Kolloquium FB10 und Sondervorträge

Coxeter groups arise in many areas of mathematics including topology, algebraic geometry, representation theory, and combinatorics. They are much studied and generally well-understood. Closely related to Coxeter groups are a class of groups known as Artin groups. The best known examples of these groups are the braid groups. While easy to define, Artin groups are much more mysterious than Coxeter groups with many key properties unknown and basic questions unanswered. In this talk, I will introduce Artin groups, discuss some of the open problems, and survey some recent work, using both combinatorial and geometric methods, to approach these problems.

Given some large positive integer N, what is the largest possible size of a subset of {1,...,N} which does not contain a three-term arithmetic progression (i.e. without three distinct elements x,y,z satisfying x+z=2y)? Similarly, given a prime p and a large positive integer n, what is the largest possible size of a subset of the vector space F_p^n which does not contain a three-term arithmetic progression (i..e without three distinct vectors x,y,z satisfying x+z=2y)? These are long-standing problems in additive combinatorics. This talk will explain the known bounds for these problems, give an overview of some of the proof techniques, and discuss additional applications of these techniques to other additive combinatorics problems.