A two variable Vandermonde decomposition ...

Date: Thu., Sept. 26, 2024, 5:00pm (Eastern Time)
Speaker: Rodrigo A. Perez, Indiana University, Indianapolis
Title: A two variable Vandermonde decomposition of q-binomials emerging from a complex dynamics problem
Abstract: When a holomorphic function $f:C \to C$ has a fixed point f(0)=0 with derivative λ=f'(0) of unit size, the question arises of conjugating f to the rotation z maps to λ z. This is possible when the argument of λ has good approximation properties; eg, when it is Diophantine. The largest domain of conjugation is known as a Siegel disk.
A famous open problem is to give bounds on the size of Siegel disks. As a concrete case, if $Arg(\lambda)$ is the Golden Ratio, does the Siegel disk contain a disk of radius 1/4?
In the talk I will explore a circle of ideas emerging from our approach to this problem: The value 1/4 is connected to the growth of Catalan numbers enumerating binary trees. A consequence of this correspondence is a 2 variable version of the Vandermonde convolution for (deformed) q-binomials. The work is a joint collaboration with M. Aspenberg, Lund University.