Title: The Kakeya Conjecture over the Heisenberg Group.

Abstract: The minimal Hausdorff dimension of Kakeya sets in R^n is an interesting open problem where the two-dimensional case can be proven simply and directly, but even obtaining bounds on the higher dimension analogues can require immense machinery. The difficulty of the general case has inspired analysts to look at Kakeya sets from a variety of different viewpoints; some of the most remarkable cases include discretized and finite field variants, other infinite fields with additional structure, and the multilinear case.

In this talk, we explore a construction of the Kakeya set in the first Heisenberg obtained via translations of a 2-dimensional Kakeya set. The additional structure that the sub-Riemannian manifold induced by H^1 provides allows us to utilize tools that have proven ineffective on the more general three-dimensional case. We will formally introduce the Kakeya conjecture in both R^n and H^n, then develop crucial tools and definitions from geometric measure theory which provide insight into these problems.