Larry Guth

Abstract:   In the last five years, several hard problems in combinatorics have been solved by using polynomials in an unexpected way. Some of the proofs are very short - so I can present a complete problem and proof in the first half of the colloquium. The problems I will talk about have to do with the incidence geometry of lines: one considers a large number of lines in R^n and tries to understand the possible intersection patterns. In the second half of the colloquium we will discuss this field and I will try to describe some of the main ideas and challenges. At the end of the talk, we will put the new methods in perspective within the field.