Symplectix 2 juillet 2021

 

Séminaire actuellement dans le cadre du trimestre IHP.
Pour plus d'info voir ici: https://indico.math.cnrs.fr/event/5767/

10h45-11h45 : Nancy Hingston, (the College of New Jersey).   

Title : String topology and self-intersections

Abstract: 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. For example, If the k-fold coproduct of a homology class X on LM is nontrivial, then every representative of X contains a loop with a (k+1)-fold self-intersection.

No knowledge of loop products or string topology will be assumed.  

 

13h45-14h45 : Leonid Polterovich (Tel Aviv)

Title: Symplectic cohomology and ideal valued measures

Abstract: 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 some applications to symplectic rigidity.  Necessary preliminaries will be explained.  A work in progress with Adi Dickstein, Yaniv Ganor, and Frol Zapolsky.

 

15h15-16h15 : Felix Schlenk (Neuchâtel)

Title : Symplectically knotted cubes

Abstract : 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 $C^4 = D^2 x D^2$ is very different. For instance, for the open ball $B^4$ of capacity 1, there exists an explicit decreasing sequence $c_1, c_2, \dots \to 1/3$ such that for $c < c_k$ there are at least $k$ symplectic embeddings of the closed cube $C^4(c)$ of capacity $c$ into $B^4$ that are not isotopic. Furthermore, there are infinitely many non-isotopic symplectic embeddings of $C^4(1/3)$ into $B^4$.

A similar result holds for several other targets, like the open 4-cube, the complex projective plane, the product of two equal 2-spheres, or a monotone product of such manifolds and any closed monotone toric symplectic manifold.
 
The proof uses exotic Lagrangian tori.
 
This is joint work with Joé Brendel and Grisha Mikhalkin.