Algebraic and Probabilistic examples in combinatorial geometry

Indiana University, Bloomington IN

Investigators

Abstract

DMS 0432237 Nets Katz Indiana University Algebraic and Geometric Examples in Combinatorial Geometry ABSTRACT We will study a collection of problems of similar flavor coming from different parts of mathematics. The problems are all combinatorial in nature and involve estimates on the overlap between various geometric objects. We will attempt to produce examples in these problems by both algebraic and probabilistic methods. A great deal of mathematical analysis involves handling the infinite. Here however we engage in a deep study of the finite. To be precise, we study the ways in which finite collections of geometric objects like rectangles, patterns of points in a grid, and vectors on a high dimensional sphere can be arranged to achieve certain unusual effects. We will attempt to discover new and better such arrangements. One way of doing this is by using a known algebraic structure to suggest a construction. Another possibility is to choose the example randomly within certain constraints.