Topology Seminar

Abstract:

In the first part of my talk I will address the following enumerative problem: How many ways are there to glue a 2k-gon into a genus g surface? It is relatively easy to enumerate these ways for a small k, for example, if k=2, there are two ways to obtain a sphere and one way to obtain a torus. It turns out, that this problem is related to Theory of Random Matrices: evaluating corresponding moments of eigenvalue distribution of Gaussian random matrices gives a solution for our problem. In the talk I am going to elaborate on this relation and give a short sketch of the proof. There are several variations of the polygon gluing problem with some restrictions, and in the second part of my talk I will consider one of
such variation, which is applicable in Computational Biology. Namely, a glued graph, embedded into a surface, is a dual (in some specific sense) to the so-called Breakpoint graph, which is used in Bioinformatics to evaluate "evolutionary" distances between genomes.

I am going to explain, what an evolutionary distance is and how a breakpoint graph can help evaluate it. I am also going to show the application of a topological recursion to compute evolutionary distance distribution by utilizing properties of the breakpoint graph.