This proposal concerns problems in number theory, algebraic geometry, and representation theory in three broad areas. First, the PI is focusing on topics related to cohomology of arithmetic groups and allied areas, such as geometry of locally symmetric spaces and their compactifications, modular curves and rings of modular forms, and special values of L-functions. Second, he is investigating the geometry of canonical Kazhdan-Lusztig cells in affine Weyl groups. Finally, he is joining F. Hirzebruch and D. Zagier in the updating of their book, The Atiyah-Singer theorem and elementary number theory.
This proposal deals with number theory, algebraic geometry, and representation theory. Number theory is the study of the properties of the whole numbers, and is the oldest branch of mathematics. Algebraic geometry studies geometric figures that can be defined by the simplest of equations, namely polynomials. Representation theory is the systematic study of symmetry, through the development of simple mathematical objects that encode the fundamental irreducible pieces of symmetry. Today the questions and phenomena addressed by these subjects serve as driving forces in much of contemporary mathematics research. Moreover, the subjects have contributed many applications in such diverse areas as codes and data transmission, robotics, chemistry, physics, and theoretical computer science.
