Algebra Seminar talk
2009-04-15
Katherine F Stevenson
Fundamental groups in characteristic p
Abstract:
The talk focuses on field extensions and covering spaces. The goal is to point out an important connection between topology and field theory. This connection also touches on analysis and algebraic geometry and is both a beautiful result and a powerful tool. We will look at Galois extensions and covering spaces on the finite level first and then discuss how the connection extends to the infinite level, connecting absolute Galois groups to fundamental groups.
The rest of the talk addresses how this correspondence has been used to prove theorems in algebraic geometry over fields of arbitrary characteristic. We will discuss connections to Hilbert's inverse Galois Problem, Shafarevich's Conjecture for $G_{Q^{\text{cyclic}}}$, and Abhyankar's 1957 Conjecture on the fundamental group of a curve. Then we will focus on recent results on the profinite group structure of absolute Galois groups.