Dr. Lukas Geyer (Dept. of Mathematical Sciences, MSU)

10/25/21  4:10pm

Abstract:

Starting in the 1980s with the groundbreaking work of Douady and Hubbard on the dynamics of polynomials in the complex plane, there has been a long-running and quite successful project to describe the dynamics of hyperbolic polynomials and rational functions by simple combinatorial and topological models. (There has also been progress, though much slower, toward the hyperbolicity conjecture, which posits that hyperbolicity is generic for polynomials and rational functions.) It turns out that in complex dynamics, the orbits of the critical points play a pivotal role, and that much of the classification project comes down to classifying maps with finite critical orbits.

While the polynomial case is by now fully understood, many questions still remain open for rational maps. I will give a non-technical overview of one of the main tools in the theory, namely Thurston maps and Thurston's classification theorem. Next I will describe the successful combinatorial classification for rational and anti-rational maps with all critical points fixed, as well as one of the open problems (about finiteness of global curve attractors) where the classification sheds some new light and gives at least partial results. I will also explain how anti-rational maps arise from problems in gravitational lensing, and how these results in complex dynamics lead to novel gravitational lensing configurations.

If time permits, I will say a few words about related problems and examples for transcendental entire and meromorphic functions. This is currently a very active field where many questions remain wide open.

This talk should be accessible to graduate students with some basic background in complex analysis.