Clay Research Award 2011

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.

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.

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