Georgia Institute of Technology College of Sciences

Let C(G) denote the set of lengths of cycles in a graph G. In this talk I shall present the recent proofs of two conjectures of P. Erdos on cycles in sparse graphs. In particular, we show that if G is a graph of average degree d containing no cycle of length less than g, then as d -> \infty then |C(G)| = \Omega(d^{\lfloor (g - 1)/2 \rfloor}). The proof is then adapted to give partial results on three further conjectures of Erdos on cycles in graphs with large chromatic number. Specifically, Erd\H{o}s conjectured that a triangle-free graph of chromatic number k contains cycles of at least k^{2 - o(1)} different lengths as k \rightarrow \infty. We define the {\em independence ratio} of a graph G by \iota(G) := \sup_{X \subset V(G)} \frac{|X|}{\alpha(X)}, where \alpha(X) is the independence number of the subgraph of G induced by X. We show that if G is a triangle free graph and \iota(G) \geq k, then |C(G)| = \Omega(k^2 \log k). This result is sharp in view of Kim's probabilistic construction of triangle-free graphs with small independence number. A number of salient open problems will be presented in conclusion. This work is in part joint with B. Sudakov. Abstract

In a wide range of applications, we deal with long sequences of slowly changing matrices or large collections of related matrices and corresponding linear algebra problems. Such applications range from the optimal design of structures to acoustics and other parameterized systems, to inverse and parameter estimation problems in tomography and systems biology, to parameterization problems in computer graphics, and to the electronic structure of condensed matter. In many cases, we can reduce the total runtime significantly by taking into account how the problem changes and recycling judiciously selected results from previous computations. In this presentation, I will focus on solving linear systems, which is often the basis of other algorithms. I will introduce the basics of linear solvers and discuss relevant theory for the fast solution of sequences or collections of linear systems. I will demonstrate the results on several applications and discuss future research directions.

I will discuss certain geometric properties of random matrices with independent logarithmically concave columns, obtained in the last several years jointly with O. Guedon, A. Litvak, A. Pajor and N. Tomczak-Jaegermann. In particular I will discuss estimates on the largest and smallest singular values of such matrices and rates on convergence of empirical approximations to covariance matrices of log-concave measures (the Kannan-Lovasz-Simonovits problem).

Convex algebraic geometry is an emerging field at the interface of convex optimizationand algebraic geometry. A primary focus lies on the mathematical underpinnings ofsemidefinite programming. This lecture offers a self-contained introduction. Startingwith elementary questions concerning multifocal ellipses in the plane, we move on todiscuss the geometry of spectrahedra and orbitopes, and we end with recent resultson the convex hull of a real algebraic variety.