Mathematics Department

Operator Algebras and Dynamics Seminar

Spring 2015

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

  • Apr
    24
  • Toeplitz Matrices, Palindromes and Spin Systems II
    Prof. Geoffrey Price
  • Apr
    13
  • Toeplitz Matrices, Palindromes and Spin Systems I
    Prof. Geoffrey Price
  • Apr
    10
  • Presentations of Topological Full Groups

    View Abstract

    We describe defining relations for the commutator subgroups of topological full groups of minimal subshifts. We establish that the word problem in a topological full group is solvable if and only if the language of the underlying subshift is recursive.
  • Apr
    02
  • Symbolic Dynamics and Entropy via Conley Index Theory
    Prof. Sarah Day
    College of William and Mary

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    Conley index theory, a generalization of Morse theory using algebraic topology, may be used in a computational framework to prove the existence of dynamics of various types. When searching for highly complicated dynamics, however, the Conley index may also become highly complicated and difficult to interpret. We present an automated approach to processing Conley index information for discrete-time dynamical systems. This approach produces a topologically semi-conjugate symbolic system whose entropy serves as a lower bound for the entropy of the system under study. Recent modifications of the original approach published in 2006 produce symbolic systems that capture more of the complexity encoded by the index, in some cases leading to substantial increases in computed lower bounds on system entropy. Sample results will be shown for the 2-dimensional Henon map and the infinite-dimensional Kot-Schaffer map. This is joint work with Rafael Frongillo.
  • Mar
    30
  • Multivariable dynamics and non-selfadjoint operator algebras
    Prof. Chris Ramsey
    University of Virginia

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    Dynamical systems and operator algebras have had a love affair since the days of Murray and von Neumann. However, this has been a somewhat one-sided relationship as much of the information encoded in the dynamics is lost in the crossed product. Enter a certain non-selfadjoint operator algebra called the semicrossed product. I aim to show that these semicrossed products are isomorphic if and only if their associated dynamical systems are conjugate.
  • Mar
    11
  • Equivalent operator categories
    Prof. David Sherman
    University of Virginia

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    Leaving rigorous definitions to the talk, operator categories are natural classes that include C*-algebras, operator systems, hereditary manifolds, operator algebras, Jordan operator algebras, etc. I will show how to associate the following features to any such category: an operator topology, a representation theory, and a convexity/dilation theory. It turns out that if one of these features agrees for a pair of categories, then all three do, in which case the categories are called equivalent. I will discuss some equivalences, along the way obtaining new observations about Arveson's hyperrigidity and maybe even triangles.
  • Feb
    27
  • The Unique Pseudo-Expectation Property for C*-Inclusions. II
    Time: 03:45 PM

    View Abstract

    Continuation of last week's lecture.
  • Feb
    20
  • The Unique Pseudo-Expectation Property for C*-Inclusions
    Time: 03:45 PM

    View Abstract

    A pseudo-expectation for a C*-inclusion (C,D) is a generalization of a conditional expectation. Whereas (C,D) may not have any conditional expectations, it must have at least one pseudo-expectation. One would expect the existence of a unique pseudo-expectation for (C,D) to be related to structural properties of the inclusion. In this talk, based on recent joint work with David Pitts, we investigate the unique pseudo-expectation property for C*-inclusions (C,D). After formally defining the property, we present some general results about it, in particular an order-theoretic characterization when D is abelian. Then we provide a number of examples of C*-inclusions with the unique pseudo-expectation property. Of special interest are the cases of abelian inclusions and W*-inclusions. Finally we relate the unique pseudo-expectation property to other properties of C*-inclusions, particularly norming in the sense of Pop, Sinclair, and Smith.
  • Jan
    16
  • Inverse limits and strange attractors
    Piotr Oprocha
    AGH University of Science and Technology, Krakow, Poland

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    In 1990 Barge and Martin presented a method of construction of global attractors of planar homeomorphisms in terms of inverse limits. This technique can also be extended to obtain attractors arising as inverse limits of degree one map of the circle. That way we can obtain attractors with very strange topological structure, such as pseudoarc or pseudocircle. In this talk we are going to survey some known results on dynamics on various types of continua that can be obtained as attractors. We are also going to mention some examples of maps on these spaces that cannot be constructed as shift homeomorphisms on inverse limit and present a few open problems that arise. At the end we are going to present recent results obtained jointly with Jan Boro\'nski. Among others we are going to explain how to obtain a pseudocircle as an attractor of map on a tori with a non-unique rotation vector on it. We will also comment on entropy and other dynamical properties of attractors obtained by this technique.
  • Jan
    09
  • On the simplicity of twisted k-graph C*-algebras
    Prof. Alex Kumjian
    University of Nevada

    View Abstract

    Let $\Lambda$ be a row-finite k-graph with no sources. It is well known that $C^*(\Lambda)$ is simple iff $\Lambda$ is aperiodic and cofinal. Given a categorical 2-cocycle c with vaues in $\mathbb{T}$ one may form the twisted k-graph C*-algebra, $C^*(\Lambda, c)$ . We use groupoid techiniques to characterize the simplicity of $C^*(\Lambda, c)$ generalizing recent work of Sims, Whitehead and Whittaker.
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