K-point congurations and multilinear generalized Radon transforms

Abstract: 

As big data sets have become more common the interest in finding and understanding patterns in them has grown. A classical question on patterns (i.e. configurations), known as the Erdos distinct distance problem, asks what is the least number of distinct distances determined by N points in the plane. A continuous analog of this is the Falconer distance problem. Although originally formulated around distance, both problems also relate to configurations since the distance between two points can be thought of as a 2-point configuration. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for higher order configurations. For example a triangle can be viewed as a 3-point configuration which then naturally leads to the question of what is the least number of distinct triangles determined by N points in the plane. In this talk we will give a brief history of Erdos and Falconer type questions for k-point configurations and then present recent Falconer type theorems for a wide class of k-point configurations in any dimension. Techniques from geometric measure theory and analysis are used to establish these recent theorems. The key step is to obtain bounds on multilinear analogs of the generalized Radon transforms introduced by Phong and Stein. In the talk we will also present these operators and their estimates.