Philip Gressman

Title: Nonconcentration, $L^p$-Improving Estimates, and Multilinear Kakeya

Abstract: In this talk I will discuss a series of related results which relate the problem of establishing $L^p$-improving estimates for Radon-like transforms in intermediate dimensions to recent algebraically formulated results generalizing multilinear Kakeya. The missing link in this chain is a new kind of inequality which quantifies the limited extent to which neighborhoods of algebraic varieties can contain large product sets. We call these new inequalities nonconcentration inequalities and explore the role that these inequalities play in the theory of geometric averaging operators and in understanding generalizations of Hausdorff measure which include Oberlin's affine Hausdorff measure.