We will discuss some recent work on quantifying the gaps between eigenvalues of sparse random matrices.  Before these results, it was not even known if the eigenvalues were distinct for this class of random matrices.  We will also touch upon how these results relate to random graphs, the graph isomorphism problem and nodal domains.  This is joint work with Van Vu and Patrick Lopatto.

Consider an adjacency matrix of a bipartite, directed, or undirected Erdos-Renyi random graph. If the average degree of a vertex is large enough, then such matrix is invertible with high probability. As the average degree decreases, the probability of the matrix being singular increases, and for a sufficiently small average degree, it becomes singular with probability close to 1. We will discuss when this transition occurs, and what the main reason for the singularity of the adjacency matrix is. This is a joint work with Anirban Basak.

Hashing is a technique widely used in coding theory (an area of computer science) and in cryptography. Although hashing is an interesting mathematical object, it is surprisingly little known to the "mainstream" mathematicians. I will focus on one specific result on hasing, namely the Leftover Hash Lemma. We will state it as a result in extremal combinatorics, give a probabilistic proof of it, and relate it to another fundamental result in extremal combinatorics, the Sauer-Shelah Lemma.

A phase transition is a sharp change in the behavior of a mathematical model as one of its parameters varies. This talk describes a striking phase transition that takes place in conic geometry. First, we will explain how to assign a notion of "dimension" to a convex cone. Then we will use this notion of "dimension" to see that two randomly oriented convex cones share a ray with probability close to zero or close to one. This fact has implications for many questions in signal processing. In particular, it yields a complete solution of the "compressed sensing" problem about when we can recover a sparse signal from random measurements. Based on joint works with Dennis Amelunxen, Martin Lotz, Mike McCoy, and Samet Oymak.