Mathematics Department

Operator Algebras and Dynamics Seminar

Spring 2019

All talks are from 3:45-4:45 p.m. in the Seminar room, unless otherwise specified.

• Apr
22
• Old and new theorems on closures and convex hulls of unitary orbits of self-adjoint operators
David Sherman
University of Virginia
Time: 03:45 PM

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This will be a selective survey about unitary orbits of self-adjoint operators. I will start with convex hulls of matrices, where the key tool is majorization. Then I will discuss closures associated to infinite-dimensional operators, and I will give a new description of the weak* closure. Finally I will discuss new results about weak* closures for self-adjoint elements in von Neumann factors. Perhaps most interesting is a noncommutative Lyapunov phenomenon": the type I (atomic) case turns out to be qualitatively different from types II and III, in which the closure is automatically convex and again described by majorization. This is joint work with Chuck Akemann.
• Apr
16
• A Brief History on Some Fermi-Ulam Models
Jing Zhou
University of Maryland
Time: 03:45 PM

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I will go over some Fermi-Ulam models that have been considered in the past and recently, including but not limited to works by Laederich&Levi, Pustyl'nikov, de Simoi&Dolgopyat, Shah&Turaev&Rom-Kedar, as well as my own results. I'll also briefly cover the proofs in some of the cases.
• Apr
02
• Group actions on product systems and K-theory
Valentin Deaconu
Time: 03:45 PM

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Product systems $Y$ over various semigroups were introduced by N. Fowler, inspired by work of W. Arveson. We will recall the definition of $Y$ and introduce group actions and crossed products $Y\rtimes G$. One motivation is group actions on higher rank graphs. We generalize a result of C. Schafhauser for a row-finite and faithful product system $Y$ indexed by ${\mathbb N}^k$ concerning the $K$-theory of the crossed product by the gauge action $\gamma$. The main result is $K_*({\mathcal O}_A(Y)\rtimes_\gamma{\mathbb T}^k)\cong \varinjlim_{n \in {\mathbb N}^k} (K_*(A),[Y_n]), where$[Y_n]$denotes the homomorphism induced by$Y_n\$ via Fredholm operators. We apply this result to a product system constructed from group representations.
• Mar
25
• Ergodicity on Fractal Spaces via Hyperbolic Geometry
Anton Lukyanenko
George Mason University
Time: 03:45 PM

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Continued fractions on the real numbers have far-reaching applications, including connections to dynamics, Diophantine approximation, and hyperbolic geometry. Their generalizations, both in R and in higher dimensions have been a topic of extensive study over the last few decades. A central question has been the extent to which all points have the same CF-based properties, i.e. whether the associated Gauss map is ergodic. I will discuss an approach used by Hensley, based on the use of transfer operators, to argue that the Gauss map for complex continued fractions is ergodic; with connections to the work of Mauldin, Urbanski, and others. Then, I will describe the recent generalization of the theory to a more general class of spaces, where instead one can use hyperbolic geometry to prove ergodicity, extending the classical approach of Artin and Series.
• Feb
25
• Mixing properties for infinite local complexity tiling substitutions
Robbie Robinson
George Washington University
Time: 03:45 PM

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Discrete substitution and finite local complexity substitution tiling dynamical systems can be weakly mixing but not strongly mixing. We discuss examples of infinite local complexity substitution tilings (in d=1 and d=2) that are mixing of all orders and have Lebesgue spectrum. We compare to results on other types of dynamical systems including interval exchanges and rank 1.
• Jan
28
• Local weak∗-Convergence, algebraic actions, and a max-min principle
Ben Hayes
University of Virginia
Time: 03:45 PM

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This talk will be concerned with algebraic actions, which are actions of a countable, discrete, group G on a compact group X. When G is sofic, a natural question is when the topological entropy and measure entropy of these actions agree. I will show that there is a complete lattice of "generalized subgroups" of X and a max-min principle in this complete lattice which essentially gives a complete answer to when this equality occurs.