| Description |
1 online resource (xiv, 454 pages) |
| Physical Medium |
polychrome |
| Description |
text file |
| Summary |
This book presents the relationship between classical theta functions and knots. It is based on a novel idea of Razvan Gelca and Alejandro Uribe, which converts Weil's representation of the Heisenberg group on theta functions to a knot theoretical framework, by giving a topological interpretation to a certain induced representation. It also explains how the discrete Fourier transform can be related to 3- and 4-dimensional topology. Theta Functions and Knots can be read in two perspectives. People with an interest in theta functions or knot theory can learn how the two are related. Those intere. |
| Bibliography |
Includes bibliographical references and index. |
| Contents |
Preface; Contents; 1. Prologue; 1.1 The history of theta functions; 1.1.1 Elliptic integrals and theta functions; 1.1.2 The work of Riemann; 1.2 The linking number; 1.2.1 The definition of the linking number; 1.2.2 The Jones polynomial; 1.2.3 Computing the linking number from skein relations; 1.3 Witten's Chern-Simons theory; 2. A quantum mechanical prototype; 2.1 The quantization of a system of finitely many free one-dimensional particles; 2.1.1 The classical mechanics of finitely many free particles in a one-dimensional space; 2.1.2 The Schrodinger representation; 2.1.3 Weyl quantization. |
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2.2 The quantization of finitely many free one-dimensional particles via holomorphic functions2.2.1 The Segal-Bargmann quantization model; 2.2.2 The Schrodinger representation and the Weyl quantization in the holomorphic setting; 2.2.3 Holomorphic quantization in the momentum representation; 2.3 Geometric quantization; 2.3.1 Polarizations; 2.3.2 The construction of the Hilbert space using geometric quantization; 2.3.3 The Schrodinger representation from geometric considerations; 2.3.4 Passing from real to Kahler polarizations; 2.4 The Schrodinger representation as an induced representation. |
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2.5 The Fourier transform and the representation of the symplectic group Sp(2n, R)2.5.1 The Fourier transform defined by a pair of Lagrangian subspaces; 2.5.2 The Maslov index; 2.5.3 The resolution of the projective ambiguity of the representation of Sp(2n, R); 3. Surfaces and curves; 3.1 The topology of surfaces; 3.1.1 The classification of surfaces; 3.1.2 The fundamental group; 3.1.3 The homology and cohomology groups; 3.1.4 The homology groups of a surface and the intersection form; 3.2 Curves on surfaces; 3.2.1 Isotopy versus homotopy; 3.2.2 Multicurves on a torus. |
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3.2.3 The first homology group of a surface as a group of curves3.2.4 Links in the cylinder over a surface; 3.3 The mapping class group of a surface; 3.3.1 The definition of the mapping class group; 3.3.2 Particular cases of mapping class groups; 3.3.3 Elements of Morse and Cerf theory; 3.3.4 The mapping class group of a closed surface is generated by Dehn twists; 4. The theta functions associated to a Riemann surface; 4.1 The Jacobian variety; 4.1.1 De Rham cohomology; 4.1.2 Hodge theory on a Riemann surface; 4.1.3 The construction of the Jacobian variety. |
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4.2 The quantization of the Jacobian variety of a Riemann surface in a real polarization4.2.1 Classical mechanics on the Jacobian variety; 4.2.2 The Hilbert space of the quantization of the Jacobian variety in a real polarization; 4.2.3 The Schrodinger representation of the finite Heisenberg group; 4.3 Theta functions via quantum mechanics; 4.3.1 Theta functions from the geometric quantization of the Jacobian variety in a Kahler polarization; 4.3.2 The action of the finite Heisenberg group on theta functions; 4.3.3 The Segal-Bargmann transform on the Jacobian variety. |
| Note |
4.3.4 The algebra of linear operators on the space of theta functions and the quantum torus. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Functions, Theta.
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Functions, Theta. |
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Knot theory.
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Knot theory. |
| Genre/Form |
Electronic books.
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Electronic books.
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| Other Form: |
Print version: Gelca, Rǎzvan, 1967- Theta functions and knots. [Hackensack] New Jersey : World Scientific, 2014 9789814520577 (DLC) 2014005501 (OCoLC)849787561 |
| ISBN |
9789814520584 (electronic book) |
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9814520586 (electronic book) |
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9789814520577 (hardcover : alkaline paper) |
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9814520578 (hardcover : alkaline paper) |
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1306942004 (e-book) |
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9781306942003 (e-book) |
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