The next International Congress of Mathematicians (ICM) will take place in 2014 in Seoul, Korea. The present question is meant to gather brief overviews of the work of the plenary speakers for the ICM 2014.
More precisely, anybody who feels qualified to give a short description of the work of one of the plenary speakers at ICM 2014 is invited to put that in an answer below, or to add to the existing answers. Ideally, there would be a single answer dedicated fully to each speaker. Answers which summarize this thread are also very welcome.
Most importantly, this thread is meant to be informative and educational for anyone in the mathematical community. Therefore, please strive to make the answers broadly accessible.
Background: A similar, very successful MathOverflow question was asked a few years ago concerning the work of the plenary speakers for the International Congress of Mathematicians in 2010. Please look there to get an idea of what could be achieved with the present question.
List of plenary speakers at ICM 2014
For completeness, here is a list of the scheduled plenary speakers at ICM 2014, (copied from here):
- Ian Agol, University of California, Berkeley, USA
- James Arthur, University of Toronto, Canada
- Manjul Bhargava, Princeton University, USA
- Alexei Borodin, Massachusetts Institute of Technology, USA
- Franco Brezzi, IUSS, Pavia, Italy
- Emmanuel Candes, Stanford University, USA
- Demetrios Christodoulou, ETH-Zürich, Switzerland
- Alan Frieze, Carnegie Mellon University, USA
- Jean-François Le Gall, Université Paris-Sud, France
- Ben Green, University of Oxford, UK
- Jun Muk Hwang, Korea Institute for Advanced Study, Korea
- János Kollár, Princeton University, USA
- Mikhail Lyubich, SUNY Stony Brook, USA
- Fernando Codá Marques, IMPA, Brazil
- Frank Merle, Université de Cergy-Pontoise/IHES, France
- Maryam Mirzakhani, Stanford University, USA
- Takuro Mochizuki, Kyoto University, Japan
- Benoit Perthame, Université Pierre et Marie Curie, France
- Jonathan Pila, University of Oxford, UK
- Vojtech Rödl, Emory University, USA
- Vera Serganova, University of California, Berkeley, USA
Best Answer
Misha Lyubich
He graduated from Kharkiv University (Ukraine) in 1980 and now works at SUNY, Stony Brook, as the Director of the Math. Institute. His work belongs to holomorphic dynamics (iteration of holomorphic maps). He made major contributions to all parts of holomorphic dynamics (rational functions, entire functions and holomorphic maps in several dimensions). His work is nicely described in his web page
http://www.math.sunysb.edu/~mlyubich/
One of his principal results is easy to state. Consider the dynamical system $x\mapsto x^2+c$ on the real line. Then for almost every $c\in[−2,1/4]$, the quadratic map $f_c(x)=x^2+c$ is either regular or stochastic.
"Regular" means almost all orbits are attracted to an attracting cycle. "Stochastic" means that there exists an ergodic invariant measure which is absolutely continuous with respect to Lebesgue measure.
This is a complete qualitative description of the nature of chaos in the real quadratic family.
This result actually completes a long line of development in which many people participated.
There is a nice non-technical exposition of this fundamental result in his paper The quadratic family as a qualitatively solvable model of chaos. Notices Amer. Math. Soc. 47 (2000), no. 9, 1042–1052.
Of his early famous results, I will mention existence and uniqueness of the measure of maximal entropy for rational maps of the Riemann sphere $P^1$, and discovery that the map $z\mapsto e^z$ of the complex plane is not ergodic with respect to the Lebesgue measure.