Borys Kuca

Structure of 2-step nilpotent ergodic averages

Multiple ergodic averages play a central role in transferring results from ergodic theory to combinatorics and number theory. Of particular interest are averages with polynomial iterates that arise in the polynomial Szemerédi theorem and its extensions. Their norm convergence was established by Walsh 15 years ago, but his method gave virtually no information on the structure of the limits. Building on recent advances in the abelian setting, I will describe new limiting formulas for polynomial averages induced by 2-step nilpotent measure-preserving actions. I will also present new (positive and negative) results on the joint ergodicity conjecture and introduce various open problems. The talk is based on joint work with Andreas Koutsogiannis and Wenbo Sun.