Michael Christ (UC Berkeley)

Title: Near Equality in Affine-Invariant Inequalities

Abstract:
Three classical inequalities in geometry and analysis are those
of Brunn-Minkowski, Riesz-Sobolev, and Young. In each, the size
of a set is quantified using Lebesgue measure. 
Each of these inequalities has the rare feature of invariance
under the full group of all affine symmetries of Euclidean space.

Each inequality is known to be an exact equality only for sets with particular structure.
For each, this is a class of sets which can (and should) be regarded as
continuum analogues of finite arithmetic progressions of minimal rank.

In the first lecture, these inequalities and their long histories will be reviewed.
We will introduce recent results which characterize those sets that nearly achieve exact equality. 
These results rely on an interplay between analysis, the geometry of Euclidean space,
and --- crucially --- on structural input from additive combinatorics. 
Two fundamental results of additive combinatorics will be reviewed.

In the second lecture, we will discuss an outline and key ideas 
for the characterization of near equality in the one-dimensional Riesz-Sobolev inequality.
If time permits, one element of the analysis of the Brunn-Minkowski inequality
in dimensions greater than one will be introduced.