Introduction to Incidence Geometry
Speaker: Larry Guth, MIT
Location: Warren Weaver Hall 1302
Date: Monday, December 16, 2013, 4 p.m.
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.