Math 221b: Seiberg-Witten Theory and 4-manifolds, Spring 2002
Monday, Wednesday, Thursday, Room 210
The course will give an introduction to gauge theory, focussing on Seiberg-Witten theory and applications to the topology of 4-dimensional manifolds. The speed at which I will go through the material depends on various factors, most notably the background of the students attending.
Here is a brief outline of what I hope to cover.
Book: Notes on Seiberg-Witten Theory, by Liviu Nicolaescu. Published by the AMS, Graduate Studies in Mathematics vol. 28. The book is on sale at the AMS bookstore until January 31 for only $35; afterwards it will be back to $59 (or $47 for AMS members.)
- Introduction to 4-manifolds. Basic invariants such as intersection form, characteristic classes. Survey of Freedman's results on topological classification. Discussion of basic problems of smooth classification.
- The Dirac operator. Clifford algebras, spin and spin^c structures, Weitzenbock formulas. Statement of index theorem in special cases.
- Seiberg-Witten equations. Group of automorphisms (gauge group) of the equations and construction of moduli space. Basic analytic results (slice theorem, simple estimates leading to compactness theorems).
- Seiberg-Witten invariants. Dimension of moduli space and orientations; perturbations and smoothness.
- Special calculations. Seiberg-Witten invariant of 4-torus and complex projective space.
- Applications. Embedded surfaces and Thom conjecture; Milnor's unknotting conjecture.
- Further applications. As time and interest of students permit.
I will develop as much of this as I need as the course goes along.
- Certainly 121ab and 110a or equivalent knowledge is required. 110b or equivalent would also be useful.
- Some experience with the following (roughly in increasing order of sophistication) would be helpful:
- Principal bundles, vector bundles, and connections.
- Basics of characteristic classes (first and second Chern classes, second Stiefel-Whitney class).
- Basic elliptic operator theory (Sobolev spaces and various embedding theorems).
- Estimates for elliptic operators; Hodge theorem.
Books related to Seiberg-Witten theory and 4-manifolds:
- Moore, John Douglas. Lectures on Seiberg-Witten invariants. Second edition. Lecture Notes in Mathematics, 1629.
- Morgan, John W. The Seiberg-Witten equations and applications to the topology of smooth four-manifolds. Mathematical Notes, 44. Princeton University Press, Princeton, NJ, 1996.
- Marcolli, Matilde. Seiberg-Witten gauge theory. Texts and Readings in Mathematics, 17.
Hindustan Book Agency, New Delhi, 1999.
- Gompf, Robert E.; Stipsicz, András I. $4$-manifolds and Kirby calculus. Graduate Studies in Mathematics, 20. American Mathematical Society, Providence, RI, 1999
- Kirby, Robion. The Topology of 4-manifolds. Lecture Notes in Math. 1374.
- Patrick Shanahan. The Atiyah-Singer Index Theorem, an Introduction. Lecture Notes in Math. 638. (For 2bc)
- Blaine Lawson and Marie-Louise Michelson. Spin Geometry. Princeton University Press. (For 2abcd).
- Spivak, vol. II. (For 2a)
- Milnor, Characteristic classes. (2b, Appendix C explains Chern classes in terms of curvature.)
- Donaldson, S. K. The Seiberg-Witten equations and $4$-manifold topology. Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 1, 45--70.
- Bennequin, Daniel Monopôles de Seiberg-Witten et conjecture de Thom (d'après Kronheimer, Mrowka et Witten). (French) [Seiberg-Witten monopoles and the Thom conjecture (following Kronheimer, Mrowka and Witten)] Séminaire Bourbaki, Vol. 1995/96. Astérisque No. 241 (1997), Exp. No. 807, 3, 59--96.
- Kotschick, Dieter The Seiberg-Witten invariants of symplectic four-manifolds (after C. H. Taubes). Séminaire Bourbaki, Vol. 1995/96. Astérisque No. 241 (1997), Exp. No. 812, 4, 195--220.
- Kronheimer, P. B. Embedded surfaces and gauge theory in three and four dimensions. Surveys in differential geometry, Vol. III (Cambridge, MA, 1996), 243--298, Int. Press, Boston, MA, 1998.
- Kronheimer, P. B.; Mrowka, T. S. The genus of embedded surfaces in the projective plane. Math. Res. Lett. 1 (1994), no. 6, 797--808.
- Witten, Edward Monopoles and four-manifolds. Math. Res. Lett. 1 (1994), no. 6, 769--796.