博碩士論文 100221008 詳細資訊




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姓名 陳明裕( Ming-yu Chen)  查詢紙本館藏   畢業系所 數學系
論文名稱
(Witten Deformation and Morse Inequalities)
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摘要(中) 本文主要內容是報告用Witten形變的方法去證明Morse inequalities。主要的證明技巧在是由 Bismut 和 Lebeau 所提出的局部化的解析方法。
摘要(英) In this thesis, we give a report on the analytic approach to
the proof of the Morse inequalities based on Witten deformation method. The analytic technique is the analytic localization method of Bismut and Lebeau. We mainly follow [18] of W. Zhang.
關鍵字(中) ★ 維騰形變
★ 莫爾斯不等式
關鍵字(英) ★ Witten deformation
★ Morse inequality
論文目次 1 Introduction...1
2 Preliminary...1
2.1 Review of the de Rham Cohomology...1
2.2 Review of the Hodge Theorem...2
2.3 Review of the Morse Inequalities...4
2.4 Example...5
3 Witten Deformation Method and Morse Inequalities...6
3.1 Witten Deformation...6
3.2 Hodge Theorem for (Omega^*(M),d_{Tf} )...7
3.3 Proof of Morse Inequalities by Witten Deformation...8
3.4 Cli ord Operators...9
3.5 An Estimate Outside of U_{xin zero( abla f)}U_x...11
3.6 On the Critical Points of f for Behavior of D^2_{Tf}...12
4 Proposition 2...13
4.1 Decomposition of the Witten Deformation Operator...13
4.2 Estimate of D_{T,1}, D_{T,2} and D_{T,3}...14
4.3 Estimate of D_{Tf}...15
4.4 Prove of Proposition 2...18
4.4.1 Lemma 3...18
4.4.2 Proof of Proposition 2...19
5 Appendix...22
Reference...24
參考文獻 [1] Aldo Andreotti and Theodore Frankel, The Lefschetz Theorem on Hyperplane Sections, Annals of Mathematics Second Series, Vol. 69, No. 3 Published by: Annals of Mathematics, (1959), pp. 713-717.
[2] R. Bott and L. Tu, Diff erential Forms in Algebraic Topology. Graduate Text in Math. Vol. 82,Springer-Verlag, 1982, pp. 13-18.
[3] J.-M. Bismut and G. Lebeau, Complex Immersions and Quillen metrics, Springer-Verlag Publ. Math. IHES. 74 (1991), Chapter IX.
[4] J.-M. Bismut, The Witten complex and degenerate Morse inequalities, J. Diff . Geom. 23 (1986), 207-240.
[5] Kung-Ching Chang, Infi nite Dimensional Morse Theory And Its Applications, Montral, Qubec, Canada : Les Presses de l'Universit de Montral, 1985.
[6] R. Forman, Morse-Theory for cell-complexes, Adv. Math. 134 (1998), 90-145.
[7] R. Forman, A user's guide to discrete Morse theory, Sem. Loth. de Comb. 48 (2002).
[8] Lennie Friedlander, Riesz projections, available at
http://math.arizona.edu/~friedlan/teach/528/proj.pdf
[9] Martin Guest, Morse Theory in the 1990s. Invitations to geometry and topology, 146207, Oxf. Grad. Texts Math., 7, Oxford Univ. Press, Oxford, 2002.
[10] Xiaonan Ma, Holomorphic Morse inequalities and Bergman Kernels, Progress in Mathematics, 254. Birkhuser Verlag, Basel, 2007.
[11] J.Milnor, Morse Theory based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J., 1963, pp. 6 , pp.32-37.
[12] Yukio Matsumoto ; translated by Kiki Hudson, Masahico Saito. An Introduction to Morse Theory, Providence, R.I. : American Mathematical Society, 2002, pp. 32.
[13] Jost, Riemannian Geometry and Geometric Analysis, 6ed, Springer-Verlag, Berlin-Heidelberg-New York, 2011, pp. 167-169.
[14] John Roe, Elliptic Operators, Topology and Asymptotic Methods, 2ed, Addison Wesley Longman, 1998, pp. 119-124.
[15] E. Witten, Supersymmetry and Morse theory. J. Diff . Gemo. 17 (1982), 661-692.
[16] F.W.Warner. Foundations of Diff erentiable Manifolds and Lie Groups. GTM 94, Springer-Verlag, Berlin-Heidelberg-New York, (1983), pp. 220-225.
[17] Rachel Elana Zax, Simplifying Complicated Simplicial Complexes: Discrete Morse Theory and its Applications, available at
http://www.math.harvard.edu/theses/senior/zax/zax.pdf
[18] Weiping Zhang, Lectures on Chern-Weil Theory and Witten Deformations, World Scienti c Publishing Co., Inc., River Edge, NJ, 2001.
指導教授 黃榮宗(Rung-Tzung Huang) 審核日期 2014-1-21
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