Tova Brown

Mathematics Graduate Student
University of Arizona

Welcome to my research page!

Here are the slides from a talk I recently gave to advanced undergraduates that explain a little bit about the family of maps that I propose to enumerate and how this count is related to random matrix theory.

Here is the paper I wrote for my oral comprehensive exam. It gives a more thorough introduction to the maps I am interested in enumerating and the connection to random matrix theory, as well as a derivation of the loop equations of random matrix theory. This lays the groundwork for a method I hope to use to study problems of geodesic distance on maps.

A short introduction to my area

Interest in the enumeration of planar graphs started back in the 1960s with the work of William T. Tutte, a Canadian mathematician and code-breaker. Physicists became interested in the subject because certain kinds of interesting mathematical graphs correspond to the Feynman diagrams of quantum mechanics. More recently, mathematicians have become interested again because of a remarkable connection between the combinatorial enumeration, matrix integrals, and other developing areas of math.

I became interested in this area for that reason too: that questions, ideas, techniques, and structure from several different areas of math are playing together very closely here. I started out learning the most accessible part of the project first, that is, some of the topology and combinatorics that are essential to defining the graphs that we propose to count and what it means to count them. Now I am immersed in learning the analysis and random matrix theory, from the probability measures underlying our matrix spaces to the asymptotic expansions we obtain and the orthogonal polynomials that pop up everywhere.

A computing project

My dissertation work moved into study of discrete dynamical systems, as each of the map enumeration problems discussed above can be viewed as one small and very special orbit within a discrete dynamical system. I wrote simulations in Python 2 to investigate a number of questions about these systems (for three different enumeration problems). The code found here is for a recurrence problem for recurrence coefficients for orthogonal polynomials. It is a small bundle of code to investigate a few basics of this system. Run Example1 or Example 2 . Both use code found in this file , and to plot you'll also need this file .

And here is a set of notes I wrote to explain the mathematical problem and the code.

Research I did as an undergrad

I spent the summer of 2009 at Clarkson University in Potsdam, NY. Under the guidance of Dr. Aaron Luttman, I studied spectral preserver problems in bounded linear operators on Banach spaces. This work continued for the next academic year and became my senior thesis as well as a published paper in the Annals of Functional Analysis (linked to here).

In the summer of 2008 I participated in the University of Arizona's Arizona Summer School on Computational Group Theory. This was a 4-week summer school and introduction to research. Along with three other students, I studied the strong symmetric genus of groups acting on compact Riemann surfaces. We extended a classification using the computer algebra system GAP and published our results (full text available here on arXiv) in the Houston Journal of Mathematics.