Georgia Institute of Technology College of Sciences

Discrepancy theory, also referred to as the theory of irregularities of distribution, has been developed into a diverse and fascinating field, with numerous closely related areas, including, numerical integration, Ramsey theory, graph theory, geometry, and theoretical computer science, to name a few. Informally, given a set system S defined over an n-item set X, the combinatorial discrepancy is the minimum, over all two-colorings of X, of the largest deviation from an even split, over all sets in S. Since the celebrated ``six standard deviations suffice'' paper of Spencer in 1985, several long standing open problems in the theory of combinatorial discrepancy have been resolved, including tight discrepancy bounds for halfspaces in d-dimensions [Matousek 1995] and arithmetic progressions [Matousek and Spencer 1996]. In this talk, I will present new discrepancy bounds for set systems of bounded ``primal shatter dimension'', with the property that these bounds are sensitive to the actual set sizes. These bounds are nearly-optimal. Such set systems are abstract, but they can be realized by simply-shaped regions, as halfspaces, balls, and octants in d-dimensions, to name a few. Our analysis exploits the so-called "entropy method" and the technique of "partial coloring", combined with the existence of small "packings".

The total diameter of a closed planar curve C is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of C. Furthermore, when C is convex or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds in the convex case only when C is a circle. We also generalize these results to m dimensional submanifolds of R^n, where the "area" will be defined in terms of the mod 2 winding numbers of the submanifold about the n-m-1 dimensional affine subspaces of R^n.

First-passage percolation is a model of a random metric on a infinite network. It deals with a collection of points which can be reached within a given time from a fixed starting point, when the network of roads is given, but the passage times of the road are random. It was introduced back in the 60's but most of its fundamental questions are still open. In this talk, we will overview some recent advances in this model focusing on the existence, fluctuation and geometry of its geodesics. Based on joint works with M. Damron and J. Hanson.

The three objects in the title come together in the study of ergodic properties of geodesic flows on flat surfaces. I will go over how these three things are intimately related, state some classical results about the unique ergodicity of translation flows and present new results which generalize much of the classical theory and also apply to non-compact (infinite genus) surfaces.