Geometry of Langlands Duality

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The project is directed towards building ideas that are expected to underlie and explain various duality phenomena in mathematics (the Langlands program) and physics (gauge theory and string theory). In each case, "duality" means that one phenomenon has two equally valid descriptions. However, at the current level of understanding these seem unrelated. Two distinct descriptions of the same object suggest that a more comprehensive framework would serve to give a conceptual explanation of the observed dualities. The aim of this project is to develop such a framework. The mathematical origin of these developments is the so called Langlands program, which is the modern view on classical number theory. The Langlands program functions as a bridge between mathematics (representation theory, algebraic geometry, homotopy theory) and physics (gauge theory, string theory). Graduate students will be involved in the research.In more technical terms, this project will extend the formulation of the geometric Langlands program from one dimension to arbitrary dimensions. Going beyond dimension one requires a reformulation of the geometric Langlands program that better matches physical theories in higher dimensions. The basis of this project is the construction of a conjectural "inner cohomology in algebraic geometry," including noncommutative coefficients. Here, "inner" means that the values of this cohomology would be objects in derived algebraic geometry rather than simply sets. As a consequence, geometric class field theory in arbitrary dimension should be Poincare duality for this inner cohomology and the geometric Langlands program should be an extension of Poincare duality to non-commutative cohomology. A classical problem is that cohomology with coefficients in a non-abelian group is not defined beyond degree two and does not have much content beyond degree one. The point of view taken in this project is that higher cohomology with coefficients in a reductive group can be defined as certain local distributions on cohomology with coefficients in tori.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.