Additive Combinatorics is a branch of mathematics concerned with the relationships between various properties of sets of integers, and more generally of subsets of other algebraic structures. It has developed rapidly over the last thirty years and, in the process, has revealed links with several other branches of mathematics, from dynamical systems to group theory. Many of the fundamental theorems of the subject can be proved using Fourier analysis, but other results require a kind of " higher-order Fourier analysis " where linear phase functions are replaced by polynomial phase functions and their generalizations. This year's lecture considered linear results and will be followed, in 2022, by a lecture examining results that require higher-order methods.
Two highlights of the lecture were Roth's theorem and Freiman's theorem. The former states that, for any real number c > 0, if n is sufficiently large and A is a subset of {1,2,...,n} of size at least cn, then A necessarily contains an arithmetic progression of length 3. The second describes the structure of a set of integers whose sum is small : the sum of a set A is the set of all sums x + y such that x and y are both elements of A. A third result presented in the lecture was a theorem I proved that serves as an introduction to additive non-abelian Combinatorics.