Lectures on Chern-Weil theory and Witten deformations /
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| Author / Creator: | Zhang, Weiping, 1964- |
|---|---|
| Imprint: | River Edge, N.J. : World Scientific, ©2001. |
| Description: | 1 online resource (xi, 117 pages). |
| Language: | English |
| Series: | Nankai tracts in mathematics ; 4 Nankai tracts in mathematics ; v. 4. |
| Subject: | |
| Format: | E-Resource Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/11218184 |
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| 100 | 1 | |a Zhang, Weiping, |d 1964- |0 http://id.loc.gov/authorities/names/n2007071188 | |
| 245 | 1 | 0 | |a Lectures on Chern-Weil theory and Witten deformations / |c Weiping Zhang. |
| 260 | |a River Edge, N.J. : |b World Scientific, |c ©2001. | ||
| 300 | |a 1 online resource (xi, 117 pages). | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 490 | 1 | |a Nankai tracts in mathematics ; |v 4 | |
| 504 | |a Includes bibliographical references and index. | ||
| 588 | 0 | |a Print version record. | |
| 505 | 0 | |a Ch. 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. 1.7. Bott vanishing theorem for foliations. 1.8. Chern-Weil theory in odd dimension. 1.9. References -- ch. 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 -- ch. 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 -- ch. 4. Poincaré-Hopf index formula: an analytic proof. 4.1. Review of Hodge theorem. 4.2. Poincaré-Hopf index formula. 4.3. Clifford actions and the Witten deformation. 4.4. An estimate outside of [symbol]. 4.5. Harmonic oscillators on Euclidean spaces. 4.6. A proof of the Poincaré-Hopf index formula. 4.7. Some estimates for [symbol]. 4.8. An alternate analytic proof. 4.9. References -- ch. 5. Morse inequalities: an analytic proof. 5.1. Review of Morse inequalities. 5.2. Witten deformation. 5.3. Hodge theorem for ([symbol]). 5.4. Behaviour of [symbol] near the critical points of f. 5.5. Proof of Morse inequalities. 5.6. Proof of proposition 5.5. 5.7. Some remarks and comments. 5.8. References -- ch. 6. Thom-Smale and Witten complexes. 6.1. The Thorn-Smale complex. 6.2. The de Rham map for Thom-Smale complexes. 6.3. Witten's instanton complex and the map [symbol]. 6.4. The map [symbol]. 6.5. An analytic proof of theorem 6.4. 6.6. References -- ch. 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. | |
| 520 | |a 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. | ||
| 650 | 0 | |a Chern classes. |0 http://id.loc.gov/authorities/subjects/sh97008471 | |
| 650 | 0 | |a Index theorems. |0 http://id.loc.gov/authorities/subjects/sh85064860 | |
| 650 | 0 | |a Complexes. |0 http://id.loc.gov/authorities/subjects/sh85029372 | |
| 650 | 7 | |a MATHEMATICS |x Topology. |2 bisacsh | |
| 650 | 0 | 7 | |a Index theorems. |2 cct |
| 650 | 0 | 7 | |a Complexes. |2 cct |
| 650 | 0 | 7 | |a Chern classes. |2 cct |
| 650 | 7 | |a Chern classes. |2 fast |0 (OCoLC)fst00853646 | |
| 650 | 7 | |a Complexes. |2 fast |0 (OCoLC)fst00871597 | |
| 650 | 7 | |a Index theorems. |2 fast |0 (OCoLC)fst00968961 | |
| 655 | 4 | |a Electronic books. | |
| 655 | 0 | |a Electronic books. | |
| 776 | 0 | 8 | |i Print version: |a Zhang, Weiping. |t Lectures on Chern-Weil theory and Witten deformations. |d River Edge, N.J. : World Scientific, ©2001 |w (DLC) 2001046629 |
| 830 | 0 | |a Nankai tracts in mathematics ; |v v. 4. |0 http://id.loc.gov/authorities/names/n2001000055 | |
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| 928 | |t Library of Congress classification |a QA613.618 .Z43 2001eb |l Online |c UC-FullText |u https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=e000xna&AN=91479 |z eBooks on EBSCOhost |g ebooks |i 12286193 | ||