Programme

Horaire quotidienne

  • 9:30-10:30 == Kristen Hendricks
  • 10:30-11:00 == pause café
  • 11:00-12:00 == Francesco Lin
  • 12:00-13:30 == Lunch
  • 13:30-14:30 == Claudius Zibrowius
  • 14:30-15:00 == pause café
  • 15:00-16:00 == Matthew Stoffregen

Mini-cours

Involutive Heegaard Floer homology and homology cobordism*

Conférencière: Kristen Hendricks

Résumé: In 2013, C. Manolescu used a Pin(2)-equivariant version of the gauge-theoretic invariant of 3-manifolds Seiberg-Witten Floer homology to disprove the remaining outstanding cases of the Triangulation Conjecture. Seiberg-Witten Floer homology is identified with Ozsvath and Szabo’s Floer-theoretic 3-manifold Heegaard Floer homology, which is noted for its unusual computational accessibility. Involutive Heegard Floer homology is a variant theory of Heegaard Floer homology which incorporates the data of a conjugation symmetry on the theory; it is inspired by Manolescu’s work on Pin(2)-SWFH and conjecturally identified with Z_4-equivariant SWFH.

In these lectures we review some general background and basic aspects of Heegaard Floer homology and then introduce involutive Heegaard Floer homology, discuss its structure and properties, and examine some of its recent applications to the study of the integer homology cobordism group. Since one of the noteworthy advantages of both Heegaard Floer and its involutive variant is computational friendliness, there will be some focus on understanding examples and sample computations.

Tentative schedule is as follows:

(1) Background on Heegaard Floer homology, introduction to involutive Heegaard Floer

(2) The involutive correction terms, basic computations and properties

(3) Iota-complexes and connected Heegaard Floer homology

(4) Almost-iota-complexes and applications to the homology cobordism group

Monopole Floer homology

Conférencier: Francesco Lin

Résumé: In this lecture series, we’ll define monopole Floer homology, which is an invariant of three-manifolds obtained by studying “infinite dimensional Morse theory”. After discussing its original motivation as a tool for understanding gluing formulas for Seiberg-Witten invariants of four-manifolds, we’ll introduce the relevant notions of differential geometry and topology needed in the construction. Finally, we’ll discuss its definition, trying to highlight the analogies and differences between Morse and Floer theory.

1) Motivation: gluing results for Seiberg-Witten invariants

2) The Seiberg-Witten equations on a three-manifold

3) S1-equivariant Morse homology

4) Floer homology, and why it works.

Algebraic Aspects of Pin(2)-equivariant Floer homology

Conférencier: Matthew Stoffregen

Résumé: In 2013, Manolescu introduced Pin(2)-equivariant monopole Floer homology, an enhancement of the monopole Floer homology of Kronheimer-Mrowka, and used it to disprove the high-dimensional triangulation conjecture. Since then, this theory has been further developed, with a generalization to all closed three-manifold due to F. Lin, along with new applications to surgeries and intersection forms. Compared to its cousin, the involutive Heegaard-Floer homology introduced by Hendricks and Manolescu, it has the advantage of a relationship to the classical Rokhlin invariant, but a main issue remains the computational inacessibility of the Pin(2)-theory.

In these lectures, we start with background in equivariant topology and introduce Pin(2)-equivariant monopole Floer homology, focusing on its formal properties. With these properties in hand, we will go through Manolescu’s disproof of the Triangulation Conjecture. Along the way, we develop machinery to further study the homology cobordism group, discuss some additional applications, and give computational tools for working with the Pin(2)-theory.

Lecture 1: Equivariant topology, Borel and Tate homology

Lecture 2: The Triangulation Conjecture and Local Equivalence

Lecture 3: Connected Homology

Lecture 4: Connected Sums and the Local Equivalence Group

Immersed curves in reduced Khovanov homology

Conférencier: Claudius Zibrowius

Résumé: The goal of this series of lectures is to reinterpret reduced Khovanov homology in terms of the Lagrangian intersection Floer theory of the 3-punctured disc. In particular, we will introduce an immersed curve invariant of 4-ended tangles which is equivalent to the reduced Khovanov homology of such tangles.

At the heart of this reinterpretation lies a classification result for chain complexes over a certain cobordism category. In the first half of the lectures, we will discuss a general framework in which to state and prove this classification result. In the second half, we will see how it gives rise to the immersed curve invariant of 4-ended tangles—in fact, a whole family of such invariants—and we will discuss some of their properties.