2005 Clay Research Awards Announced

September 23, 2005

CAMBRIDGE, MA - The Clay Mathematics Institute (CMI) announces the recipients of the 2005 Clay Research Award: Manjul Bhargava, of Princeton University and Nils Dencker, of Lund University. The awards will be presented at CMI's annual meeiting, held this year at Oxford University on October 11. Following the presentation of the awards, there will be talks on the awardees' work and then a public lecture by Sir Andrew Wiles. See www.claymath.org for further information.

CMI recognizes 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, namey, 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.

CMI recognizes Nils Dencker for his complete resolution of a conjecture made by F. Treves and L. Nirenberg in 1970. This conjecture posits an essentially geometric necessary and suffcient condition, "Psi", for a pseudo-differential operator of principal type to be locally solvable, i.e., for the equation Pu = f to have local solutions given a finite number of conditions on f. Dencker's work provides a full mathematical understanding of the surprising discovery by Hans Lewy in 1957 that there exist a linear partial differential operator -- a one-term, third-order perturbation of the Cauchy-Riemann operator -- which is not local 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.

Manjul Bhargava

Manjul Bhargava was born in Hamilton, Ontario, Canada but spent most of his early years in Long Island, New York. He received his A.B. in Mathematics summa cum laude from Harvard University in 1996 and his Ph.D. from Princeton University under the advisorship of Andrew Wiles in 2001. After brief visiting positions at the Institute for Advanced Study and Harvard University, he joined the faculty of Princeton University as Professor of Mathematics in 2003. An accomplished tabla player whose primary research interests lie in number theory, representation theory, and algebraic geometry, Bhargava has received numerous awards and honors, including three Derek Bok Awards for Excellence in Teaching (1993-95), the Hoopes Prize for Excellence in Scholarly Work and Research from Harvard University (1996), the AMS-MAA-SIAM Morgan Prize for Outstanding Undergraduate Research in Mathematics (1997), the MAA Merten M. Hasse Prize for Exposition (2003), a Packard Foundation Fellowship in Science and Engineering (2004), and the AMS Blumenthal Award for the Advancement of Pure Mathematics (2005). Bhargava was also the Clay Mathematics Institute's first five-year Research Fellow (2000-05). He was named one of Popular Science magazine's "Brilliant 10" in 2002.

Nils Dencker

Nils Dencker was born 1953 in Lund, Sweden. He received his Ph.D. from Lund University in 1981 under the direction of Lars Hormander. After spending 1981-1983 as a C.L.E. Moore instructor at MIT, he returned to Lund University. He has been the Director of Studies at the Department of Mathematics 2001-2003. Dencker received in 2003 the Gaarding prize from the Royal Physiographic Society in Lund. He is currently the vice chairman of the Swedish Mathematical Society.

Dencker's research interests lie in the microlocal analysis of partial differential equations and the calculus of pseudodifferential operators. He has studied the propagation of polarization (vector valued singularities) for systems of partial differential equations, for example in double refraction. Dencker has also studied the pseudospectra (spectral instability) of semi-classical partial differential equations and the solvability of partial differential equations.

CMI

The Clay Mathematics Institute (CMI) is a private, non-profit foundation based in Harvard Square which is dedicated to increasing and disseminating mathematical knowledge. CMI aims to further the recognition of the beauty, power and universality of mathematical thought. Its programs include an annual summer school, conferences and workshops, public lectures, programs for talented high school students, and support for individual researchers, notably the Clay Research Fellows and Clay Senior Scholars. Please see www.claymath.org. Contact: James Carlson, President, or David Ellwood, Research Director. (617) 995 2600, Email: carlson@claymath.org; ellwood@claymath.org