Mathematics Colloquium

First moment obstructions are certificates via simple union bounds of the inexistence of target structures in a random system. In recent years, first moment obstructions have been proven to be the main driving force across multiple fundamental phenomena in probability and combinatorics.

I will highlight new perspectives on first moment obstructions as a measure of complexity of a collection of structures. These perspectives open up new avenues towards understanding sets with small doubling, which are fundamental objects in additive combinatorics. As a demonstration, I will describe an essentially complete resolution of a conjecture of Alon, Balogh, Morris and Samotij on sharp enumeration of sets with small doubling.