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This is my suggested followup to this question, which I thought resulted in some very worthwhile summaries. Now that we know the Fields Medalists and the other prizewinners, it would be nice for anybody who feels qualified to give a brief overview of the work (or aspects of the work) of one of them. The ICM has already posted brief overviews, but I would like to see the work presented in more detail or from another perspective.

For reference, the Fields Medalists are Elon Lindenstrauss, Ngô Bảo Châu, Stanislav Smirnov and Cedric Villani. The other prizewinners are Daniel Spielman (Nevanlinna Prize), Yves Meyer (Gauss Prize) and Louis Nirenberg (Chern Medal). Congratulations to all of them!

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closed as no longer relevant by Steve Huntsman, Loop Space, S. Carnahan Aug 20 '10 at 9:33

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In line with what Davidac897 and Thierry said, is this question really necissary? – B. Bischof Aug 20 '10 at 0:42
I missed the detailed information when I looked at the ICM website. Sorry. We probably don't need this question. – Peter Shor Aug 20 '10 at 2:16
No worries, I'd thought this would be a good queston, but the ICM website was fast, and Terry Tao already wrote an answer, and it doesn't look like we were going to get anything better than those two things here. – Noah Snyder Aug 20 '10 at 15:06

I found the "work profiles" by Julie Rehmeyer on the ICM website to be good for a very high level view. But I like the idea of getting many different perspectives.

I can only speak to Dan Spielman. His most widely known work is in two areas

(1) Smoothed Analysis of Algorithms: The simplex algorithm for linear programming is fast in practice, but slow in theory. This gap is bothersome to theorists. "Smoothed analysis" provides a new method of analyzing the running time of algorithms by looking at how fast an algorithm runs on random perturbations of worst case inputs. Spielman and Shang-Hua Teng proved that the simplex algorithm runs "fast in theory" in their framework. This created a degree of excitement and inspired much further work. My impression is that while smoothed analysis is a very nice model the analysis is not necessarily easy to do and hence worst case analysis still predominates. But it is a useful tool to have for those anomalous cases like the simplex algorithm.

(2) Fast Error Correcting Codes Low Density Parity Codes have been known since the 60s, but were rarely used because they were considered too computationally inefficient. In the mid 90s new algorithms were discovered by Spielman and others which made these codes appear more attractive. In particular Spielman invented codes based on expander graphs and proved that they could be encoded and decoded particularly fast. In many cases these codes can almost achieve the theoretical bound for information transmission. This is a large and very active area of research in Electrical Engineering and many of the advances came from that community. These codes are now considered competitive with the best and are widely used in practice.

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There is very detailed information already available on the ICM 2010 website. I'm not too sure we can add much to that, at least unless someone comes up with a more specific question.

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Terence Tao has put a nice post about the Fields Medailists on his blog.

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