|DATE||April 22 (Thu), 2021|
|INSTITUTE||University of Chicago|
|TITLE||Sausages and Butcher Paper II (KIAS Quadranscentennial Lectures)|
|ABSTRACT||Zoom Link: https://us02web.zoom.us/j/82708491826
Poster : https://drive.google.com/file/d/1aUn9IV-SRJHajhMXDMUF8b4pg2qnK36k/view?usp=sharing
The shift locus S_d is the space of conjugacy classes of degree d polynomials f(z) in one complex variable for which all the critical points tend to infinity under repeated application of f. When d=2 this is the complement of the Mandelbrot set. Although S_d is a very complicated space geometrically, it turns out one can get a surprisingly concrete description of its topology; for example, S_2 is homeomorphic to an open annulus (this is equivalent to the famous theorem of Douady-Hubbard that the Mandelbrot set is connected). I would like to discuss two very explicit ways to capture the topology of S_d, one via the combinatorics of laminations (Butcher paper) and one via algebraic geometry (sausages). As a corollary of this explicit description one can show that S_d is a K(pi,1) with the homotopy type of a complex of half its real dimension.