Activities This Week

Topological recursion is a remarkable universal recursive procedure that has been found in many enumerative geometry problems, from combinatorics of maps, to random matrices, Gromov-Witten invariants, Hurwitz numbers, Mirzakhani’s hyperbolic volumes of moduli spaces, knot polynomials. A recursion needs an initial data: a spectral curve, and the recursion defines the sequence of invariants of that spectral curve. There is a duality in topological recursion which allows one to obtain closed formulas for the invariants of the recursion and which has implications in free probability theory and integrable hierarchies. In the talk I will survey recent progress in the topic with the examples from Hurwitz numbers theory, Hodge integrals and combinatorics of maps.

The talk is based on the joint works with A. Alexandrov, P. Dunin-Barkowski, M. Kazarian and S. Shadrin.

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.