Operator Algebras and Dynamics Seminars
Fall 2018
All talks are from 3:45-4:45 p.m. in the Seminar room, unless otherwise specified.
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Nov26
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Continuous Orbit Equivalence and Continuous Cocycle RigidityKostya MedynetsTime: 03:45 PM
View Abstract
In the 1980s, Mike Boyle proved that whenever two minimal Z-actions on the Cantor set are continuous orbit equivalent, they are automatically conjugate. Results of this nature are referred to as continuous orbit equivalence rigidity. In the talk, we will discuss the phenomena of continuous orbit equivalence rigidity and continuous cocycle rigidity and what systems are known to exhibit them.
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Nov05
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Problems in representations of quantum informationTanner CrowderNRLTime: 03:45 PM
View Abstract
There is a mismatch between the standard Hilbert space formalism of quantum mechanics and the current communication problems in quantum information technology. When quantum mechanics was developed, the mathematical language was not intended for informatics; instead, it was better equipped to handle concepts like the quantization of electromagnetic radiation. Consequently, basic problems like quantitatively measuring information flow through a quantum channel remain ambiguous. The Bloch representation of quantum information provides an elegant description for 2 level quantum systems, e.g. the qubit. Each qubit can be uniquely represented by a 3 vector in the unit ball, and every quantum operation is uniquely represented by an affine map that maps the ball to itself. The elegance of the Bloch representation lies in its ability to shed the cumbersome Hilbert space notions in quantum mechanics and rely on linear algebra, allowing for manageable calculations of information theoretic quantities. We will discuss the utility of the Bloch representation, the difficulties of extending the Bloch representation to higher dimensions, and some of the recent advancements in the area.
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Oct25
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Limit theorems and generalized baker’s transformationsSeth ChartTowson UniversityTime: 03:45 PM
View Abstract
A real valued random variable is a function from some measure space to the real numbers. A measurable mapping of the space into itself is a measurable dynamical system. A random variable together with a mapping of its domain produce a sequence of random variables by a standard construction. The Birkhoff Ergodic Theorem can be viewed as a Strong Law of Large Numbers for the sequence of random variables under some mild hypotheses on the random variable and mapping. It is natural to wonder when a Central Limit Theorem is available for the sequence of random variables. One of the key ingredients for a Central Limit Theorem is sufficiently rapid decay of correlations. Generalized baker’s transformations are a family of area preserving mappings of the unit square. These mappings provide a rich set of examples of both uniformly hyperbolic and non-uniformly hyperbolic mappings. In this talk, we will explore some examples where a Central Limit Theorem or a generalization thereof can be obtained. In particular, we will analyze generalized baker’s transformations where the rate of decay of correlations is slow and consequently a Generalized Central Limit Theorem holds with convergence to a non-normal stable distribution.
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Oct01
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Faithfulness vs. Uniqueness, for Pseudo-ExpectationsVrej ZarikianUSNATime: 03:45 PM
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Sep24
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Faithfulness vs. Uniqueness, for Pseudo-ExpectationsVrej ZarikianUSNATime: 03:45 PM
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Sep17
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Approximate Transitivity of the Natural Action of Finite Permutations of N on {0,1}^NMitch BakerUSNATime: 03:45 PM
View Abstract
In 1980, Connes and Woods introduced the notion of approximate transitivity (AT) for dynamical systems arising in the context of von Neumann algebras (i.e. Connes’ flow of weights) – for the purpose of characterizing which approximately finite dimensional factors could be written as an infinite tensor product of type I factors (ITPFI). The latter had played a major role in the development of the theory of von Neumann algebras, but their connection to general factors had long remained mysterious. Later on, straight-away dynamical systems researchers investigated how this property related to properties studied in the context of standard dynamical systems (e.g. entropy, rank one, etc.), obtaining some interesting results - but so far proving approximate transitivity has been quite difficult, and relatively few classes are known even now. We present a new infinite class of natural dynamical systems arising from product spaces and the classical action of permutations of finite length on them (originally motivated by our study of group-invariant type III factors and their flow of weights), and show that they are approximately transitive.
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Sep10
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Approximate Transitivity of the Natural Action of Finite Permutations of N on {0,1}^NMitch BakerUSNATime: 03:45 PM
View Abstract
In 1980, Connes and Woods introduced the notion of approximate transitivity (AT) for dynamical systems arising in the context of von Neumann algebras (i.e. Connes’ flow of weights) – for the purpose of characterizing which approximately finite dimensional factors could be written as an infinite tensor product of type I factors (ITPFI). The latter had played a major role in the development of the theory of von Neumann algebras, but their connection to general factors had long remained mysterious. Later on, straight-away dynamical systems researchers investigated how this property related to properties studied in the context of standard dynamical systems (e.g. entropy, rank one, etc.), obtaining some interesting results - but so far proving approximate transitivity has been quite difficult, and relatively few classes are known even now. We present a new infinite class of natural dynamical systems arising from product spaces and the classical action of permutations of finite length on them (originally motivated by our study of group-invariant type III factors and their flow of weights), and show that they are approximately transitive.
