Alex Iosevich

Title: On the distribution of simplexes in thin subsets of Euclidean space

Abstract: We are going to describe some recent results of the following type. How large does the Hausdorff dimension of a subset of Euclidean space (Or a Riemannian manifold) need to be to ensure that it contains vertices of an equilateral simplex, or that the Lebesgue measure of the set of non-congruent simplexes determined by this set is positive? The key problem in this area is the Falconer distance problem where several significant advances took place in the past two years. Interaction between discrete and continuous aspects of these problems will be emphasized throughout.