Summer School
Dates: from may 30th to june 10th 2011
Lectures
The full program is now available for download
First week
- Patrick Massot
- Topological methods in 3--dimensional contact geometry
This will be an introduction to Giroux's theory of convex surfaces in
contact 3-manifolds and its simplest applications. The first goal is to
explain why all the information about a contact structure in a
neighborhood of a generic surface is encoded by finitely many
curves on the surface. Then we will describe the bifurcations that
happen in generic families of surfaces (with one or sometimes two
parameters). Hopefully, time will permit to use this to convince the
audience that the standard contact structure on S^3 is tight (Bennequin)
and that all tight contact structures on S^3 are isotopic to it
(Eliashberg). There are no prerequisite from contact topology, the
course will start with the definition of contact structures.
- Frédéric Bourgeois
- Contact homology, symplectic homology and Legendrian surgery
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The first part of the course will be an overview of several holomorphic
curves invariants for symplectic and contact manifolds, such as contact
homology and symplectic homology. The relationship between these
invariants will then be explained; this will lead to a common algebraic
framework. We will finish with a description of the effect of
Legendrian surgery on these invariants.
- Fran Presas
- Open books and Lefschetz pencils in contact geometry.
- The goal of this course is to introduce the notion of open book and Lefschetz pencils decomposition and to show some of the uses of them in contact topology. We will discuss the constructions in dimension 3 manifolds in detail and we will give an outlook of what can be done in higher dimensional manifolds.
Lectures:
- Definition of open book. Adapted open books. Convex functions. Equivalence of convex functions and open books.
- Open books in 3 dimensions (1). Existence.
- Open books in 3 dimension (2). Uniqueness up to stabilizations. Applications.
- Open books and symplectic Lefschetz pencils. Relations with holomorphically fillable contact manifolds.
- Contact Lefschetz pencils. Existence and some old and new results.
Second week
- Denis Auroux
- An introduction to Fukaya categories
The first half of this course will be an overview of Lagrangian Floer
homology: the Lagrangian intersection problem, holomorphic discs, the
Floer differential, product structures, and A-infinity relations. This
will lead to an (incomplete, but sufficient for many purposes)
definition of the Fukaya category.
In the second half, we will take a closer look at several flavors of
Fukaya categories, and some applications. In particular, we will discuss
how Dehn twists and connected sums give rise to exact triangles, and
briefly present Seidel's work on Fukaya categories of Lefschetz
fibrations. If time permits, we will finish by a short introduction to
wrapped Fukaya categories and their relation to symplectic field theory.
- Gordana Matic
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Title: Contact invariants in Heegaard Floer Homology
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In the first half of the course we will give a brief introduction to
Heegaard Floer Homology. We will then explain how to define
invariants of contact structures in Heegaard Floer Homology for
closed manifolds, and show some applications of the invariant. In the
second half of the course we will concentrate on manifolds with
sutured boundary. We will introduce partial open book decompositions
and see how they lead to a contact invariant in Sutured Floer
Homology. We will finish with some examples, calculations and
applications.
- Norbert A'Campo (Singularities)
- Klaus Niederkruger
- Introduction to contact topology in higher dimensions
- This series of talks will be mostly focused on fillability questions
for higher dimensional contact manifolds.
The first two talks will give an overview of some basic examples and
theorems known so far, comparing them with analogous results in
dimension three. We will also explain several constructions and
operations that lead to non-fillable manifolds.
The second half of this series will explain how to use holomorphic
curves with boundary to prove the fillability results stated in the
first half of the course.
No knowledge of holomorphic curves will be required, and many
properties will only be quoted.
The aim is that at the end of the talks, the audience has some
intuitive knowledge how to read off holomorphic curve properties from
certain topological information of the contact manifold.
List of participants
Full list of participants for the special trimester.
