Let L be a compact Lagrangian in a Weinstein manifold
obtained from a subcritical one by attaching a handle along a Legendrian V. We will see how to associate to L a filling of a satelite of V and how this one induces a representation of the Chekanov-Eliashberg algebra of V. We will show that Legendrian contact homology linearised with respect to this representation recovers the Floer...
I will show that the standard contact structure on the real projective spaces RP^{4k+1} is not Liouville fillable using a classical argument on degeneration of moduli spaces of holomorphic spheres. A stronger result has been obtained by Zhengyi Zhou using more algebraic methods. This is a joint work with Klaus Niederküger
Mixing triangulation (in the sense of triangulated categories) with persistence (as in persistence modules) leads to a class of interesting pseudo-metrics in a variety of examples: metric spaces, Tamarkin categories, filtered topological spaces, Fukaya
categories. I will discuss some generalities concerning this machinery and how it specifically applies to the symplectic context. The talk is...
A multiple cut operation on a symplectic manifold produces a collection of cut spaces, each containing relative normal crossing divisors. We explore what happens to curve count-based invariants when a collection of cuts is applied to a symplectic manifold. The invariant we consider is the Fukaya algebra of a Lagrangian submanifold that is contained in the complement of relative divisors. The...
I will define a Floer complex associated to a pair of transverse Lagrangian cobordisms in the symplectization of a contact manifold, by a count of SFT pseudo-holomorphic discs. Then I will show that this complex is endowed with an A_\infty structure. Moreover, I will describe a continuation element in the complex associated to a cobordism L and a small transverse push-off of L.
String topology studies the algebraic structure of the homology of the free loop space of a manifold. I'll describe joint work with Nathalie Wahl about string topology operations, and about what these operations compute. We have simplified, chain-level definitions for the "loop" or "string" product and coproduct. The new definitions make possible new links between geometry and loop products....
I will discuss three "big fiber theorems", the Centerpoint Theorem from combinatorics, the Gromov Maximal Fiber Theorem from topology, and the Non-displaceable Fiber Theorem by Entov and myself, from a unified viewpoint provided by Gromov's ideal-valued measures.
The latter theory, in the symplectic context, is combined with relative symplectic cohomology developed by Varolgunes, yielding...
While by a result of McDuff the space of symplectic embeddings of a closed 4-ball into an open 4-ball is connected,
the situation for embeddings of cubes 𝐶4=𝐷2×𝐷2 is very different. For instance, for the open ball 𝐵4 of capacity 1, there exists an explicit decreasing sequence 𝑐1,𝑐2,⋯→1/3 such that for 𝑐<𝑐𝑘 there are at least 𝑘 symplectic embeddings of the closed cube 𝐶4(𝑐) of capacity 𝑐 into...
I will present a set-up for orbifold Lagrangian Floer theory,
We introduce the notion of dihedral twisted sectors for Lagrangians, intersections of Lagrangians. Then I explain their role in the construction of Floer theory. It is in collaboration with Bohui Chen and Bai-Ling Wang.
We discuss joint work with M. Sullivan where we show that a contactomorphism cannot squeeze some fixed Legendrian knot into an arbitrarily small neighbourhood of a non-Legendrian knot, under the additional constraint that the two knots become isotopic inside the neighbourhood, and that the contact manifold is tight. The techniques used are Giroux's theory of convex surfaces combined with...
