I will discuss two new (unrelated) phenomena. (1) Taking maximal averages of functions has connections to transcendental number theory and (2) the Ulam sequence (1,2,3,4,6,8,11,...) defined via additive combinatorics has very strange distribution behavior when multiplied with 2.571....

# Ergodic Theory & Statistical Mechanics

This weekly seminar welcomes many visiting speakers and also provides a forum for results by Princeton/IAS faculty and graduate students. While most talks focus on one of the two namesake topics, there tends to be overlap with other fields. Most recently, talks have included material on celestial mechanics, number theory, probability theory, the Schrodinger equation, spectral theory and Teichmueller theory.

For more information about this seminar, contact Yakov Sinai or Jon Fickenscher

**Please click on seminar title for complete abstract.**

##### New interactions between Analysis and Number Theory

Yale University

##### TBA - Elon Lindenstrauss

Einstein Institute of Mathematics, Hebrew University of Jerusalem

##### TBA - Alex Kontorovich

Rutgers University, IAS

##### TBA - Ilya Vinogradov

Princeton University

##### Positive metric entropy arises between some KAM tori

The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still have a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori.

Penn State University)