Viewing archives for Research Award Winners

Tristan Buckmaster

The 2019 Clay Research Award to Tristan Buckmaster, Philip Isett (Caltech) and Vlad Vicol (NYU) is made in recognition of the profound contributions that each of them has made to the analysis of partial differential equations, particularly the Navier-Stokes and Euler equations. 

Buckmaster and Vicol changed perceptions of the nature of weak solutions to the Navier-Stokes equations by proving that such solutions can be remarkably wild – severely non-smooth and highly non-unique.

With a series of breakthroughs, Isett finally settled Onsager’s 1949 Conjecture that, in turbulent regimes, below a specific regularity threshold, solutions to Euler’s equations can be energy dissipative.

Both works solve longstanding problems of fundamental importance. In the course of doing so, they introduce novel ideas in hard analysis, building on the program of De Lellis and Székelyhidi, inspired by the pioneering work of Nash and Gromov.

Vlad Vicol

The 2019 Clay Research Award to Tristan Buckmaster (Princeton), Philip Isett (Caltech) and Vlad Vicol is made in recognition of the profound contributions that each of them has made to the analysis of partial differential equations, particularly the Navier-Stokes and Euler equations. 

Buckmaster and Vicol changed perceptions of the nature of weak solutions to the Navier-Stokes equations by proving that such solutions can be remarkably wild – severely non-smooth and highly non-unique.

With a series of breakthroughs, Isett finally settled Onsager’s 1949 Conjecture that, in turbulent regimes, below a specific regularity threshold, solutions to Euler’s equations can be energy dissipative.

Both works solve longstanding problems of fundamental importance. In the course of doing so, they introduce novel ideas in hard analysis, building on the program of De Lellis and Székelyhidi, inspired by the pioneering work of Nash and Gromov.

Wei Zhang

The 2019 Clay Research Award to Wei Zhang (MIT) is made in recognition of his ground-breaking work in arithmetic geometry and arithmetic aspects of automorphic forms.

His landmark contributions include a proof of the global Gan-Gross-Prasad conjecture for a wide class of automorphic representations of unitary groups, a proof with Zhiwei Yun of a higher-order generalization of the Gross-Zagier formula over function fields, and a proof of Kolyvagin’s conjecture on the structure of Selmer groups for a large class of elliptic curves over Q. 

He has formulated an arithmetic version of the Gan-Gross-Prasad Conjecture –a vision for a far-reaching generalization of the Gross-Zagier formula over number fields – and pioneered a relative trace formula approach to its proof.He recently achieved a major step in this program by proving the Arithmetic Fundamental Lemma.

Image courtesy of Allegra Boverman

Clifford Taubes

The 2008 Clay Research Award was made to Clifford Taubes for his proof of the Weinstein conjecture in dimension three.

The Weinstein conjecture is a conjecture about the existence of closed orbits for the Reeb vector field on a contact manifold. A contact manifold is an odd-dimensional manifold with a one-form A such that A wedged with the n-th exterior power of dA is everywhere nonzero. In particular, the kernel of A is a maximally nonintegrable field of hyperplanes in the tangent bundle. The Reeb vector field generates the kernel of dA and pairs to one with A. Alan Weinstein asked some thirty years ago whether this vector field must, in all cases, have a closed orbit. (The unit sphere in complex n-space with A the annihilator of the maximal complex subspace of the real tangent space is an example of a contact manifold and contact 1-form. In this case, the orbits of the Reeb vector field generate the circle action whose quotient gives the associated complex projective space.) Note, by contrast, that there exist non-contact vector fields, even on the 3-sphere, with no closed orbits. These are the counter-examples (due to Schweitzer, Harrison and Kuperberg) to the Seifert conjecture. Hofer affirmed the Weinstein conjecture in many 3-dimensional cases, for example the three-sphere and contact structures on any 3 dimensional, reducible manifold. Taubes’ affirmative solution of the Weinstein conjecture for any 3-dimensional contact manifold is based on a novel application of the Seiberg-Witten equations to the problem.

Alex Eskin

The 2007 Clay Research Award was made to Alex Eskin for his work on rational billiards and geometric group theory, in particular, his crucial contribution to joint work with David Fisher and Kevin Whyte in establishing the quasi-isometric rigidity of sol.

Michael Harris

The 2007 Clay Research Award was made to Michael Harris and Richard Taylor for their work on local and global Galois representations, partly in collaboration with Laurent Clozel and Nick Shepherd-Barron, culminating in the solution of the Sato-Tate conjecture for elliptic curves with non-integral j-invariants.

Richard Taylor

The 2007 Clay Research Award was made to Michael Harris and Richard Taylor for their work on local and global Galois representations, partly in collaboration with Laurent Clozel and Nick Shepherd-Barron, culminating in the solution of the Sato-Tate conjecture for elliptic curves with non-integral j-invariants.

Christopher Hacon

The 2007 Clay Research Award was made to Christopher Hacon and James McKernan for their work in advancing our understanding of the birational geometry of algebraic varieties in dimension greater than three, in particular, for their inductive proof of the existence of flips.

James McKernan

The 2007 Clay Research Award was made to Christopher Hacon and James McKernan for their work in advancing our understanding of the birational geometry of algebraic varieties in dimension greater than three, in particular, for their inductive proof of the existence of flips.

Claire Voisin

The 2008 Clay Research Award was made to Claire Voisin for her disproof of the Kodaira conjecture.

The Kodaira conjecture was formulated in 1960, when Kunihiko Kodaira showed that any compact complex Kaehler surface can be deformed to a projective algebraic surface. For the proof, Kodaira used his classification theorem for complex surfaces. The conjecture asks whether Kaehler manifolds of higher dimension can be deformed to a projective algebraic manifold. Voisin constructs counterexamples: in each dimension four or greater, there is a compact Kaehler manifold which is not homotopy equivalent to a projective one. For dimension at least six, she gives examples which are also simply connected. A later result gives a substantial strengthening: in any even dimension ten or greater, there exist compact Kaehler manifolds, no bimeromorphic model of which is homotopy equivalent to a projective algebraic variety. Distinguishing the homotopy type of projective and non-projective Kaehler manifolds is achieved through novel Hodge-theoretic arguments that place subtle restrictions on the topological intersection ring of a projective manifold.

Andrew Wiles

The 1999 Clay Research Award was made to Andrew Wiles for his role in the development of number theory.

Edward Witten

The 2001 Clay Research Award was made to Edward Witten for a lifetime of achievement, especially for pointing the way to unify apparently disparate fields of mathematics and to discover their elegant simplicity through links with the physical world.