Upcoming Events
Title: Uniform pointwise ergodic theorems via ultraproducts
Abstract: We take a new point of view on weighted pointwise ergodic theorems by viewing the sequence (f(Tnx))n >= 1 as a vector in an ultraproduct space. This point of view allows us to directly transfer mixing properties of the underlying transformation to mixing properties of a shift operator on the ultraproduct space. In this new context, weighted pointwise ergodic theorems correspond to identifying functionals in the dual space that annihilate the starting vector. This allows us to not only recover the uniform Wiener-Wintner theorem of Lesigne, but to also prove a uniform pointwise ergodic theorem for ergodic, mild mixing, and strong mixing measure preserving transformations. Furthermore, our methods are general enough to apply to any L1-L\infty contraction on a Bochner space Lp(X,\mu;E) constructed from a reflexive Banach space .
In this talk we will discuss bounded weight modules, i.e., modules that decompose as direct sums of weight spaces and whose sets of weight multiplicities are uniformly bounded. Our main focus will be on the direct limits of classical Lie (super)algebras. In particular, we will present the classification of the simple bounded weight modules over $\mathfrak{sl} (\infty)$, $\mathfrak{o} (\infty)$, $\mathfrak{sp} (\infty)$, as well as over their super-analogs. A key role in the study plays the theory of weight modules over Weyl and Clifford superalgebras of infinitely many variables. The talk is based on joint works with I. Penkov and V. Serganova.
Nathan Dalaklis (Dynamics Seminar, TBD)