I'll define a new symplectic object called a pumpkin domain and I'll construct its Fukaya category. This simultaneously generalizes the wrapped Fukaya category of a Liouville domain and the Fukaya-Seidel category of a Lefschetz fibration. Pumpkin domains come with a natural geometric gluing operation ; at the level of Fukaya categories, it corresponds to a certain pushout. After describing this, I'll give some simple applications and a conjectural connection to Legendrian contact homology.
Contact homology is a powerful invariant of contact manifolds introduced by Eliashberg--Givental--Hofer. The definition involves certain counts of pseudo-holomorphic curves, however these are usually only "virtual" counts since the moduli spaces of such curves are often not cut out transversally. I will discuss one way to construct these counts rigorously.
La géométrie algébrique dérivée est une théorie récente dont le but est de pouvoir réaliser des opérations de nature homotopique dans un contexte algébrique. Dans l’exposé, on se concentrera sur un aspect bien précis, celui des intersections dérivées de cycles algébriques, à travers trois exemples :
— l’exemple d’une intersection de deux courbes, qui fournit une interprétation de la formule des Tor
de Serre ;
— le cas l’auto-intersection de la diagonale d’un schéma algébrique lisse, qui encode l’isomorphisme
de Hochschild-Kostant-Rosenberg ;
L'invariant d'Alexander L2 est un invariant de nœuds introduit par Li et Zhang en 2006, que l'on peut voir comme une certaine torsion L2 sur un complexe de chaînes L2 associé à l'extérieur du nœud. Il peut aussi être construit depuis une présentation du groupe du nœud, à l'aide du calcul de Fox, similairement au polynôme d'Alexander. Dans mon exposé je présenterai cette construction après quelques rappels sur les invariants de nœuds et la théorie des invariants L2, puis je présenterai plusieurs propriétés de l'invariant d'Alexander L2, notamment le fait qu'il détecte le nœud trivial.
Dans cet exposé, je parlerai de plongements symplectiques de domaines toriques en dimension quatre et d'un nouveau résultat qui relie l'espace des trajectoires des billes sur une table de billard avec un domaine torique. J'expliquerai comment certaines capacités symplectiques qui dérivent de l'homologie de contact plongée peuvent être utilisées pour montrer que certains plongements sont optimaux.
Embedded contact homology is an invariant of a three-manifold isomorphic to Heegaard Floer homology and Seiberg-Witten Floer homology. However, ECH chain complex depends on the choice of a contact form on the manifold and is of interest for studying symplectic geometric properties (e.g. ECH capacities). Extending the work of Hutchings-Sullivan, we combinatorially describe the ECH chain complexes of toric contact manifolds such as T^3 with a T^2-invariant contact form.
Lagrangian Floer cohomology is the best tool to study the intersection theory of Lagrangian submanifolds of a given symplectic manifold. Unfortunately it cannot be defined in general, instead there is a more complicated algebraic invariant, a filtered A-infinity algebra, known as the Fukaya algebra, introduced by Fukaya-Oh-Ohta-Ono . In this talk we will review its constructions and describe the Fukaya algebra of the product of two Lagrangians. For this we will first have to define the tensor product of A-infinity algebras.
In recent years, low dimensional topologists have become interested in the study of "generic" smooth maps to surfaces. The approach is similar to Morse theory, only with two dimensional target. In this talk, I will discuss a specific problem in the 4-dimensional context which is analogous to the (uniqueness of) cancellation of critical points of Morse functions. I will also indicate applications to certain pictorial descritpions of 4-manifolds in terms of curve configurations on surfaces. This is joint work with Kenta Hayano.
If we have are two commuting symplectomorphisms of a symplectic manifold, each one of them induces an automorphism of Floer cohomology of the other one. I will show that the supertraces of these two automorphisms are equal, developing a suggestion by Paul Seidel. As a particular case, I will explain that if a symplectomorphism f commutes with a symplectic involution, the dimension of HF(f) is bounded below by a topological quantity: the Lefschetz number of the restriction of f to the fixed locus of the involution.
We show that the Hamiltonian isotopy class of the symplectic Dehn twist depends on the parametrisation used in the construction. Moreover, this result is used to construct non-trivial symplectomorphisms of T^(S^n x S^1) having compact support, as well as non-standard symplectic structures on T^(S^n x S^1) coinciding with the standard structure outside of a compact set. (joint with Jonathan Evans)
