Functional transcendence of Bottcher coordinates

This week, we will have two talks instead of one. Both are informal, describing work in progress.

Thursday, May 15, 2025
4:00-6:00pm
Science Center Room 232

Sina Saleh (Harvard)

Title: Functional transcendence of Bottcher coordinates

Abstract: A result due to Becker and Bergweiler shows that if f is not an exceptional polynomial, then the associated Böttcher coordinate Phi which sends an open subset of the basin of infinity to a disk D_r of radius r < 1 must be a transcendental function. In other words, there cannot exist a polynomial Q such that Q(z,Phi(z)) = 0. It follows similarly that the inverse Psi of Phi is also a transcendental function. In this talk, we will investigate a similar question: Suppose C is an algebraic curve through (0,0) in (D_r)^2 given by a local parametrization (a(t),b(t)). When can there exist a polynomial P such that P(Psi(a(t)), Psi(b(t))) = 0? We will address this question and consider higher-dimensional versions.

Alex Kapiamba (Harvard)

Title: Punctures in COR curves

Abstract: Many examples of families of rational maps are defined by critical orbit relations. In this talk we will look at some methods of studying degeneration in these families, focusing on the Per_n(0) curves in the moduli space of quadratic rational maps.