Larry Guth: Polynomial methods in incidence geometry

Incidence geometry is a part of combinatorics studying the possible intersection patterns of lines, circles, or other simple shapes. For example, given L lines in the plane, what is the maximum possible number of points that lie in at least r lines? This problem was solved up to a constant factor by a fundamental theorem of Szemeredi and Trotter from the early 80's. Nevertheless, there are many simple open problems - for example, if we replace lines by circles in the problem above.

We will describe recent work in this area using polynomials in a somewhat unexpected way. This approach was influenced by work in error-correcting codes, where interesting ideas about polynomials are used to make good codes and good methods of decoding. These methods have led to good new estimates about the intersection patterns of lines in 3-dimensional space.