Work of plenary speakers at ICM 2014

Question

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):

1. Ian Agol, University of California, Berkeley, USA
2. James Arthur, University of Toronto, Canada
3. Manjul Bhargava, Princeton University, USA
4. Alexei Borodin, Massachusetts Institute of Technology, USA
5. Franco Brezzi, IUSS, Pavia, Italy
6. Emmanuel Candes, Stanford University, USA
7. Demetrios Christodoulou, ETH-Zürich, Switzerland
8. Alan Frieze, Carnegie Mellon University, USA
9. Jean-François Le Gall, Université Paris-Sud, France
10. Ben Green, University of Oxford, UK
11. Jun Muk Hwang, Korea Institute for Advanced Study, Korea
12. János Kollár, Princeton University, USA
13. Mikhail Lyubich, SUNY Stony Brook, USA
14. Fernando Codá Marques, IMPA, Brazil
15. Frank Merle, Université de Cergy-Pontoise/IHES, France
16. Maryam Mirzakhani, Stanford University, USA
17. Takuro Mochizuki, Kyoto University, Japan
18. Benoit Perthame, Université Pierre et Marie Curie, France
19. Jonathan Pila, University of Oxford, UK
20. Vojtech Rödl, Emory University, USA
21. Vera Serganova, University of California, Berkeley, USA

Answer 1

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.

Answer 2

Takuro Mochizuki

Mochizuki is probably most famous for proving a conjecture of Kashiwara. People with masochistic tendencies can read on...

Recall that Beilinson, Bernstein, Deligne (and Gabber) proved things like the decomposition theorem and hard Lefschetz for semisimple perverse sheaves of geometric origin. The last assumption is needed since their technique involves reducing to the $\ell$-adic case in positive characteristic. A while back Kashiwara conjectured that these results should hold more generally for semisimple perverse sheaves (geometric or not) on complex varieties, or more generally for semisimple holonomic (not necessarily regular) $D$-modules. This conjecture is now a theorem due to Mochizuki.
The proof is long and complicated, but here is a quasi-explanation based on my limited understanding. When $L$ is a semisimple local system on a smooth projective variety $X$, it corresponds, thanks to work of Corlette-Simpson, to a so called harmonic bundle. Suffices it to say that this means that $L$ carries operators similar to $d,\partial, \bar \partial$ on a Kahler manifold, and the usual proof of hard Lefschetz carries over. But this is not enough, since a perverse sheaf is only generically a local system. In other words, $L$ may only be defined on a Zariski open subset of $X$; constructing the appropriate harmonic object is much more delicate and due, and in this generality, to Mochizuki. I won't attempt to say more other than to mention that Mochizuki develops the machinery of twistor modules (due to Simpson and Sabbah) to the point where he can show that holomonic semisimple $D$-modules correspond to such things. And this plays a key role in the resolution of the conjecture.

Answer 3

Alexei Borodin

He graduated from University of Pennsylvania in 2001, was at Caltech for several years and is now at MIT. Quoting the blurb here, "Borodin studies problems on the interface of representation theory and probability that link to combinatorics, random matrix theory, and integrable systems."

The aforementioned areas come together in the study of the KPZ universality class, a family of random processes whose fluctuations converge to Tracy-Widom distributions when appropriately rescaled. Conjecturally, these processes should be robust to minor perturbations in much the same way as the central limit theorem says any reasonable averaging process has Gaussian fluctuations. He has been studying problems related to KPZ from the beginning of his career. Many of these results are based on asymptotic analysis of random representation theoretic objects. One of his major breakthroughs (with Ivan Corwin) is the introduction of Macdonald processes. These provide a unifying framework for several of the models known to belong to the KPZ universality class, and allow for certain perturbations to the models.

There are lecture notes by Borodin and Gorin on these topics with an excellent introduction for those new to these ideas.

Answer 4

James Arthur

is a number theorist and representation theorist, working on the theory of automorphic representations (the Langlands program). He has proved many deep results on automorphic representations and their L-functions.

One (extremely pedestrian) way of understanding this theory, and some of Arthur's contributions to it, is this. Automorphic representations are some kind of analytic widgets, defined for reductive groups (e.g. the general linear group $GL(n)$) over number fields, which are a far-reaching generalization of modular forms. At least some of these are expected to be related to representations of Galois groups, in a functorial way. This last condition ("Langlands functoriality") roughly means that "natural" operations on Galois representations -- e.g. direct sums, tensor products, restricting to a subgroup, inducing up from a subgroup, etc -- should correspond to operations on automorphic representations; and one can hope to look for these purely in the automorphic world (even when, as in most cases, we can't prove anything about the link to Galois reps).

Arthur's work has confirmed many cases of these functoriality conjectures for automorphic representations. For instance his 1989 book with Clozel shows that automorphic representations of $GL(n)$ over a number field have operations corresponding to induction and restriction on the Galois side. This year he published a monograph on automorphic representations of orthogonal and symplectic groups, which shows (among a wealth of other things!) that there is a correspondence between these and auto reps of $GL(n)$ satisfying a suitable self-duality condition (analogous to the fact that a Galois representation into $GL(n)$ which is self-dual will land inside an orthogonal or symplectic group).

Answer 5

Jean-François Le Gall

One of his striking recent results is that random triangulations of the sphere with $n$ faces converge (in the Gromov-Hausdorff sense) as $n\rightarrow\infty$, after appropriate rescaling, to a limiting random metric space, the so-called Brownian map. This was mentioned in an answer to another question on this site.

Answer 6

Demetrios Christodoulou

A summary of his career can be found on his autobiography for the Shaw prize. In short, his works to date has focused on the use of geometric methods to resolve problems in nonlinear PDEs, and much of his work is in the field of mathematical general relativity. Some of the highlights:

• He proved the nonlinear stability of Minkowski space with Sergiu Klainerman, which established the use of a fully intrinsic-geometrical method of applying the vector field method which allows proving weighted energy estimates that respect the intrinsic null structure of the Lorentzian metric induced by the quasilinear coefficients. The same method is crucial in his later work on formation of trapped surfaces in general relativity and also formation of shocks in three dimensional fluids, and is now a (somewhat) standard method in the geometric analysis of quasilinear hyperbolic equations.

• In a series of papers from late 1980s to 1999, he examined the spherically symmetric collapse of self-gravitating scalar field. This is a model problem for gravitation waves (one cannot study pure gravitation waves in spherical symmetry due to Birkhoff's theorem). He was able to prove for this model both the existence of naked singularity solutions and their instability, thereby showing a version of the weak cosmic censorship conjecture for generic solutions and showing that genericity in the statement is required.

• His work on shock formation in fluids is essentially the study of blow-up behaviour of small data solutions to a quasilinear wave equation where the null condition fails. Alinhac has already previously shown that such solutions will blow-up generically in finite time, found the asymptotic lifespan as the size of data goes to zero, and described the local geometry at the first blow-up point as that of shock formation (that the main blow-up mechanism is essentially that of Burger's equation, the crossing of characteristic surfaces). The main improvements are (a) a new proof using only finitely many derivatives; Alinhac required $C^\infty$ data (b) full tracking of geometric quantities along the evolution and (c) description of the singular boundary of the globally hyperbolic domain [or physically, description of the shock front]. This improvements are done with a view toward perhaps being able to extend the notion of "entropy weak solutions" of 1+1 dimensional conservation laws to fluids in higher spatial dimensions.

• His work on formation of trapped surfaces gave the first rigorous proof that a black hole can form dynamically from the focussing of gravitational waves. In this work he introduced the short pulse method (which grew out of his work on self gravitating scalar fields), which, in one guise, can be thought of as a way of getting large data, long time existence for quasilinear wave equations when some version of null condition is available. (Basically, one way of reading the null condition, which is so successful in small data theory, is that wave packets can only weakly self interact. In particular, this prevents the self-reinforcing ODE type blow-up to occur, at least to leading order. If one carefully keeps track of the full hierarchy of sizes from the nonlinear effects [roughly speaking the weak self interaction of the wave packets generate other wave packets in other directions, and we need to keep track of all of them], especially the "anomalous" (compared to linear theory) terms, one can get long-time existence while also extracting information unavailable from naive linear analysis.)

Remark: the short pulse method is in some sense dual to the peeling estimates implied by the nonlinear stability of Minkowski space. These peeling estimates show, in particular, that even in the small data regime, the asymptotic behaviour of gravitational waves cannot be captured purely as the behaviour of a linear spin-2 field. There will necessarily be some non-linear effects which manifest at top order. This "Christodoulou memory effect" in principle may affect the efforts of experimentally observing gravitation radiation.

Answer 7

Ian Agol

...supplied in the comments a link to the question Elevator pitch for the Virtual Fibering Theorem, which has two very nice answers discussing his (and others') work.

Answer 8

Emmanuel Candes

He is famous for his contribution to Compressive sensing. Famously he, in collaboration with T. Tao and using geometric functional analysis, proved that if a vector, e.g. a digital image, is sparse in a fixed basis, then one can reconstruct the vector with very high accuracy, in $\ell_2$ metric, from a small number of random measurements and only by solving a linear program.

E. Candes , T. Tao , "Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies", IEEE Trans. Inform. Theory, Nov. 2006

This paper, along with the work of David Donoho [1], founded the theoretical foundation of compressive sensing.

[1] D. Donoho , "Compressed Sensing", IEEE Trans. Inform. Theory, Mar. 2006

Answer 9

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.