Olof Sisask: Solutions to systems of linear equations modulo primes, via higher-order Fourier analysis

A fundamental theorem of Szemeredi asserts that even if one is allowed to delete 99.9% of the integers, substantial additive structure must remain -- in particular, one will be left with arithmetic progressions of arbitrary length. One can study similar questions in other groups, a particularly natural setting being that of Z/pZ, the integers modulo a prime. The aim of this talk is to describe some additive combinatorial results in this setting -- in particular about the minimal number of k-term progressions left in Z/pZ after a deletion such as the above -- and in the process to illustrate some notions from the developing theory of higher-order Fourier analysis. From joint work with Pablo Candela.