Master Classes

Holomorphic curves and applications to enumerative geometry, symplectic and contact topology

October 26-November 2, 2012, Institut de Recherche de Mathématique Avancée, Strasbourg

Pre-courses (October 26-27)

Basics in Analysis (M. Damian).

Sobolev spaces, Fredholm theory, Elliptic operators.

Elementary notions in symplectic and contact geometry (E. Opshtein).

Moser, Darboux, Weinstein and Gray's theorem. Hamiltonian group.

A bit of topology (V. Kharlamov).

cellular homology and De Rahm cohomology, linear fiber bundles, Chern class...

Main lectures (October 29-November 2)

Pseudo-holomorphic curves in Symplectic and Contact Topology (C. Wendl)

Lecture Notes.

A pseudoholomorphic curve can be defined as a map from a Riemann surface to a symplectic manifold that satisfies a nonlinear PDE generalizing the standard Cauchy-Riemann equations of complex analysis. In a seminal paper [3] in 1985, Gromov showed that the topology of the space of holomorphic curves in a symplectic manifold can be used to prove many deep results about the manifold's symplectic structure, such as his famous "non-squeezing" theorem. In the early 90's, the work of Floer, Eliashberg [2] and Hofer [4] extended the applications of holomorphic curves into the odd-dimensional cousin of symplectic geometry, known as contact geometry. Contact manifolds arise naturally as regular level surfaces of Hamiltonians in symplectic manifolds, or as boundary components of symplectic cobordisms: the latter furnish a natural setting in which to study punctured holomorphic curves with cylindrical ends, producing a more TQFT-style picture that yields invariants of contact structures.

The goal of this lecture series will be to give an overview of the basic theory and a few of its classic applications in symplectic and contact topology. I will spend the first three lectures surveying the necessary technical apparatus, which involves a mixture of global analysis with methods from the theory of elliptic PDEs. The fourth lecture will then explain a beautiful application due to McDuff [5], in which the presence of a single symplectically embedded sphere in a closed symplectic 4-manifold allows one to deduce what that manifold is up to symplectomorphism. In the final lecture, I will sketch the extension of these methods to study contact manifolds, including applications to the Weinstein conjecture and symplectic fillability.

Most (but not all) of the material we'll discuss is covered in the lecture notes [7], and some further references are listed below. In particular, the book [6] is an invaluable resource for learning the subject.

LECTURE 1: Introduction and local properties.

Overview of main goals, the nonlinear Cauchy-Riemann equation, elliptic regularity, similarity principle, unique continuation.

LECTURE 2: Moduli spaces.

Moduli spaces of unparametrized curves, Fredholm theory and generic transversality, automatic transversality.

LECTURE 3: Compactness and intersection theory.

Gromov's compactness theorem, positivity of intersections, adjunction formula.

LECTURE 4: Some applications.

Exceptional spheres, McDuff's results on rational/ruled symplectic 4-manifolds [5]

LECTURE 5: Introduction to punctured holomorphic curves.

Contact manifolds and symplectizations, symplectic cobordisms, punctured holomorphic curves and the Hofer energy, SFT compactness [1], sketch of application to symplectic fillability [2] and the Weinstein conjecture [4].

REFERENCES :

Real and open Gromov-Witten invariants (J.-Y. Welschinger)

I will explain how to extract Gromov-Witten type invariants from a counting of pseudo-holomorphic disks with boundary on oriented Lagrangian surfaces in closed symplectic 4-manifolds. Or when these manifolds have a real structure, that is an anti-symplectic involution, from a counting of rational real pseudo-holomorphic curves.

Some topological problems in Symplectic Geometry (E. Opshtein)

A symplectic form on an even-dimensional manifold is a non-degenerate closed 2-form. These structures naturally appear in cotangent spaces (the phase space of classical mechanics), or in algebraic geometry, in the framework of projective, or more generally Kahler, manifolds. Although locally, symplectic structures are very flexible, without invariant, the poor compatibility between closedness and non-degeneracy tend to rigidify the global structure. This competition local flexibility versus global rigidity leaves the structure halfway between a purely topological object (like a volume form) or a much more rigid one (like a Riemanian structure). The pseudo-holomorphic curves, introduced by Gromov in 1985, are a central tool to understand this competition.

This lecture is an introduction to symplectic topology, that is global problems apearing in symplectic geometry. I will illustrate the above-mentioned competition and the use of pseudo-holomorphic curves on various kind of problems about the topology of symplectic manifolds.

LECTURE 1 : Some problems in symplectic geometry.

I will state some classical problems of symplectic topology : classes represented by symplectic forms, topology of symplectic manifolds, fixed points of hamiltonian diffeomorphisms, topology of Lagrangian submanifolds. I will also explain some h-principle type statements.

LECTURE 2 : Constructions of symplectic manifolds.

Classical examples, blowing-up, Gompf construction.

LECTURE 3 : Biran decomposition.

Biran's theorem, Lefschetz-type property, sub-criticality. Applications to ellispoid embeddings.

LECTURE 4 : Gromov's non-squeezing.

Proof of the theorem and its application to the definition of the notion of symplectic homeomorphisms.

LECTURE 5: Balls embeddings and isotopies.

Role of the pseudo-holomorphic curves in the problem of balls isotopies. And more generally in the description of some diffeomorphism groups.

Talks

U. Frauenfelder : Contacting the moon.

In this talk I first introduce the restricted three body problem. I then explain a joint result with P.Albers, G.Paternain and O.van Koert which tells us that energy hypersurfaces of the restricted three body problem are contact for energies below and slightly above the first critical one. Finally I show how combined with holomorphic curve techniques this can be used to obtain global surfaces of sections.

P. Massot : Holomorphic curves and filling obstructions

Some contact manifolds can easily be seen as the boundary of a compact symplectic manifold. However it's much harder to prove that a given contact manifold is *not* such a boundary. In this talk I will explain how certain domains in contact manifolds obstruct existence of a symplectic filling by spawning, in any hypothetical filling, families of holomorphic curves which have nowhere to run hence provide a contradiction.

F. Schlenk : Explicit maximal symplectic packings of a four ball by equal balls

In important work, Gromov, McDuff-Polterovich and Biran computed, for each k, the percentage of the volume of the 4-ball that can be symplectically filled by k equal balls. Their constructions, that are not explicit, use holomorphic spheres in the blow-up of the complex projective plane. I will explain elementary and explicit constructions that give such maximal packings for k less than 9 and for squares.