Mathematical Sciences: Fundamental Groups and Algebraic Geometry

点击翻译按钮获取中文摘要

Arapura DMS 96-23184 This awards supports the research of Professor D. Arapura to work in algebraic geometry. He will study how the fundamental group of a smooth projective variety affects the birational geometry of the space. In particular, he wants to investigate to what extent geometric conditions on the space of representations of the fundamental group can be used to prove the nonvanishing of certain birational invariants of the underlying variety. This research is in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which blossomed to the point where it has, in the past 10 years, solved problems that have stood for centuries. Originally, it treated figures defined in the plane by the simplest of equations, namely polynomials. Today, the field uses methods not only from algebra, but also from analysis and topology, and conversely it is extensively used in those fields. Moreover it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics.