This project is devoted to the study of the geometric Langlands program. This is a relatively new field whose basic feature is the employment of methods of modern algebraic geometry (e.g., the theory of perverse sheaves, algebraic stacks, D-modules, conformal field theory) for investigation of various problems in representation theory and in number theory. The purpose of the geometric Langlands program is to relate local systems on an algebraic curve with respect to some reductive group to perverse sheaves on the moduli space of principle bundles on that curve with respect to the Langlands dual group. In this proposal three particular directions of study are suggested: 1) Construction of automorphic sheaves for GL(n) using a geometric analog of the Whittaker model; 2) The study of automorphic sheaves that correspond to local systems induced from the maximal torus, i.e. the case of Eisenstein series; 3) Construction of automorphic sheaves for GL(2) that correspond to 2-dimensional ramified local systems. Progress in each of the above directions will contribute to a better understanding of the general structure of the Langlands correspondence and of the related number-theoretic and representation-theoretic problems.
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 integral numbers and is the oldest branch of mathematics. From the outset, problems in number theory have furnished a driving force in creating new ideas in all areas of mathematics. The Langland's program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of integral numbers. The object of this proposal is to explore the applications of geometric techniques within the Langlands program.
