Mathematical Sciences: Perverse Sheaves, D-Modules and Their Applications

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Dr. Vilonen intends to continue his work on perverse sheaves, D-modules and their applications to representation theory. There are projects in three different areas. The first area of research is on problems relating to the micro-local structure of perverse sheaves, Riemann-Hilbert correspondence for irregular D-modules (and E-modules) and a direct construction of mixed perverse sheaves. The second area of research is on the Langlands conjecture. Here the investigator intends to work on a geometric form of the Langlands conjecture. This geometric formulation implies the Langlands conjecture for function fields. The third area of research is on representation theory of Lie groups. The goal here is to understand explicitly the structure of (g,K)-modules. These labors are all part of a large effort involving numerous mathematicians to perfect algebraic and analytic tools for relating geometry of spaces to properties of structures which "live" on these spaces.