Operator Algebras and Dynamics Seminars
Spring 2019
All talks are from 3:454:45 p.m. in the Seminar room, unless otherwise specified.

Apr22

Old and new theorems on closures and convex hulls of unitary orbits of selfadjoint operatorsDavid ShermanUniversity of VirginiaTime: 03:45 PM
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This will be a selective survey about unitary orbits of selfadjoint operators. I will start with convex hulls of matrices, where the key tool is majorization. Then I will discuss closures associated to infinitedimensional operators, and I will give a new description of the weak* closure. Finally I will discuss new results about weak* closures for selfadjoint 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.

Apr16

A Brief History on Some FermiUlam ModelsJing ZhouUniversity of MarylandTime: 03:45 PM
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I will go over some FermiUlam 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&RomKedar, as well as my own results. I'll also briefly cover the proofs in some of the cases.

Apr02

Group actions on product systems and KtheoryValentin DeaconuU. NevadaRenoTime: 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 rowfinite 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.

Mar25

Ergodicity on Fractal Spaces via Hyperbolic GeometryAnton LukyanenkoGeorge Mason UniversityTime: 03:45 PM
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Continued fractions on the real numbers have farreaching 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 CFbased 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.

Feb25

Mixing properties for infinite local complexity tiling substitutionsRobbie RobinsonGeorge Washington UniversityTime: 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.

Jan28

Local weak∗Convergence, algebraic actions, and a maxmin principleBen HayesUniversity of VirginiaTime: 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 maxmin principle in this complete lattice which essentially gives a complete answer to when this equality occurs.