Generalized First-Order Conformal Invariants

Series: Logic Seminar

Speaker: Taneli Huuskonen (University of Helsinki, Mathematics)

Title: Generalized First-Order Conformal Invariants

Date: Tuesday, October 20, 1998

Time: 2:30 PM

Place: 113 McAllister Building

Abstract:  

Let G be a complex domain, that is, an open connected subset of the
complex plane.  The analytic functions defined in G form a ring,
denoted by H(G), under pointwise addition and multiplication.  It is a
classical result that two such rings are isomorphic iff the underlying
domains are conformally equivalent, that is, can be mapped to each
other by an analytic bijection.  It turns out that many traditional
conformal invariants can be expressed in terms of the first-order
theory of the ring.  In fact, it is independent of ZFC whether the
first-order theory of H(G) actually determines the domain G up to
conformal equivalence.

We look at some of the highlights along the path to this result and
study the possibilities of generalizing it to various subrings of the
rings H(G).