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

LaTeX notes: Up to talk 5. Will be slowly updated. [link]

Schedule and handwritten notes: 

  • Theory:
  • Course overview and preview of results. Introduction to gauge theory, including review of principal/associated bundles, connections, and curvature. Speaker: Shing Tak Lam. [notes]
  • Spin groups and Spin-structures. Spinc groups and Spinc-structures. Speaker: João Camarneiro. [notes 1, 2]
  • Connections on Spin-bundles and Spinc-bundles. Dirac operators. Speaker: Simi Hellsten. [notes]
  • Seiberg–Witten theory. SW equations for 4-manifolds & key technical results. Speaker: Andraž Čepič. [notes]
  • Seiberg–Witten theory, cont. Further technical results and basic computations. Speaker: Shing Tak Lam. [notes]
  • Applications:
  • 4-manifolds: intersection forms, Donaldson’s theorem, and SW invariants. Speaker: Simi Hellsten. [notes]
  • K¨ahler surfaces: recap of K¨ahler geometry, and applications of SW invariants. Speaker: João Camarneiro. [notes]
  • 3-manifolds: SW invariants of 3-manifolds and interpretations. Speaker: Tudor Caba. [notes]
  • Symplectic geometry: recap of symplectic 4-manifolds, and applications SW invariants. Speaker: João Camarneiro. [notes]
  • The Thom conjecture: summary of the minimal genus problem, proof of simple case of Thom conjecture using SW invariants. Speaker: Simi Hellsten. [notes]