Representation Theory and Mathematical Physics

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ABSTRACTPrincipal Investigator: Kapranov, Mikhail Proposal Number: DMS - 0925341 Institution: Yale University Title: Representation Theory and Mathematical PhysicsLie groups and their representations are a fundamental area of mathematics, with connections to geometry, topology, number theory, physics, combinatorics, and many other areas. Representation theory is one of the cornerstones of the Langlands program in number theory, dating to the 1970s. Zuckerman's work on derived functors, the translation principle, and coherent continuation lie at the heart of the modern theory of representations of Lie groups. One of the major unsolved problems in representation theory is that of the unitary dual. The fact that there is, in principle, a finite algorithm for computing the unitary dual relies heavily on Zuckerman's work.In recent years there has been a fruitful interplay between mathematics and physics, in geometric representation theory, string theory, and other areas. New developments on chiral algebras, representation theory of affine Kac-Moody algebras, and the geometric Langlands correspondence will be some of the focal points of the conference. Recent developments in the geometric Langlands program point to exciting connections between certain automorphic representations and SYZ type fibrations in geometric mirror symmetry. Problems about these representations may now be amenable to new geometric techniques and insights developed for mirror symmetry.