The 2011 Clay Research Award has been made toJean-François Quint and Yves Benoist for their spectacular work on stationary measures and orbit closures for actions of non-abelian groups on homogeneous spaces.
This work is a major breakthrough in homogeneous dynamics and related areas of mathematics. In particular, Benoist and Quint proved the following conjecture of Furstenberg. Let H be a Zariski dense semisimple subgroup of a Lie group which acts by left translations on the quotient of G by a discrete subgroup with finite covolume. Consider a probability measure m on H whose support generates H. Then any m-stationary probability measure for such an action is H-invariant.
The 2011 Clay Research Award has been made to Yves Benoist and Jean-François Quint for their spectacular work on stationary measures and orbit closures for actions of non-abelian groups on homogeneous spaces.
This work is a major breakthrough in homogeneous dynamics and related areas of mathematics. In particular, Benoist and Quint proved the following conjecture of Furstenberg. Let H be a Zariski dense semisimple subgroup of a Lie group which acts by left translations on the quotient of G by a discrete subgroup with finite covolume. Consider a probability measure m on H whose support generates H. Then any m-stationary probability measure for such an action is H-invariant.
The 2011 Clay Research Award has been made to Jonathan Pila for his resolution of the André-Oort Conjecture in the case of products of modular curves.
This work gives the first unconditional proof of fundamental cases of these general conjectures beyond the original theorem of André concerning the product of two such curves. The foundational techniques that Pila developed to achieve this breakthrough range from results in real analytic geometry, which give sharp upper bounds for the number of rational points of bounded height on certain analytic sets, to the use of O-minimal structures in mathematical logic.
Xinyi Yuan received his PhD from Columbia University in 2008 under the supervision of Shou-Wu Zhang. In his 2006 preprint, “Big Line Bundles over Arithmetic Varieties,” he proves an arithmetic analogue of a theorem of Siu and derives, among other consequences, a natural sufficient condition for when the orbit under the absolute Galois group is equidistributed. Xinyi was apponted as a Clay Research Fellow for a term of three years beginning July 2008.
Peter Scholze obtained his PhD in 2012 under the supervision of Michael Rapoport at the Universität Bonn. After working about the cohomology of Shimura varieties and the Langlands program, his PhD thesis was about a theory of perfectoid spaces, which gives a method to compare objects in mixed characteristic with objects in equal characteristic p, with applications to p-adic Hodge theory and the weight-monodromy conjectures. Peter was appointed as a Clay Research Fellow for a term of five years beginning July 2011.
The 2014 Clay Research Award was made to Peter in recognition of his many and significant contributions to arithmetic algebraic geometry, particularly in the development and applications of the theory of perfectoid spaces.
Sucharit Sarkar received his PhD from Princeton University in 2009 under the guidance of Zoltan Szabo. His research area is in low dimensional topology. His dissertation addressed topics in Heegaard Floer homology for 3-manifolds and knots inside 3-manifolds. Sucharit was appointed as a Clay Research Fellow for a term of five years beginning July 2009.
Sophie Morel received her PhD from Université Paris-Sud in 2005 under the supervision of Gérard Laumon. Her thesis, Complexes d’intersection des compactifications de Baily-Borel – le cas des groupes unitaires sur Q is an important step forward in the Langlands program. She develops a theory of weight truncation on varieties over finite fields with which she derives a simple description of the intersection complexes on the Baily-Borel compactifications of certain Shimura varieties over finite fields. From this in turn she obtains a formula for the trace of the Frobenius endomorphism on the Euler characteristic of the intersection cohomology. Sophie was appointed as a Clay Research Fellow for a term of five years beginning October 2006.
Davesh Maulik received his PhD in 2007 from Princeton University under the supervision of Rahul Pandharipande. His mathematical interests include algebraic geometry and its connections with symplectic geometry, mathematical physics, and combinatorics. His research focus is in the area of Gromov-Witten theory and enumerative geometry. Davesh was appointed as a Clay Research Fellow for a term of five years beginning July 2007.
Adrian Ioana was born in Romania (1981), he received his B.S. from Unversity of Bucharest and his PhD in 2007 from UCLA under the direction of Professor Sorin Popa. His research interests are von Neumann algebras and orbit equivalence. Adrian, together with Jesse Peterson and Sorin Popa, have proved the existence of a II_1 factor with trivial outer automorphism group. He also showed that any countable group which contains a free subgroup has uncountably many non orbit equivalent actions. Adrian was appointed as a Clay Research Fellow for a term of three years beginning August 2008.
Tim Austin received his PhD in 2010 from the University of California, Los Angeles, under the supervison of Terence Tao. His interests cover ergodic theory, metric geometry and geometric group theory. He has developed new techniques for the analysis of certain nonconventional ergodic averages associated with the phenomenon of multiple recurrence, and has shown how to construct examples of infinite discrete gorups with various novel geometric properties. Tim was appointed as a Clay Research Fellow for a term of five years beginning July 2010.
Mohammed Abouzaid received his PhD from the University of Chicago in 2007 under the supervision of Paul Seidel. His thesis used techniques from tropical geometry to give a new approach to the homological mirror symmetry conjecture for toric varieties. He is interested in symplectic topology and its interactions with algebraic geometry and differential topology. Mohammed was apponted as a Clay Research Fellow for a term of five years beginning July 2007.