Activities This Week

Finite Rokhlin dimension, a generalization of the Rokhlin property, is a regularity property for actions of certain groups on C-algebras. The main interest in Rokhlin dimension was its use to establish various permanence properties: for example, if the C-algebra acted on has finite nuclear dimension and the action has finite Rokhlin dimension then the crossed product again has finite nuclear dimension. As such, the main interest in Rokhlin dimension was to show that it is finite, and not much attention was paid to its actual value. In particular, while it is known that there are actions with positive finite Rokhlin dimension (that is, have finite Rokhlin dimension but do not have the Rokhlin property, which corresponds to Rokhlin dimension zero), there were no examples of actions of finite groups with finite Rokhlin dimension greater than 2. I’ll discuss a recent preprint in which we provide examples of actions of finite groups on simple AF algebras with arbitrarily large finite Rokhlin dimension. This shows that Rokhlin dimension is not just a tool to establish regularity results, but is an interesting invariant for group actions, which in a sense measures the complexity of the action.

This is joint work with N. Christopher Phillips.

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.

The Gelfand-Kazhdan formal geometry is a way of describing geometric structures on a smooth manifold M in terms of the jet bundle. The works of Fedosov, Nest-Tsygan and Bezrukavnikov-Kaledin put the problem of classifying deformation quantizations of, respectively, smooth, holomorphic and algebraic symplectic manifolds into the context of formal geometry. They showed that, if the Hodge filtration on the cohomology of the symplectic manifold splits, the set of deformation quantizations of M could be identified with a certain subset of $H^2(M)[[h]]$ via the so-called period map. In the talk I want to describe an upgrade of the period map from a map between sets to a morphism between suitably defined deformation functors. This upgrade could be used to reprove the Fedosov-Nest-Tsygan-Bezrukavnikov-Kaledin theorems, to help classify quantizations without the Hodge filtration splitting condition, and to connect the period map with the so-called Rozansky-Witten invariants.

Let $(\lambda_{1},…,\lambda_{d})=\lambda\in(0,1)^{d}$ be with $\lambda_{1}>…>\lambda_{d}$ and let $\mu_{\lambda}$ be the distribution of the random vector $\sum_{n\ge0}\pm (\lambda_{1}^{n},…,\lambda_{d}^{n})$, where the $\pm$ are independent fair coin-tosses. Assuming $P(\lambda_{j})\ne 0$ for all $1\le j\le d$ and nonzero polynomials with coefficients $\pm1,0$, we show that $\operatorname{dim}\mu_{\lambda}=\min \big(d,\dim_{L}\mu_{\lambda} \big)$, where $\dim_{L}\mu_{\lambda}$ is the Lyapunov dimension. This extends to higher dimensions a result of Varjú from 2018 regarding the dimension of Bernoulli convolutions on the real line. Joint work with Haojie Ren.