Topics in Arithmetic Algebraic Geometry

Columbia University, New York NY

Investigators

Abstract

The investigator and his collaborators study the following topics in arithmetic algebraic geometry: the uniformity of small points for family of heights, the arithmetic Kodaira-Spencer class for arithmetic surfaces with applications to the effective Mordell conjecture, the Gross-Zagier type formula for Shimura varieties with applications to the Birch and Swinnerton-Dyer conjecture and the Andre-Oort conjecture. This is a project in a subfield of mathematics known as number theory. Many of these questions are motivated by the philosophy that algebraic information can be obtained by geometric methods. At the center of this project is the use of a symmetric group, or algebraic group, which is an object that is both algebraic and geometric in nature. These symmetric groups arise naturally in physics and chemistry. It is not too ambitious to say that the solution to the problems in this proposal will one day affect research in cryptography, theoretical physics, and quantum computing.