TITLE: Work of plenary speakers at ICM 2014
QUESTION [89 upvotes]: 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

REPLY [12 votes]: Fernando Coda-Marques
Together with Andre Neves he proved the Willmore conjecture. This states that the Clifford torus in the 3-sphere minimizes the Willmore energy, with energy $2\pi^2$. To prove this conjecture, they show that for 5-parameter sweepouts of $S^3$ by surfaces, with certain boundary conditions, the min-max area is $2\pi^2$. This theorem has other consequences besides the Willmore conjecture.