Keith Rogers

Title: Improved bounds for Kakeya in intermediate dimensions using semialgebraic geometry

Abstract: We will consider the degree to which unit length tubes of diameter $\delta$, which point in directions which are separated by $\delta$, can be compressed. The Kakeya conjecture can be formulated as a lower bound on the measure of any set that contains the tubes. On the one hand, we will see that the bound holds whenever the containing set is a neighbourhood of a real algebraic variety, confirming a conjecture of Guth. The proof employs tools from semialgebraic geometry including Gromov's algebraic lemma and Tarski's projection theorem. On the other hand, we will see that the conjectured bound holds if there is no algebraic structure at all, using polynomial partitioning. Balancing between the two cases yields improved bounds in certain intermediate dimensions. This is joint work with Jonathan Hickman and Nets Katz.