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

维洛宁博士打算继续他的研究 层、D-模及其在表示中的应用 理论 有三个不同领域的项目。 研究的第一个领域是关于 反常层的微观局部结构 不规则D-模（和E-模）的对应和 混合反常层的直接构造。 第二个研究领域是朗兰兹猜想。 在这里，研究人员打算对几何形式的工作， 朗兰兹猜想 这个几何公式意味着 函数域的朗兰兹猜想。 研究的第三个领域是表征理论 李群。 这里的目标是明确地理解 （g，K）-模的结构 这些工作都是一项巨大努力的一部分， 无数数学家完善了代数和分析工具 用于将空间的几何形状与结构的性质联系起来， “活”在这些空间里。

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.