IAS Analysis Seminar Spring 2008

the density of polynomial configurations a+p_1(n),...,a+p_k(n) in sets of integers and the form of the closure of the diagonal of X^k under the "polynomial action" (T^{p_1},...,T^{p_k}), where X is a "universal" nilmanifold (that is, a compact homogeneous space of a "universal" nilpotent Lie group) and T is a translation of X. It is easy to describe this diagonal in the case all p_i are linear, and not easy when p_i are not linear.

of stiffness and stationary measures, and show how under fairly general assumptions stationary measures can be classified. A key ingredient is a result of Bourgain related to the sum product phenomena on the reals. In particular, we prove the following: let A, B be two non commuting 2x2 integer matrices of determinant one. Consider a random product X_r....X_1.y where y is a point in the two torus. We show that as r-> infinity this random product is distributed in an increasingly uniform manner. Based on joint work with Bourgain, Furman and Mozes.

arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a cube, or more generally of the form p(n) where p(n) is any integer polynomial with zero constant term. We produce a variety of new results of this type. We show that the common difference can be of the form [n^c] where c is any positive real number, or more generally of the form [a(n)] where a(x) is any function that belongs to some Hardy field and satisfies some mild growth conditions. This allows us for example to deal with the class of logarithmico-exponential functions, i.e., functions that can be constructed by a finite combination of the ordinary arithmetical symbols, the real constants, and the functions e^x, logx. This is joint work with Mate Wierdl.

bilinear averages in Ergodic theory, arising from Z^2 actions. Our techniques combine novel one and a half dimensional phase-space analysis with more standard one dimensional theory. This is joint work with Christoph Thiele.