University of Maryland
Student Complex Geometry Seminar
Time: Wednesdays, 4–5 PM
Place: Kirwan Hall, MTH 0306
This semester, the Student Complex Geometry Seminar (SCGS) will focus on Morse Theory.
The plan is to cover the basics of Morse theory in the first half of the semester and then move on to more advanced topics.
Reference: J. W. Milnor, Morse Theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, NJ, 1963 (unofficial printing).
Historical Reference and Survey on Morse Theory:
- M. Morse, The Critical Points of Functions and the Calculus of Variations in the Large, Bulletin of the American Mathematical Society 35 (1929), 38–54.
- R. Bott, Morse Theory Indomitable, Publications Mathématiques de l’IHÉS 68 (1988), 99–114.
- J. J. O’Connor and E. F. Robertson, Harold Calvin Marston Morse, MacTutor History of Mathematics Archive, University of St Andrews; last updated October 2003.
Potential Further Topics (Suggestions are welcome!)
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Morse Theory on Path Spaces, Riemannian Geometry, and Bott Periodicity
References:- J. W. Milnor, Morse Theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, NJ, 1963; Parts III–IV (unofficial printing).
- J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland Mathematical Library, Vol. 9, North-Holland, Amsterdam–Oxford, 1975; American Elsevier Publishing Co., Inc., New York, 1975; Chapter 4.
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Witten Deformation and the Analytic Approach to Morse Inequalities
References:- E. Witten, Supersymmetry and Morse Theory, Journal of Differential Geometry 17 (1982), no. 4, 661–692 (1983).
- W. P. Zhang, Lectures on Chern–Weil Theory and Witten Deformations, Nankai Tracts in Mathematics, Vol. 4, World Scientific, River Edge, NJ, 2001; Chapter 5.
- G. Marinescu, The Laplace Operator on High Tensor Powers of a Line Bundle, Habilitationsschrift; Chapter 2.
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Morse Homology (Prelude to Floer Theory?)
References:- M. Hutchings, Lecture Notes on Morse Homology (with an Eye towards Floer Theory and Pseudoholomorphic Curves), lecture notes, December 15, 2002.
- M. Audin and M. Damian, Morse Theory and Floer Homology, translated from the 2010 French original by Reinie Erné, Universitext, Springer, London, 2014; EDP Sciences, Les Ulis, 2014; Part I.
- M. Schwarz, Morse Homology, Progress in Mathematics, Vol. 111, Birkhäuser, Basel, 1993.
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Picard–Lefschetz Theory
Reference:- Voisin C. Hodge Theory and Complex Algebraic Geometry II. Schneps L, trans. Cambridge University Press; 2003. §1–3.
Tentative Schedule
| Date | Content | Reference | Speaker | Supplement |
|---|---|---|---|---|
| Sep 9 | Introduction, Hessian, Non-degenerate Critical Points, Morse Lemma | Milnor §2 | Vasanth | Milnor §2 (PDF) |
| Sep 16 | Fundamental Theorem of Morse Theory I | Milnor §3 | Jen Lorenzo | |
| Sep 23 | Fundamental Theorem of Morse Theory II | Milnor §3 | Jen Lorenzo | |
| Sep 30 | Fundamental Theorem of Morse Theory III | Milnor §3 | Jen Lorenzo | |
| Oct 7 | Reeb Theorem, Morse inequalities | Milnor §4, 5 | Xuchun | |
| Oct 14 | Existence of Morse Functions | Milnor §6 | Eric | |
| Oct 21 | Cellular decomposition of ℂPn, Lefschetz Hyperplane Theorem | Milnor §4, 7 | Yu-Chi | |
| Oct 28 | ||||
| Nov 4 | ||||
| Nov 11 | ||||
| Nov 18 | ||||
| Dec 2 | ||||
| Dec 9 |