# Seminars & Events for Minerva Lectures

##### Lecture I: Geometry and dynamics on hyperbolic surfaces

The first lecture will give some background on the geometry and dynamics on hyperbolic surfaces. I will give a brief overview of Teichm\"uller theory and properties of the mapping class groups and the space of geodesic currents. I will discuss some natural actions of the mapping class group. I will formulate some basic questions and explain some of the ideas used for studying these objects.

##### Lecture II: Dynamics on moduli spaces of hyperbolic surfaces

In the second lecture, I will discuss several natural geometric flows defined on bundles over the moduli spaces of curves. I will describe basic ergodic properties of these flows. I will discuss some open questions and some of the progress made in recent years.

##### Lecture III: Counting mapping class group orbits on hyperbolic surfaces

Let $X$ be a complete hyperbolic metric on a surface of genus $g$ with $n$ punctures. In this lecture I will discuss the problem of the growth of $s^{k}_{X}(L)$, the number of closed curves of length at most $L$ on $X$ with at most $k$ self-intersections. More generally, we investigate the properties of the orbit of an arbitrary closed curve $\gamma$ under the action of the mapping class group. I will also discuss problems regarding the distribution of the corresponding geodesics on $T^1(X)$.

##### Sparsification of Graphs and Matrices

Random graphs and expander graphs can be viewed as sparse approximations of complete graphs, with Ramanujan expanders providing the best possible approximations. We formalize this notion of approximation and ask how well an arbitrary graph can be approximated by a sparse graph. We prove that every graph can be approximated by a sparse graph almost as well as the complete graphs are approximated by the Ramanujan expanders: our approximations employ at most twice as many edges to achieve the same approximation factor. Our algorithms follow from the solution of a problem in linear algebra. Given an expression for a rank-n symmetric matrix A as a sum of rank-1 symmetric matrices, we show that A can be well approximated by a weighted sum of only O(n) of those rank-1 matrices.

##### The solution of the Kadison-Singer Problem

In 1959, Kadison and Singer posed a problem in operator theory that has reappeared in many guises, including the Paving Conjecture, the Bourgain-Tzafriri Conjecture, the Feichtinger Conjecture, and Weaver's Conjecture. I will explain how we solve the Kadison-Singer Problem by proving Weaver's Conjecture in Discrepancy Theory. I will explain the "method of interlacing polynomials" that we introduced to solve this problem, and sketch the major steps in the proof. These are the introduction of "mixed characteristic polynomials"---the expected characteristic polynomials of a sum of random symmetric rank-1 matrices, the proof that these polynomials are real rooted, and the derivation of an upper bound on their largest roots. These techniques are elementary, and should be understandable by a broad mathematical audience.

##### Ramanujan Graphs and Free Probability

We use the method of interlacing polynomials and a finite dimensional analog of free probability to prove the existence of bipartite Ramanujan graphs of every degree and number of vertices. No prior knowledge of Ramanujan graphs or free probability will be assumed. Ramanujan graphs are defined in terms of the eigenvalues of their adjacency or Laplacian matrices. In this spectral perspective, they are the best possible expanders. Infinite families of Ramanujan graphs were first shown to exist by Margulis and Lubotzky, Phillips and Sarnak using Deligne's proof of the Ramanujan conjecture. These constructions were sporadic, only producing graphs of special degrees and numbers of vertices. In this talk, we outline an elementary proof of the existence of bipartite Ramanujan graphs of very degree and number of vertices.