**Title:** Single-scale ergodic theory

**Speaker:** Terence Tao

**Speaker Info:** University of California, Los Angeles

**Brief Description:**

**Special Note**: **Note the unusual location!**

**Abstract:**

Ergodic theory has traditionally been concerned with the analysis of averages of dynamical systems in the asymptotic limit when the scale parameter N goes to infinity. However, more recently some "single-scale" variants of this theory have begun to emerge, in which the scale parameter N is large but fixed; such variants are useful in various finitary applications, for instance in counting patterns such as arithmetic progressions in the primes. (The recently established inverse conjecture for the Gowers uniformity norms can be viewed as the single-scale analogue of the Host-Kra characteristic factor theorem in ergodic theory.)In this single-scale context, one loses the ability to take classical limits (though one can use ultralimits and nonstandard analysis as a partial substitute), which deprives the theory of such basic tools as the ergodic theorem. Nevertheless, a surprising amount of the asymptotic theory still survives in the single-scale context, though sometimes with some new features. We illustrate this with the equidistribution theory of linear or polynomial sequences in tori or nilmanifolds, and with a number of other examples.

Copyright © 1997-2024 Department of Mathematics, Northwestern University.