James Carlson
Jim Carlson was President of the Clay Mathematics Institute 2003-2012.
Jim Carlson was President of the Clay Mathematics Institute 2003-2012.
The 2005 Clay Research Award was made to Nils Dencker for his complete resolution of a conjecture made by F. Trèves and L. Nirenberg in 1970. This conjecture posits an essentially geometric necessary and sufficient condition, Psi, for a pseudo-differential operator of principal type to be locally solvable, i.e., for the equation Pu = ƒ to have local solutions given a finite number of conditions on ƒ.
Dencker’s work provides a full mathematical understanding of the surprising discovery by Hans Lewy in 1957 that there exists a linear partial differential operator — a one-term, third-order perturbation of the Cauchy-Riemann operator — which is not locally solvable in this sense. The necessity of condition “Psi” was shown for operators in dimension 2 by R. Moyer in 1978 and in general by L. Hormander in 1981. The sufficiency of the condition has resisted many previous attacks.
The 2005 Clay Research Award was made to Manjul Bhargava for his discovery of new composition laws for quadratic forms and for his work on the average size of ideal class groups.
The field of composition laws had lain dormant for 200 years since the pioneering work of C.F Gauss. The laws discovered by Bhargava were a complete surprise, and led him to another major breakthrough, namely, counting the number of quartic and quintic number fields with given discriminant. The ideal class group is an object of fundamental importance in number theory. Nonetheless, despite some conjectures of Cohen and Lenstra about this problem, there was not a single proven case before Bhargava’s work. Bhargava solved the problem for the 2-part of the class groups of cubic fields, in which case, curiously, the numerical evidence had led people to doubt the Cohen-Lenstra heuristics.
Akshay Venkatesh completed his undergraduate degree at the University of Western Australia, Perth, and received his PhD from Princeton University in 2002 under the direction of Peter Sarnak. His mathematical interests center around number theory and automorphic forms. He is particularly interested inequidistribution questions on homogeneous spaces, and the interplay between ergodic and spectral techniques. Akshay was appointed as a Clay Research Fellow for a term of two years beginning 2004.
András Vasy received his PhD from the Massachusetts Institute of Technology in 1997 under the supervision of Richard Melrose. The focus of his research program is partial differential equations in the presence of an additional geometric structure, of a higher rank type. Examples include scattering theory both for N-body quantum Hamiltonians and on higher rank symmetric spaces, and wave propagation on manifolds with corners. In 2002 he received the Alfred P. Sloan Research Fellowship. András was a appointed as a Clay Research Fellow for a term of two years beginning 2004.
David Speyer received his PhD from the University of California, Berkeley in 2005 under the supervision of Bernd Sturmfels. Much of his research is in the emerging area of tropical geometry, to which he has contributed both fundamental results as well as applications, e.g., a new proof of Horn’s conjecture on eigenvalues of hermitian matrices and (with Lior Pachter) the reconstruction of phylogentic trees from subtree weights. His research interests include continuing work in tropical geometry, cluster algebras and the geometry of grassmannians and flag varieties. David was appointed as a Clay Research Fellow for a term of five years beginning June 2005.
Maryam Mirzakhani received her PhD from Harvard University in 2004 under the supervision of Curtis T. McMullen. In her thesis she showed how to compute the Weil-Petersson volume of the moduli space of bordered Riemann surfaces. Her research interests include Teichmüller theory, hyperbolic geometry, ergodic theory and symplectic geometry. Maryam was appointed as a Clay Research Fellow for a term of four years beginning July 2004.
The 2014 Clay Research Award was made to Maryam for her many and significant contributions to geometry and ergodic theory, in particular to the proof of an analogue of Ratner’s theorem on unipotent flows for moduli of flat surfaces.
Ciprian Manolescu received his PhD from Harvard University under the supervision of Peter B. Kronheimer. In his thesis he gave a remarkable gluing formula for the Bauer-Furuta invariants of four-manifolds. His research interests span the areas of gauge theory, low-dimensional topology, symplectic geometry, and algebraic topology. Ciprian was appointed as a Clay Reearch Fellow for a term of four years beginning 2004.
Bo’az Klartag received his PhD from Tel Aviv University in 2004 under the supervision of Vitali Milman. In his thesis Klartag showed that a small number of Minkowski and Steiner symmetrizations suffice to bring a convex body in n-space close to a Euclidean ball. His research interests include geometric problems in high dimension, in particular asymptotic convex geometry. Bo’az was appointed as a Clay Research Fellow for a term of three years beginning September 2005.
Elon Lindenstrauss received his PhD in 1999 from the Hebrew University under the supervision of Benjamin Weiss. His research focuses on the interface between dynamics and other fields, in particular number theory. Using ergodic theoretic and other techniques, he studied the problem of Arithmetic Quantum Unique Ergodicity, a problem at the interface between the theory of automorphic forms and mathematical physics. Elon was appointed as a Clay Research Fellow for a term of two years from September 2003.