# Fernando Roman-Garcia

University of Illinois, Urbana-Champaign

**Talk Title: **Intersection of projections and slices of sets in the Heisenberg group

**Abstract: **We study the Hausdorff dimension of the intersection of horizontal projections of subsets of the Heisenberg group, as well as dimension of intersections of sets with vertical subgroups. We will see that if A and B are Borel subsets of the nth-Heisenberg group, and both have dimension greater than m, then for a positive measure set of m-dimensional horizontal subgroups, the intersection of the images of A and B under the projection onto these subgroups have positive Hausdorff m-measure. We will also see that if A is a measurable set of Hausdorff dimension greater than m, then there is a positive measure set of vertical planes for which their left translates by almost every point in the group intersects A on a set of dimension dim(A)-m.