# Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

## Three 20 minute research talks

**Dustin Connery-Grigg (Université de Montreal)**, Geometry and topology of Hamiltonian Floer complexes in low-dimension

In this talk, I will present two results relating the qualitative dynamics of non-degenerate Hamiltonian isotopies on surfaces to the structure of their Floer complexes.

The first will be a topological characterization of those Floer chains which represent the fundamental class in $CF_*(H,J)$ and which moreover lie in the image of some chain-level PSS map. This leads to a novel symplectically bi-invariant norm on the group of Hamiltonian diffeomorphisms, which is both $C^0$-continuous and computable in terms of the underlying dynamics. The second result explains how certain portions of the Hamiltonian Floer chain complex may be interpreted geometrically in terms of positively transverse singular foliations of the mapping torus, with singular leaves given by certain maximal collections of unlinked orbits of the suspended flow. This construction may be seen to provide a Floer-theoretic construction of the `torsion-low’ foliations which appear in Le Calvez’s theory of transverse foliations for surface homeomorphisms, thereby establishing a bridge between the two theories.

**Pazit Haim-Kislev (Tel Aviv University)**, Symplectic capacities of $p$-products

A generalization of the cartesian product and the free sum of two convex domains is the $p$-product operation. We investigate the behavior of symplectic capacities with respect to symplectic $p$-products, and we give applications related to Viterbo's volume-capacity conjecture and to $p$-decompositions of the symplectic ball.

**Thibaut Mazuir (University of Paris)**, Higher algebra of A-infinity algebras in Morse theory

In this short talk, I will introduce the notion of $n$-morphisms between two A-infinity algebras. These higher morphisms are such that 0-morphisms correspond to standard A-infinity morphisms and 1-morphisms correspond to A-infinity homotopies. Their combinatorics are then encoded by new families of polytopes, which I call the n-multiplihedra and which generalize the standard multiplihedra.

Elaborating on works by Abouzaid and Mescher, I will then explain how this higher algebra of A-infinity algebras naturally arises in the context of Morse theory, using moduli spaces of perturbed Morse gradient trees.