My Columbia colleague Patrick Gallagher passed away a few months ago at the age of 84. He had only recently retired, and for many years was the longest serving member of the department and an important part of its institutional memory. On October 10 there will be a memorial conference here at Columbia.

It’s too bad Pat didn’t live to see the latest from Terry Tao, who describes recent results which are related to old work of Gallagher’s by “Our proof of this theorem proceeds more or less along the same lines as Gallagher’s calculation, but now with *k* allowed to grow slowly with *x*.”

Turning to other topics, Peter Scholze continues to come up with new ideas about the foundations of mathematics at a pace far too fast for me to fool myself into thinking I might be able to follow what he’s doing. In recent months he has run a course on “Condensed Mathematics”, which involves new ideas about topology developed with Dustin Clausen. For a video of a recent talk where he explains this, see here.

On the Langlands/representation theory front, some interesting things are:

- A course run by Geordie Williamson in Sydney earlier this year.
- If you’re interested in what Williamson is up to, you might want to watch videos from “somewhere in Russia” last month, or “somewhere on Long Island” last week. Lecture notes are here.
- Also in Russia last month, at the Skoltech Center for Advanced Studies, where my Columbia colleague Igor Krichever is director, there was a skoltech summer school on mathematical physics, with some of the lectures on video here.
- Several new papers addressing different points of view about geometric Langlands are out, including a very recent one from Etingof, Frenkel and Kazhdan. For some background on the relation of this to work by Langlands himself, see here and here. For some geometric Langlands-related work of a different sort, see here and here.

Finally, the 2019 PCMI was devoted to the topic of *Quantum Field Theory and Manifold Invariants*, videos are here.

**Update**: Also on the Langlands/representation theory/quantum front, this week at Northeastern there’s a conference going on (rumored to be celebrating the fact that Etingof and Okounkov share the same birthday, and are now 50). Videos are starting to appear here, including one of Edward Frenkel discussing the paper with Etingof and Kazhdan mentioned above.

Breakthrough Prizes have been announced (they seem to be moving earlier each year, wasn’t the announcement in December a few years back). The recipients are Alex Eskin in maths and, unsurprisingly, the Event Horizon Telescope collaboration in Physics.

anon,

That is weird. In the past they kept this a secret until the prize ceremony near the end of the year. Maybe they’re trying to get in before the Nobels, announced in a few weeks.

I also still don’t understand why they’ve inverted their previous practice of giving the “special prize” for current experimental results (e.g. LIGO/2016 and Higgs/2013) and the regular prize for (often failed) theory. This year the “special prize” went to the failed SUGRA unification idea and the regular prize to a current experimental result (EHT).

There’s some value in making an award to an experimental collaboration like EHT (as opposed to certain individuals). Besides the problem of awards for failed theory, for the rest of the math-physics awards they’re making, I think we’d be better off without them. They’re no different than any number of other awards out there, other than embodying the misguided idea that what math and physics research need is more Hollywood glitz.

Eskin has co-authored papers with 7 Fields medalists. I’d be surprised if anyone else has a Fields number (to coin a term) as high as 5.

@social network. Eskin clearly deserved the prize. Those seven are Margulis, McMullen, Kontsevich, Okounkov, Lindenstrauss, Avila, and Mirzhakhani.

Except for Margulis, the other 6 Fields medals came after 1998, and four of them mostly worked in dynamical systems. Also, Fields medals for work in dynamical systems (or closely related work) include Yoccoz in 1994 and Venkatesh in 2018. Let’s not forget that Sinai got an Abel prize in 2014. Seems like a very prestigious field in mathematics.

@jamie, presumably the “Fields number” positively correlates with (among other things) strength as a mathematician. If you are strong enough to collaborate with top people in the field, and they keep coming back to write more papers with you, that’s a pretty good sign that you are strong in your own right. At least that’s how I interpreted it.

Dynamical systems, at least in the USA, has had the reputation of being more social than other areas of math. Between that and it being a field that “relates to everything” it makes sense for collaboration numbers to be higher there.

Bourgain seems to have had at least five medalists (Bombieri, Figalli, Lindenstrauss, Tao, Venkatesh) as co-authors (six, if you count Connes; they were both editors of a conference proceedings). I only checked the ones that I though the most likely to have co-authored with him, so I may have missed someone.

All,

Maybe that’s enough discussion of the new “Fields number” metric.