Fundamental Groups and Absolute Galois Groups

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

The Principal Investigator studies fundamental groups and absolute Galois groups of varieties, especially in finite characteristic. In particular, he investigates the fundamental groups of affine curves over an algebraically closed field of finite characteristic; the absolute Galois groups of function fields of curves over non-algebraically closed fields of finite characteristic; and the fundamental groups and absolute Galois groups of higher dimensional varieties over algebraically closed fields of arbitrary characteristic. This study is in large part guided by possible generalizations of Abhyankar's Conjecture for curves, which the Principal Investigator has proven. A major goal is to determine the properties of the fundamental groups and absolute Galois groups in question, and to determine which finite groups are quotients of these profinite groups, in order to obtain greater insight into the algebra and geometry of these spaces. Methods include formal patching, embedding problems, cohomology, valuation theory, and group theory. The area of this project relates aspects of algebra and geometry in a way that makes it possible to solve problems that would be intractible in either field alone. The linkage between the algebra and geometry arises from geometric spaces that are defined by algebraic equations. These spaces can have complicated patterns of symmetry, which manifest themselves both in the algebra and the geometry. This project seeks to understand what types of symmetries can occur in the context of these spaces, and how spaces with one type of symmetry can relate to a given space with another type of symmetry. These spaces are in many cases defined with respect to an algebraic system in which a particular prime number plays a special role; and the properties of those spaces can be interpreted as giving information about the solvability of equations involving prime numbers.