Stein Manifolds, Contact Structures and Knots (Morlet Chair – François Lalonde)

INTRODUCTION

In the last two decades, several spectacular results obtained, among others, by Donaldson, Eliashberg, Floer, Giroux, Gromov, Kronheimer-Mrowka, Ozsváth-Szabó, Taubes and Witten generated impressive developments in low-dimensional symplectic, contact and geometric topology. Such developments interacted with each other in extremely unexpected and fruitful ways, as for instance in Kronheimer-Mrowka’s proof of the « Property P » conjecture on knots in S3: this result uses essentially Eliashberg’s results about concave symplectic fillings of contact structures, but also Instanton Floer homology and Witten’s conjecture concerning the relation between Donaldson and Seiberg-Witten invariants.

GENERAL DESCRIPTION

The format of this small workshop consists of three main lecture series complemented by a small number of talks by young scientists. The target audience of each of the 3 lecture series is graduate students and postdocs working in areas close to the topics chosen by the speakers.

LECTURE SERIES

One of the underlying themes of the developments alluded to above consists of various kinds of homology theories obtained as infinite-dimensional analogues of Morse homology, constructed using pseudo-holomorphic curves or solutions of various kinds of PDE’s on smooth manifolds. One of the homology theories with the largest number of applications to low-dimensional geometry and topology is Ozsváth-Szabó’s Heegaard Floer homology. In this workshop, Matt Hedden will give a series of lectures describing some of those applications.

Until very recently, all the applications of Heegaard Floer homology have been in dimensions three or four. One of the latest very interesting developments is the announcement by Vincent Colin of an extension of Heegaard Floer homology as a homology theory for contact manifolds of arbitrary dimensions. Vincent Colin will give a series of lectures on this exciting new work.

Last but not least, one of the fundamental open problems in contact topology, i.e. the existence question for contact structures in arbitrary dimensions, has just been solved by Borman, Murphy and Eliashberg. Emmy Murphy will give a series of lectures on and around this amazing result.

• Ana Garcia Lecuona (Aix-Marseille Université)
• Paolo Lisca (University of Pisa)
• András Stipsicz (Hungarian Academy of Sciences)

LECTURES

• Vincent Colin (Université de Nantes)

Open Book Decompositions and Floer Type Homologies

• Matthew Hedden (Michigan State University)

Knot Theory, Complex Curves, and Heegaard Floer Homology

• Emmy Murphy (MIT)

Existence of Contact Structures and Over-twistedness in all Dimensions

TALKS

• Maciej Borodzik (University of Warsaw)

Heegaard Floer Homologies and Rational Cuspidal Curves

• Daniele Celoria (University of Florence)

Grid Homology in Lens Spaces: Coecients and Computations

• Marco Golla (University of Pisa)

Dehn Surgery and Rational Homology Ball

• Ana Garcia Lecuona (Aix-Marseille University)

Splice Links and Colored Signatures

• Burak Özbagci (Koç University Istanbul)

Contact Open Books with Exotic Pages

• Juanita Pinzon Caicedo (Georgia University)

Trisections, Braids and Two-fold Covers of S4

• Wojciech Politarczyk (UAM Poznań)

Equivariant Khovanov Homology

• Andy Wand (University of Glasgow)

Tightness and Open Book Decompositions