Gauge-Theoretic Invariants

This reading group aims to serve as an introduction to gauge theoretic invariants in topology, focusing primarily on the Seiberg–Witten invariants in low-dimensional topology. The focus will accordingly be more on the intuition and motivation behind gauge-theoretic invariants, and on their use in modern topology; details of analysis will be taken on trust. No prior background in gauge theory will be necessary, but familiarity with differential topology will be expected.

Organizer: Simi Hellsten

Notes: Up to talk 1. [link]

Schedule: 

  • Theory:
  • Course overview and preview of results. Introduction to gauge theory, including review of principal/associated bundles, connections, and curvature. Speaker: Shing Tak Lam.
  • Spin groups and Spin-structures. Spinc groups and Spinc-structures. Speaker: João Camarneiro.
  • Connections on Spin-bundles and Spinc-bundles. Dirac operators. Speaker: Simi Hellsten
  • Seiberg–Witten theory. SW equations for 4-manifolds and 3-manifolds.
  • Seiberg–Witten theory, cont. Key technical results, e.g. compactness and transversality.
  • Applications:
  • 4-manifolds: intersection forms and SW invariants.
  • 3-manifolds: SW invariants, links to knots, and conjectured relations.
  • K¨ahler surfaces: recap of K¨ahler geometry, and SW invariants.
  • Symplectic geometry: recap of symplectic 4-manifolds, and SW invariants.
  • The Thom conjecture