Mathematical Sciences: Algebra, Number Theory and Algebraic Geometry

本研究将延续首席研究员在数论和表征理论方面的研究。Kazhdan的项目包括仿射标志流形的研究，迹公式和局部域上约化群的倒钩几何。Gross正在研究与正交群相关的Shimura变量上的某些Heegner循环以及类数公式的推广。Tate正在研究阿贝尔变体的高度配对，对Birch和Swinnerton-Dyer猜想的改进，以及某些三维正则代数的分类。这三位研究者都在桥梁领域工作，应用代数和几何技术来回答数论中的问题。最终的目标是解整数形式的方程。这些方程所定义的几何对象是研究的主要对象。

This research will continue the research of the Principal Investigators' in number theory and representation theory. Kazhdan's projects include the study of affine flag manifolds, the trace formula and the cuspidal geometry of reductive groups over local fields. Gross is studying certain Heegner cycles on Shimura varieties associated to orthogonal groups as well as generalizations of the class-number formula. Tate is working on height pairings for abelian varieties, refinements of the conjecture of Birch and Swinnerton-Dyer as well as the classification of certain 3-dimensional regular algebras. All three of these researchers work on the bridge area that applies algebraic and geometric techniques to answer questions in number theory. Ultimately the goal is to solve equations in integers. The geometric objects these equations define are the main objects of study.