Local Geometric Langlands Conjecture in Quasi-Classical Setting

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The PI will conduct research in two directions in the area of the geometric Langlands program. The Langlands program started as a series of conjectures proposed by Robert Langlands beginning in 1967. From its origins in number theory and representation theory, it developed into a framework connecting and influencing many areas of mathematics. In particular, the geometric Langlands program is formulated in terms of algebraic geometry and can be interpreted in physical terms as a duality in quantum field theory. This project introduces new geometric ideas into the area. It is likely that the techniques developed in this project will find applications in other areas of mathematics. This award provides training of graduate students through research.In more detail, the PI will work on the classical limit of the local geometric Langlands conjecture and on the duality for compactified Jacobians of formal curves. In earlier work, the PI has studied the classical limit of the global geometric conjecture; while progress in the special case of the group GL(n) was made, it became apparent that the general case will require new tools. One goal of this project is to develop such tools using local methods. At the same time, the research focuses on an essentially unexplored direction in the geometric Langlands program: while the general shape of conjectures is known (and stated in the proposal), the details are unclear at this point. The project aims to clarify the situation by using the easier case of GL(n) to test various conjectures before attempting to prove them in general.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.