Larry Guth talks: 

Thursday, March 5 (analysis seminar)

Title: A restriction estimate using polynomial partitioning

Abstract:

The restriction conjecture is an open problem in Fourier analysis first
raised by Stein in the late 60?s. It is about the Lp estimates obeyed by an
oscillatory integral operator.  Analysts have raised a number of simpler
model problems to help understand this difficult prob- lem. For instance,
functions supported on long thin tubes come up naturally in the restriction
problem. The Kakeya problem is a purely geometric problem about how long
thin tubes intersect that idealizes some of the issues in the restriction
problem. The Kakeya problem is also a hard open problem. A long thin tube
looks roughly like a line. There are some simpler model problems about the
intersection pattern of a finite set of lines in Euclidean space. Recently,
Zeev Dvir solved one of these model problems about lines in a surprisingly
short paper, using polynomials. (This was surprising because the problem
does not have any obvious connection to polynomials.) It is an open question
how much these polynomials techniques will be able to say about the Kakeya
problem or the restriction problem. In a recent paper, I made a small
improvement in the best estimate for the restriction problem using
polynomials.  In this talk, I will recall the restriction and Kakeya
problems, explain the polynomial approach to estimating the combinatorics of
lines, and explain how to modify this approach to study the restriction
problem.


Friday, March 6 (Colloquium) 
Introduction to incidence geometry.

Abstract: Incidence geometry is a branch of combinatorics that studies the
possible intersection patterns of lines, circles, and other simple shapes. 
For example, suppose that we have a set of L lines in the plane.  An r-rich
point is a point that lies in at least r of these lines.  For a given L, r,
how many r-rich points can we make?  This is a typical question in the
field, and there are many variations.  What if we replace lines with
circles?  What happens in higher dimensions?  We will give an introduction
to this field, describing some of the important results, tools, and open
problems.

We will discuss two important tools used in the area.  One tool is to apply
topology to the problem.  This tool allows us to prove results in R^2 that
are stronger than what happens over finite fields.  The second tool is to
look for algebraic structure in the problem by studying low-degree
polynomials that vanish on the points we are studying.  We will also discuss
some of the (many) open problems in the field and try to describe the nature
of the difficulties in approaching them.