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Summer school Gauge Theory and Applications -- Abstracts

Francesco Lin

  • Title: My lecture series will be titled "An introduction to monopole Floer homology"
  • Abstract: In this lecture series, we will discuss the formal properties and the construction of the invariants of three-manifolds introduced by Kronheimer and Mrowka using Seiberg-Witten theory. After introducing the relevant geometric and topological background, we will focus on recent applications, in particular the disproof of the Triangulation conjecture.

Tomasz Mrowka

  • Title: tba
  • Abstract: tba

Nikolai Saveliev

  • Title: Seiberg-Witten equations and end-periodic Dirac operators
  • Abstract: These lectures will serve as an introduction to the Seiberg-Witten invariant of smooth 4-manifolds with the rational homology of S^1 x S^3 due to Mrowka, Ruberman, and Saveliev. The invariant is a sum of two terms: one is a count of the Seiberg-Witten monopoles on the manifold X, and the other is (essentially) the index of the Dirac operator on a non-compact manifold with end modeled on the infinite cyclic cover of X. Each term is metric and perturbation dependent, but these dependencies cancel each other. Making this precise leads to some interesting index theory, including an extention of the Atiyah-Patodi-Singer index theorem to manifolds with periodic ends. The invariant will be compared to its counterpart arising in the Donaldson theory, in the spirit of the Witten Conjecture. We will also discuss some properties and applications of this invariant; many more will be covered by Daniel Ruberman and Jianfeng Lin in their lectures at the upcoming conference.

Steven Sivek

  • Title: Instantons and contact structures
  • Abstract: The first part of this minicourse will be an introduction to Donaldson invariants and instanton Floer homology. In the second half we will define the contact invariant constructed by Baldwin and myself in Kronheimer and Mrowka’s sutured instanton homology, discuss some of its key properties, and apply it to show that the rank of Khovanov homology detects the trefoils.