“Multilinear operators: a bridge between combinatorial number theory and algebraic geometry”

I will introduce a series of interesting operators in harmonic analysis that are deeply connected with ergodic theory, number theory and PDE. Various techniques used in studying these operators will be mentioned along the way. As an example, I will show why harmonic analysis can help reduce a number theory problem (Roth theorem) to an algebraic geometry problem (which is solvable bytheories of Deligne and Katz): this joint work with Li and Sawin fully answers a question of Bourgain and Chang about existence of three-term polynomial progressions in subsets of finite fields.