Séance du 15 mars 2019

Lieu: IHP, salle 201.

11:00 Alexander Ritter (Oxford)
The cohomological McKay correspondence via Floer theory
Abstract: The goal of my talk is to present joint work with Mark McLean (Stony Brook, NY), which proves the cohomological McKay correspondence using symplectic topology techniques. This correspondence states that given a crepant resolution Y of the singularity \C^n / G, where G is a finite subgroup of SL(n,\C), the conjugacy classes of G are in 1-1 correspondence with generators of the cohomology of Y. This statement was proved by Batyrev (1999) and Denef-Loeser (2002) using algebraic geometry techniques. We instead construct a certain symplectic cohomology group of Y whose generators are Hamiltonian orbits in Y to which one can naturally associate a conjugacy class in G. We then show that this symplectic cohomology recovers the classical cohomology of Y.

14:15 Francesco Lin (Princeton)
Monopole Floer homology and spectral geometry
Abstract: By studying the Seiberg-Witten equations, Kronheimer and Mrowka defined a package of invariants of three-manifolds called monopole Floer homology. In this talk, we discuss some interactions between this topological invariant and the spectral geometry of the Laplacian on the underlying Riemannian manifold, with the goal of understanding concrete examples of hyperbolic and Solv manifolds.

16:00 Lucas Dahinden (Heidelberg)
Volume growth of positive contactomorphisms
Abstract: In cooriented contact manifolds there is a natural notion of moving positively, which gives birth to the notion of positive contactomorphism. The quest of studying the dynamical behaviour of these maps includes counting chords between special (Legendrian) submanifolds, which has implications for volume growth and topological entropy. The machinery we use is symplectic homology and Rabinowitz--Floer homology. In this talk I will try to avoid the heavy part of the machinery. Instead I focus on encoding geometric data in such an action functional, and on how to extract dynamical information.

Prochaines séances: 12/04 (Benedetti, Meschler, Golovko), 10/05 (Nonenmacher, Salamon, ?), 14/06 (Macarini, ?, ?).

Autre activité symplectique à Paris:
Séminaire Nantes-Orsay

Séance du 8 février 2019

Lieu: IHP, amphi Darboux.

11:00 Steven Sivek (Imperial College)
Representations and sheaves for Legendrian knots
Abstract: We describe an A_\infty category built out of n-dimensional representations of the Chekanov-Eliashberg DGA of a Legendrian knot in R^3. The case n=1 recovers the augmentation category, which is known to be equivalent to a category of constructible sheaves of microlocal rank 1 constructed by Shende-Treumann-Zaslow. In this talk, we will discuss some evidence for the conjecture that the categories of n-dimensional representations and of microlocal rank n sheaves are equivalent for all n>1 as well. This is joint work with Baptiste Chantraine and Lenny Ng.

14:15 Cheuk Yu Mak(Cambridge)
Tropically constructed Lagrangians in mirror quintic threefolds
Abstract: In this talk, we will explain how to construct embedded closed Lagrangian submanifolds in mirror quintic threefolds using tropical curves and the toric degeneration technique. As an example, we will illustrate the construction for tropical curves that contribute to the Gromov–Witten invariant of the line class of the quintic threefold. The construction will in turn provide many homologous and non-Hamiltonian isotopic Lagrangian rational homology spheres, and a geometric interpretation of the multiplicity of a tropical curve as the weight of a Lagrangian. This is a joint work with Helge Ruddat.

16:00 Nassima Keddari (Strasbourg)
Lagrangian submanifolds and displaceability.
Abstract: The starting point of the talk is one result of M.Damian (2012) who proved a version of Audin’s conjecture for some displaceable monotone Lagrangian submanifolds. To do so, he defines a lifted version of Floer homology and the crucial argument is that this homology is zero for a displaceable Lagrangian. Not all monotone Lagrangians are displaceable, but there are some which are “almost displaceable” in the sense that they have a neighbourhood where all other Lagrangians are displaceable (for example, a great circle in the sphere). Let L_0 be one of them, this means that if we choose a Lagrangian, L, “close” to L_0, it has the same topology and is displaceable. Therefore, the idea is to apply the proof of M.Damian to this Lagrangian and then deduce the same properties for L_0. However, to do so, we need to define Floer homology and its lifted version for L, which is not monotone. We will explain these constructions and give some other consequences arising from them.

Prochaines séances: 15/03 (Albers, Lin, Ritter), 12/04 (Benedetti, Meschler, Golovko), 10/05 (Dahinden, Nonenmacher, Salamon), 14/06 (Macarini,? , ?).

Autre activité symplectique à Paris:
Séminaire Nantes-Orsay