Emmy Noether Lecture Series

Sponsored by the Minerva Research Foundation

Abstract: Projection theory asks how the size of a set in $R^n$—often measured by Hausdorff dimension—behaves under projections onto lower-dimensional spaces, both for orthogonal projections and for more general nonlinear families. One would like to understand when dimension is typically preserved and to control the (often small) exceptional set of directions or parameters where it is not.

In this lecture series we study how to find structures in an arbitrary set in $R^n$ and how those structures can be used to prove projection estimates.  We will survey discretized and multi-scale ideas initiated by Bourgain and developed further by Shmerkin, Orponen, and others, which yield recent progress on projection problems.