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Record Nr. |
UNINA9910780601303321 |
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Autore |
Zhang Weiping |
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Titolo |
Lectures on Chern-Weil theory and Witten deformations [[electronic resource] /] / Weiping Zhang |
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Pubbl/distr/stampa |
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River Edge, N.J., : World Scientific, c2001 |
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ISBN |
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Descrizione fisica |
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1 online resource (131 p.) |
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Collana |
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Nankai tracts in mathematics ; ; 4 |
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Disciplina |
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Soggetti |
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Chern classes |
Index theorems |
Complexes |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Note generali |
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Description based upon print version of record. |
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Nota di bibliografia |
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Includes bibliographical references and index. |
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Nota di contenuto |
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Contents; Preface; Chapter 1 Chern-Weil Theory for Characteristic Classes; 1.1 Review of the de Rham Cohomology Theory; 1.2 Connections on Vector Bundles; 1.3 The Curvature of a Connection; 1.4 Chern-Weil Theorem; 1.5 Characteristic Forms, Classes and Numbers; 1.6 Some Examples; 1.6.1 Chern Forms and Classes; 1.6.2 Pontrjagin Classes for Real Vector Bundles; 1.6.3 Hirzebruch's L-class and A-class; 1.6.4 K-groups and the Chern Character; 1.6.5 The Chern-Simons Transgressed Form; 1.7 Bott Vanishing Theorem for Foliations; 1.7.1 Foliations and the Bott Vanishing Theorem |
1.7.2 Adiabatic Limit and the Bott Connection1.8 Chern-Weil Theory in Odd Dimension; 1.9 References; Chapter 2 Bott and Duistermaat-Heckman Formulas; 2.1 Berline-Vergne Localization Formula; 2.2 Bott Residue Formula; 2.3 Duistermaat-Heckman Formula; 2.4 Bott's Original Idea; 2.5 References; Chapter 3 Gauss-Bonnet-Chern Theorem; 3.1 A Toy Model and the Berezin Integral; 3.2 Mathai-Quillen's Thom Form; 3.3 A Transgression Formula; 3.4 Proof of the Gauss-Bonnet-Chern Theorem; 3.5 Some Remarks; 3.6 Chern's Original Proof; 3.7 References; Chapter 4 Poincare-Hopf Index Formula: an Analytic Proof |
4.1 Review of Hodge Theorem4.2 Poincare-Hopf Index Formula; 4.3 Clifford Actions and the Witten Deformation; 4.4 An Estimate Outside |
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of Up zero(V) Up; 4.5 Harmonic Oscillators on Euclidean Spaces; 4.6 A Proof of the Poincare-Hopf Index Formula; 4.7 Some Estimates for DT,i's, 2 i 4; 4.8 An Alternate Analytic Proof; 4.9 References; Chapter 5 Morse Inequalities: an Analytic Proof; 5.1 Review of Morse Inequalities; 5.2 Witten Deformation; 5.3 Hodge Theorem for ( * (M), dTf; 5.4 Behaviour of rf Near the Critical Points of f; 5.5 Proof of Morse Inequalities; 5.6 Proof of Proposition 5.5 |
5.7 Some Remarks and Comments5.8 References; Chapter 6 Thom-Smale and Witten Complexes; 6.1 The Thom-Smale Complex; 6.2 The de Rham Map for Thom-Smale Complexes; 6.3 Witten's Instanton Complex and the Map eT; 6.4 The Map P, TeT; 6.5 An Analytic Proof of Theorem 6.4; 6.6 References; Chapter 7 Atiyah Theorem on Kervaire Semi-characteristic; 7.1 Kervaire Semi-characteristic; 7.2 Atiyah's Original Proof; 7.3 A proof via Witten Deformation; 7.4 A Generic Counting Formula for k(M ); 7.5 Non-multiplicativity of k(M); 7.6 References; Index |
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Sommario/riassunto |
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This invaluable book is based on the notes of a graduate course on differential geometry which the author gave at the Nankai Institute of Mathematics. It consists of two parts: the first part contains an introduction to the geometric theory of characteristic classes due to Shiing-shen Chern and André Weil, as well as a proof of the Gauss-Bonnet-Chern theorem based on the Mathai-Quillen construction of Thom forms; the second part presents analytic proofs of the Poincaré-Hopf index formula, as well as the Morse inequalities based on deformations introduced by Edward Witten. <br><i>Contents:</i> |
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