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Every four years at the International Congress of Mathematicians (ICM), the Fields Medal laureates are announced. Today, at the 2018 ICM in Rio de Janeiro, it was announced that the Fields Medal was awarded to Caucher Birkar, Alessio Figalli, Peter Scholze, and Akshay Venkatesh.

After the two previous congresses in 2010 and 2014, I wrote blog posts describing some of the work of each of the winners. This time, though, I happened to be a member of the Fields Medal selection committee, and as such had access to a large number of confidential letters and discussions about the candidates with the other committee members; in order to have the opinions and discussion as candid as possible, it was explicitly understood that these communications would not be publicly disclosed. Because of this, I will unfortunately not be able to express much of a comment or opinion on the candidates or the process as an individual (as opposed to a joint statement of the committee). I can refer you instead to the formal citations of the laureates (which, as a committee member, I was involved in crafting, and then signing off on), or the profiles of the laureates by Quanta magazine; see also the short biographical videos of the laureates by the Simons Foundation that accompanied the formal announcements of the winners. I am sure, though, that there will be plenty of other mathematicians who will be able to present the work of each of the medalists (for instance, there was a *laudatio* given at the ICM for each of the winners, which should eventually be made available at this link).

I know that there is a substantial amount of interest in finding out more about the inner workings of the Fields Medal selection process. For the reasons stated above, I as an individual will unfortunately be unable to answer any questions about this process (e.g., I cannot reveal any information about other nominees, or of any comparisons between any two candidates or nominees). I think I can safely express the following two personal opinions though. Firstly, while I have served on many prize committees in the past, the process for the Fields Medal committee was by far the most thorough and deliberate of any I have been part of, and I for one learned an astonishing amount about the mathematical work of all of the shortlisted nominees, which was an absolutely essential component of the deliberations, in particular giving the discussions a context which would have been very difficult to obtain for an individual mathematician not in possession of all the confidential letters, presentations, and other information available to the committee (in particular, some of my preconceived impressions about the nominees going into the process had to be corrected in light of this more complete information). Secondly, I believe the four medalists are all extremely deserving recipients of the prize, and I fully stand by the decision of the committee to award the Fields medals this year to these four.

I’ll leave the comments to this post open for anyone who wishes to discuss the work of the medalists. But, for the reasons above, I will not participate in the discussion myself.

*[Edit, Aug 1: looks like the ICM site is (barely) up and running now, so links have been added. At this time of writing, there does not seem to be an online announcement of the composition of the committee, but this should appear in due course. -T.]*

*[Edit, Aug 9: the composition of the Fields Medal Committee for 2018 (which included myself) can be found here. -T.]*

As is now widely reported, the Fields medals for 2010 have been awarded to Elon Lindenstrauss, Ngo Bao Chau, Stas Smirnov, and Cedric Villani. Concurrently, the Nevanlinna prize (for outstanding contributions to mathematical aspects of information science) was awarded to Dan Spielman, the Gauss prize (for outstanding mathematical contributions that have found significant applications outside of mathematics) to Yves Meyer, and the Chern medal (for lifelong achievement in mathematics) to Louis Nirenberg. All of the recipients are of course exceptionally qualified and deserving for these awards; congratulations to all of them. (I should mention that I myself was only very tangentially involved in the awards selection process, and like everyone else, had to wait until the ceremony to find out the winners. I imagine that the work of the prize committees must have been extremely difficult.)

Today, I thought I would mention one result of each of the Fields medalists; by chance, three of the four medalists work in areas reasonably close to my own. (Ngo is rather more distant from my areas of expertise, but I will give it a shot anyway.) This will of course only be a tiny sample of each of their work, and I do not claim to be necessarily describing their “best” achievement, as I only know a portion of the research of each of them, and my selection choice may be somewhat idiosyncratic. (I may discuss the work of Spielman, Meyer, and Nirenberg in a later post.)

On Thursday, UCLA hosted a “Fields Medalist Symposium“, in which four of the six University of California-affiliated Fields Medalists (Vaughan Jones (1990), Efim Zelmanov (1994), Richard Borcherds (1998), and myself (2006)) gave talks of varying levels of technical sophistication. (The other two are Michael Freedman (1986) and Steven Smale (1966), who could not attend.) The slides for my own talks are available here.

The talks were in order of the year in which the medal was awarded: we began with Vaughan, who spoke on “Flatland: a great place to do algebra”, then Efim, who spoke on “Pro-finite groups”, Richard, who spoke on “What is a quantum field theory?”, and myself, on “Nilsequences and the primes.” The audience was quite mixed, ranging from mathematics faculty to undergraduates to alumni to curiosity seekers, and I severely doubt that every audience member understood every talk, but there was something for everyone, and for me personally it was fantastic to see some perspectives from first-class mathematicians on some wonderful areas of mathematics outside of my own fields of expertise.

Disclaimer: the summaries below are reconstructed from my notes and from some hasty web research; I don’t vouch for 100% accuracy of the mathematical content, and would welcome corrections.

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