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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
    16
  • Jing Zhou
    University of Maryland
    Time: 03:45 PM
  • Apr
    01
  • Group actions on product systems and K-theory
    Valentin Deaconu
    U. Nevada-Reno
    Time: 03:45 PM

    View Abstract

    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

    View Abstract

    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.
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