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Mathematics Department

Operator Algebras and Dynamics Seminars

Fall 2015

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

  • Dec
    14
  • UHF Factors, Dynamical Properties, and a Theorem of Powers
    Prof. Mitch Baker
    USNA
    Time: 03:45 PM

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    We present an old result proved by Powers and used in his famous paper on UHF algebras in order to compare the concepts of ergodicity and strong mixing in Dynamical Systems to a kind of asymptotic factorization of states in a non-commutative setting.
  • Dec
    07
  • Integrable Measure Equivalence and Quasi-isometric Rigidity of Nilpotent Groups
    Michael Cantrell
    University of Illinois, Chicago
    Time: 03:45 PM

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    Integrable measure equivalence is an equivalence relation on countable groups that is finer than measure equivalence. While all amenable groups form one ME class, they split in to infinitely many IME classes. Our main result is that the IME classes of nilpotent groups are quite rigid. This generalizes/implies a seminal result of Pansu on the quasi-isometric rigidity of nilpotent groups, answers a question of Austin, and yields a cocycle analog of Pansu's Rademacher-type differentiation theorem for nilpotent groups. Our ergodic-theoretic approach differs from previous approaches to QI rigidity, and our main tool is a nilpotent-valued ergodic theorem.
  • Nov
    02
  • A rigidity theorem for generalized odometers. Part II
    Kostya Medynets
    United States Naval Academy

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    We show that generalized odometers are continuously orbit equivalent if and only if the sequences of finite-index subgroups defining the systems are virtually isomorphic. For minimal equicontinuous $Z^d$-systems the continuous orbit equivalence implies that the acting groups have finite index subgroups (having the same index) whose actions are piecewise conjugate. This result extends M.~Boyle's flip-conjugacy theorem originally established for $\Z$-actions. As a corollary we obtain a dynamical classification of the restricted isomorphism between generalized Bunce-Deddens $C*$-algebras. We also show that the full group associated with a generalized odometer is amenable if and only if the acting group is amenable.
  • Oct
    26
  • A rigidity theorem for generalized odometers. Part I
    Kostya Medynets
    United States Naval Academy
    Time: 03:45 PM

    View Abstract

    We show that generalized odometers are continuously orbit equivalent if and only if the sequences of finite-index subgroups defining the systems are virtually isomorphic. For minimal equicontinuous $Z^d$-systems the continuous orbit equivalence implies that the acting groups have finite index subgroups (having the same index) whose actions are piecewise conjugate. This result extends M.~Boyle's flip-conjugacy theorem originally established for $\Z$-actions. As a corollary we obtain a dynamical classification of the restricted isomorphism between generalized Bunce-Deddens $C*$-algebras. We also show that the full group associated with a generalized odometer is amenable if and only if the acting group is amenable.
  • Oct
    19
  • Dispersing billiards and the heat equation
    Peter Nandori
    University of Maryland
    Time: 03:45 PM

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    Dispersing billiard or Sinai billiard is one of the very few physical systems where chaos is well understood (see the laudation for Sinai's Abel prize in 2014). In the talk, I will briefly review the fascinating story of dispersing billiards. Then I will report on some endeavors for deriving heat equation from large billiard type systems in two toy models: for the distribution of mass when energy is fixed (joint work with D. Dolgopyat) and the evolution of the energy when mass is fixed (joint work with P. Balint, T. Gilbert, I.P. Toth, D. Szasz).
  • Oct
    05
  • Sofic Shifts and Finite State Codes
    MIDN James Talisse
    Time: 03:45 PM

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    We present a necessary and sufficient condition for transforming a sofic shift into the full shift. In particular, if the entropy of the sofic shift is greater than or equal to the entropy of the full shift then such a transformation exists, which we call a finite-state code. Additionally a specific algorithm is developed for establishing finite-state codes. This has fundamental applications in data storage.
  • Sep
    28
  • Dynamics of Some Substitution Systems
    Kelly B. Yancey
    University of Maryland, College Park
    Time: 03:45 PM

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    In this talk we will discuss substitution dynamical systems and some of their dynamical properties. Specifically we will talk about when they display the property of rigidity. Certain classes of interval exchange transformations will also be discussed in relation to substitutions. We will focus on 3-IETs.
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