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Theta Functions and Knots /
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 interpretati...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
[Hackensack] New Jersey :
World Scientific,
[2014]
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- 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.
- 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.
- 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.
- 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.
- 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.