosophical than I was expecting, contrasting their views on philosophy of science over time, but it was fascinating stuff.

There were certainly many many other interesting talks that I saw (and indeed many that I didn’t) but I’m tired and my memory is failing now and I have to make a post or Beeminder will take my money.

## Other thoughts

There were a few political statements made during the conference for example by Pavel Exner condemning the situation in Turkey (where academics were recalled/banned from travelling) and from Timothy Gowers regarding the Brexit (short version: 48% of us voted to stay). This is something that I think is appropriate for large I/ECM like conferences, which represent the whole mathematical community.

## Non-maths

Berlin is a really awesome city so even outside of the conference I did a lot of nice things, and I’m really glad I got a good excuse to visit before I finish my time in Germany. I took quite a lot of photographs (especially the Sunday before the conference when it was really nicely overcast), maybe I’ll upload some to Flickr and put a link here when I’m back in the UK and have some free time.