Subfactors, Planar Algebras and Rotations PDF Download
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Author: Michael E. Burns
Publisher:
ISBN:
Category :
Languages : en
Pages : 272
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Author: Michael E. Burns
Publisher:
ISBN:
Category :
Languages : en
Pages : 272
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Author: David Signorielli Penneys
Publisher:
ISBN:
Category :
Languages : en
Pages : 145
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This dissertation consists of three self-contained papers from my graduate work at UC Berkeley. The chapters increase in complexity from the annular Temperley-Lieb category to strongly Markov inclusions of finite von Neumann algebras to infinite index II_1-subfactors. In Chapter 2, we discuss how two copies of the cyclic category generate the annular Temperley-Lieb category. In the process, we give a presentation of the annular Temperley-Lieb category via generators and relations, and we see the cyclic category evolve from the simplicial and semi-simplicial categories. Chapter 3 is joint work with Vaughan F.R. Jones. First, we define a canonical planar *-algebra associated to a strongly Markov inclusion of finite von Neumann algebras (the notion of such an inclusion is defined within). Second, we show for an inclusion of finite dimensional C^*-algebras with the Markov trace, the canonical planar algebra is isomorphic to the graph planar algebra of the Bratteli diagram of the inclusion. We use this fact to show that a subfactor planar algebra embeds into the graph planar algebra of its principal graph. In Chapter 4, we expand upon Burns' work on rotations for infinite index II_1-subfactors. We start with a II_1-factor bimodule, and we construct a tower of centralizer algebras and a sequence of central L^2-vectors. In the finite index setting, the centralizer algebras and central L^2-vectors agree, but in the infinite index setting, these spaces can differ dramatically. We develop planar calculi for both sequences which are compatible. Interestingly, we obtain planar structure without Jones' basic construction or the resulting Jones projections! We also generalize Burns work on extremality and the existence of rotations to the bimodule setting, and we recover his main theorem. Along the way, we prove some results about relative tensor products of extended positive cones, and we give an example of an infinite index subfactor with finite dimensional higher relative commutants.
Author: Narjess Afzaly
Publisher: American Mathematical Society
ISBN: 1470447126
Category : Mathematics
Languages : en
Pages : 94
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Publisher: World Scientific
ISBN:
Category :
Languages : en
Pages : 1001
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Author: Zeph A. Landau
Publisher:
ISBN:
Category :
Languages : en
Pages : 288
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Author: Institut des hautes études scientifiques (Paris, France)
Publisher: American Mathematical Soc.
ISBN: 0821852035
Category : Mathematics
Languages : en
Pages : 695
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The work of Alain Connes has cut a wide swath across several areas of mathematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics. Specific themes covered by the articles are as follows: entropy in operator algebras, regular $C^*$-algebras of integral domains, properly infinite $C^*$-algebras, representations of free groups and 1-cohomology, Leibniz seminorms and quantum metric spaces; von Neumann algebras, fundamental Group of $\mathrm{II}_1$ factors, subfactors and planar algebras; Baum-Connes conjecture and property T, equivariant K-homology, Hermitian K-theory; cyclic cohomology, local index formula and twisted spectral triples, tangent groupoid and the index theorem; noncommutative geometry and space-time, spectral action principle, quantum gravity, noncommutative ADHM and instantons, non-compact spectral triples of finite volume, noncommutative coordinate algebras; Hopf algebras, Vinberg algebras, renormalization and combinatorics, motivic renormalization and singularities; cyclotomy and analytic geometry over $F_1$, quantum modular forms; differential K-theory, cyclic theory and S-cohomology.
Author: Paul Bruillard
Publisher: American Mathematical Soc.
ISBN: 1470440741
Category : Education
Languages : en
Pages : 240
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This volume contains the proceedings of the AMS Special Session on Topological Phases of Matter and Quantum Computation, held from September 24–25, 2016, at Bowdoin College, Brunswick, Maine. Topological quantum computing has exploded in popularity in recent years. Sitting at the triple point between mathematics, physics, and computer science, it has the potential to revolutionize sub-disciplines in these fields. The academic importance of this field has been recognized in physics through the 2016 Nobel Prize. In mathematics, some of the 1990 Fields Medals were awarded for developments in topics that nowadays are fundamental tools for the study of topological quantum computation. Moreover, the practical importance of this discipline has been underscored by recent industry investments. The relative youth of this field combined with a high degree of interest in it makes now an excellent time to get involved. Furthermore, the cross-disciplinary nature of topological quantum computing provides an unprecedented number of opportunities for cross-pollination of mathematics, physics, and computer science. This can be seen in the variety of works contained in this volume. With articles coming from mathematics, physics, and computer science, this volume aims to provide a taste of different sub-disciplines for novices and a wealth of new perspectives for veteran researchers. Regardless of your point of entry into topological quantum computing or your experience level, this volume has something for you.
Author:
Publisher:
ISBN:
Category : Dissertations, Academic
Languages : en
Pages : 806
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Author: Vaughan F. R. Jones
Publisher: Cambridge University Press
ISBN: 0521584205
Category : Mathematics
Languages : en
Pages : 178
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Subfactors have been a subject of considerable research activity for about 15 years and are known to have significant relations with other fields such as low dimensional topology and algebraic quantum field theory. These notes give an introduction to the subject suitable for a student who has only a little familiarity with the theory of Hilbert space. A new pictorial approach to subfactors is presented in a late ch apter.
Author:
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 1078
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