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.
  1. 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.
  2. The Dirac operator. Clifford algebras, spin and spin^c structures, Weitzenbock formulas. Statement of index theorem in special cases.
  3. 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).
  4. Seiberg-Witten invariants. Dimension of moduli space and orientations; perturbations and smoothness.
  5. Special calculations. Seiberg-Witten invariant of 4-torus and complex projective space.
  6. Applications. Embedded surfaces and Thom conjecture; Milnor's unknotting conjecture.
  7. Further applications. As time and interest of students permit.
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.)

Prerequisites:

  1. Certainly 121ab and 110a or equivalent knowledge is required. 110b or equivalent would also be useful.
  2. Some experience with the following (roughly in increasing order of sophistication) would be helpful:
    1. Principal bundles, vector bundles, and connections.
    2. Basics of characteristic classes (first and second Chern classes, second Stiefel-Whitney class).
    3. Basic elliptic operator theory (Sobolev spaces and various embedding theorems).
    4. Estimates for elliptic operators; Hodge theorem.
I will develop as much of this as I need as the course goes along.

Some references:

Books related to Seiberg-Witten theory and 4-manifolds:
  1. Moore, John Douglas. Lectures on Seiberg-Witten invariants. Second edition. Lecture Notes in Mathematics, 1629.
  2. 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.
  3. Marcolli, Matilde. Seiberg-Witten gauge theory. Texts and Readings in Mathematics, 17. Hindustan Book Agency, New Delhi, 1999.
  4. Gompf, Robert E.; Stipsicz, András I. $4$-manifolds and Kirby calculus. Graduate Studies in Mathematics, 20. American Mathematical Society, Providence, RI, 1999
  5. Kirby, Robion. The Topology of 4-manifolds. Lecture Notes in Math. 1374.
General background:
  1. Patrick Shanahan. The Atiyah-Singer Index Theorem, an Introduction. Lecture Notes in Math. 638. (For 2bc)
  2. Blaine Lawson and Marie-Louise Michelson. Spin Geometry. Princeton University Press. (For 2abcd).
  3. Spivak, vol. II. (For 2a)
  4. Milnor, Characteristic classes. (2b, Appendix C explains Chern classes in terms of curvature.)
Survey articles:
  1. Donaldson, S. K. The Seiberg-Witten equations and $4$-manifold topology. Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 1, 45--70.
  2. 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.
  3. 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.
  4. 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.
Original papers:
  1. Kronheimer, P. B.; Mrowka, T. S. The genus of embedded surfaces in the projective plane. Math. Res. Lett. 1 (1994), no. 6, 797--808.
  2. Witten, Edward Monopoles and four-manifolds. Math. Res. Lett. 1 (1994), no. 6, 769--796.