This fall we will have a seminar on Seiberg-Witten theory. The goal is to learn about the construction of the Seiberg-Witten invariant for 4-manifolds and to see applications of this invariant. The first two-thirds of the seminar will be about setting up the theory and defining the invariant. The last third will be about several applications to the topology and geometry of 4-manifolds.

Intended topics with references

  1. Introduction to topology of four-manifolds
  2. Clifford algebras, spin groups, spin^c groups: Chapter 2 of [1]; Section 1.3.1, 1.3.2 of [2].
  3. Spin^c structures, spinors: Sections 3.1, 3.2 of [1]; Section 1.3.3 of [2]; Chapter 2 of [3].
  4. Dirac operator, Seiberg-Witten equations and functional setup: Sections 3.3, 4.1, 4.2 of [1]; Section 1.3.4, 2.1 of [2]; Section 3.1 of [3].
  5. Elliptic PDE theory, existence, finite dimensionality: Sections 3.3, 4.3-6 of [1]; Section 1.5, 2.2.2 of [2]; Section 3.2 of [3]
  6. Compactness: Chapter 5 of [1]; Section 2.2.1 of [2]; Section 3.3 of [3].
  7. Transversality, orientability, defining the invariant: Chapter 6 of [1]; Sections 2.2.3, 2.2.4, 2.3 of [2]; Sections 3.4, 3.7 of [3].
  8. Kähler and symplectic have non-zero invariant: Chapter 7 of [1]; Chapter 3 of [2]; Sections 3.8, 3.9 of [3]; [4].
  9. Intersection form, Symplectic Thom conjecture: Section 2.4.2 of [2]; Section 3.5 of [3]; [6].
  10. Weinstein conjecture: [5]
  11. Exotic smooth structures: TBD

References

  1. Morgan, John W. 2014. The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds. Mathematical Notes, v. 44. Princeton University Press.
  2. Nicolaescu, Liviu I. 2000. Notes on Seiberg-Witten Theory. Graduate Studies in Mathematics, volume 28. American Mathematical Society. link.
  3. Moore, John Douglas. 2001. Lectures on Seiberg-Witten Invariants. 2nd ed. Lectures on Seiberg-Witten Invariants 1629. Springer. link.
  4. Kotschick, Dieter. The Seiberg-Witten invariants of symplectic four-manifolds, in Séminaire Bourbaki : volume 1995/96, exposés 805-819, Astérisque, no. 241 (1997), Talk no. 812, 26 p. link.
  5. Hutchings, Michael. 2009. ‘Taubes’s Proof of the Weinstein Conjecture in Dimension Three’. Version 2. Preprint, arXiv. link.
  6. Ozsvath, Peter, and Zoltan Szabo. 2000. ‘The Symplectic Thom Conjecture’. The Annals of Mathematics 151 (1): 93. link.

Upcoming Talks

  • December 19, 2025 | Seiberg-Witten theory (Fall 2025)

    Bas Wensink - TBA

    Time: 12:30 — Location: HFG 707

Previous Talks