Mathematical Sciences: Algebra, Number Theory and Algebraic Geometry

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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.