Mathematical Sciences: Algebra, Number Theory and Algebraic Geometry

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9622742 Kazhdan This award provides funds for the investigations of David Kazhdan and Benedict Gross in several areas of representation theory and number theory. Kazhdan will continue his work on the discrete series for GL(n) and on minimal representations over local fields. He is also addressing several problems posed by Drinfield on quantization, and on generalizations of affine Lie algebras. Gross is working on a construction of motives with Galois Group G2, via an exceptional theta correspondence, and on a generalization of Serre's conjectures on Galois representations to finite groups of Lie type. This research falls into the fields of Algebraic Geometry and Number Theory. Algebraic Geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays, the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics. Number Theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.