Colloquium Ical Atom Mailing List

Polynomial equations and polynomial maps are central objects in modern mathematics, and understanding their geometry and singularities is of great importance. In this talk, I will pitch the idea that polynomial maps can be studied by investigating analytic properties of regular measures pushed-forward by them (over local and finite fields). Such pushforward measures are amenable to analytic and model-theoretic tools, and the rule of thumb is that singular maps produce pushforward measures with bad analytic behavior. I will discuss some results in this direction, as well as some applications to group theory and representation theory. In particular, I plan to mention some recent results on local integrability of Harish-Chandra characters.

Based on joint projects with R. Cluckers, I. Glazer, J. Gordon and S. Sodin.

The Gowers uniformity k-norm on a finite abelian group measures the averages of complex functions on such groups over k-dimensional arithmetic cubes. The inverse question about these norms asks if a large norm implies correlation with a function of an algebraic origin. The analogue of the Gowers uniformity norms for measure-preserving abelian actions are the Host-Kra-Gowers seminorms, which are intimately connected to the Host-Kra-Ziegler factors of such systems. The corresponding inverse question, in the dynamical setting, asks for a description of such factors in terms of systems of an algebraic origin. In this talk, we survey recent results about the inverse question in the dynamical and combinatorial settings, and in particular how an answer in the former setting can imply one in the latter. This talk is based on joint works with Asgar Jamneshan and Terence Tao. This talk is aimed at a general audience. In particular, no prior knowledge in ergodic theory or additive combinatorics is required.

In the past twenty years a research area called “noncommutative function theory” came into being, drawing researchers and ideas from complex analysis, operator algebras, control theory, algebraic geometry and free probability (maybe I forgot some). In this talk, I will do my best to explain what this is about and why this field is in blossom.