Department of Mathematics - Mathematics Colloquium - On Besicovitch compression of curves and surfaces

Given a subset of R^d containing copies of prescribed curves or surfaces, what is a sharp estimate on the size of smallest such subset? For example, Besicovitch proved the existence of a measure zero compact set that contains a unit line segment in every direction, which is famously known as a Kakeya set. On the other hand, the famous Kakeya conjecture asserts that such Kakeya sets in R^d must have full dimension d. In this talk, we shall discuss the key geometric reasoning behind how far we can compress these curves or surfaces into a tiny subset.