Langlands Duality and Moduli Spaces of G-Bundles

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Abstract for award of Braverman DMS-0300271This proposal consists of 3 parts. The first part is joint with R.Bezrukavnikov and is devoted to the study of an analog of the geometric Langlands conjecture (as formulated by Beilinson and Drinfeld) in the case when the base field has positive characteristic. The second part is joint with M.Finkelberg and D.Gaitsgory. There the PI proposes to study the intersection cohomology of the so called Uhlebeck spaces of G-bundles on the projective plane. The answer should be useful for studying super-symmetric gauge theories in the way suggested recently by Nekrasov. In the third part the PI proposes to apply the so called "versal deformations" of formal arcs (defined recently by Drinfeld and Grinberg-Kazhdan) to the problem of construction of local L-functions for reductive groups over a local field of positive characteristic.This proposal is in the part of mathematics known as the Langlands program. The Langlands program is part of number theory. Number theory is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished a driving force in creating new mathematics in other diverse parts of the discipline. The Langlands program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. One aspect of this proposal is to explore the applications of geometric techniques within the Langlands program.