Operator Algebras and Dynamics Seminar  

Fall 2014

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

  • Nov 24
    TBA Scott Atkinson University of Virginia Time: 03:45 PM

  • Nov 21
    TBA Prof. Sergey Bezuglyi The University of Iowa Time: 03:45 PM

  • Nov 17
    TBA Prof. David Pitts University of Nebraska Time: 03:45 PM

  • Nov 10
    Symbolic dynamics for three dimensional flows with positive topological entropy Yuri Lima University of Maryland Time: 03:45 PM

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    Since Hadamard, the construction of symbolic models for dynamical systems has been successfully implemented in many scenarios. In a joint work with Sarig, we deal with flows with positive speed on three dimensional manifolds. These include geodesic flows on surfaces, and Reeb vector fields. Provided the flow has positive entropy, we code it by a suspension over a countable Markov shift. Here is an application: for almost every metric on the two-sphere, there are positive real numbers C and h such that there are at least Ce^{Th}/T closed geodesics of size at most T.
  • Nov 03
    Extendable endomorphisms of factors (part 2) Prof. Alexis Alevras USNA Time: 03:45 PM

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    An endomorphism of a factor M acting standardly on a Hilbert space H gives rise, via the conjugation operator, to an endomorphism of the commutant M'. The endomorphism is called extendable if there is a simultaneous extension of the actions on M and M' to an endomorphism of B(H). We will review recent results of Bikram, Izumi, Srinivasan and Sunder on extendability, with applications to the classification of endomorphism semigroups of factors of type II and III.
  • Oct 20
    Extendable endomorphisms of factors (part 1) Prof. Alexis Alevras USNA Time: 03:45 PM

    View Abstract

    An endomorphism of a factor M acting standardly on a Hilbert space H gives rise, via the conjugation operator, to an endomorphism of the commutant M'. The endomorphism is called extendable if there is a simultaneous extension of the actions on M and M' to an endomorphism of B(H). We will review recent results of Bikram, Izumi, Srinivasan and Sunder on extendability, with applications to the classification of endomorphism semigroups of factors of type II and III.
  • Oct 09
    Diagonality and Idempotents, Jasper's frame theory problem, and Schur-Horn theorems Prof. Gary Weiss University of Cincinnati Time: 03:45 PM

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    (Joint work with Jireh Loreaux.) We explore various relations of operators to their diagonals which study we coin diagonality and explain. This will include how to use 0-diagonality (operators with zero diagonal in some basis) to answer a frame theory question equivalent to an idempotent question of Jasper's, for which I will give some background. Then we will explore recent and past work on Schur-Horn theorems on possible diagonals a positive compact operator can have. I hope to give an overview of some core steps in the development of both. This work was inspired by work of Gohberg, Markus, Arveson, Kadison, Jasper, Fan, Fong, Herrero, and others, which will be explained.
  • Oct 03
    The Kadison-Singer Problem (part 4) Vrej Zarikian USNA Time: 03:45 PM

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    In the first 3 lectures we developed the tools of the Marcus, Spielman, and Srivastava proof of KS, culminating with a probabilistic statement about the maximum root of the characteristic polynomial of a certain class of random positive-semidefinite matrices. In this lecture we outline how, by a series of clever reductions, the aforementioned probabilistic statement implies a positive solution to Anderson's Matrix Paving Problem (known to be equivalent to KS).
  • Sep 22
    The Kadison-Singer Problem (part 3) Vrej Zarikian USNA Time: 03:45 PM

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    Continuation of the previous talks, emphasizing multilinearity properties of the mixed characteristic polynomial, and statements about "comparison with the mean".
  • Sep 15
    The Kadison-Singer Problem (part 2) Vrej Zarikian USNA Time: 03:45 PM

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    We continue to develop the tools required for Marcus, Spielman, and Srivastava's proof of the Kadison-Singer Problem. In particular, real stability and the mixed characteristic polynomial.
  • Sep 05
    The Kadison-Singer Problem (part 1) Vrej Zarikian USNA Time: 03:45 PM

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    The Kadison-Singer Problem was a famous open problem in the theory of operator algebras, posed in 1959. It was recently (and completely unexpectedly) solved (in the affirmative) by three computer scientists: Adam Marcus, Daniel Spielman, and Nikhil Srivastava. Their proof is both novel and (in a sense) elementary. In this series of lectures I will give a detailed account of their proof. The first lecture will explain the problem as well as some of its many reformulations. I will also introduce and explore the properties of "real stability" for complex polynomials, a key ingredient of the proof.
  • Aug 21
    On diagonal actions of branch groups and corresponding characters Artem Dudko Stony Brook Time: 03:45 PM Abstract

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