Toric and Tropical Techniques in Symplectic Field Theory

The techniques of symplectic and tropical geometry have given us important tools for both constructing and computing modern invariants of Kähler manifolds and symplectic manifolds that are related to holomorphic curves. These approaches are crucial for analysing the A-side in homological mirror symmetry. They also have connections to invariants in low-dimensional topology and knot theory, such as the one furnished by gauge theory. Notable examples of such techniques are as follows.

• Gromov-Witten invariants (of symplectic manifolds) that count closed curves; open Gromov-Witten invariants (of pairs formed by a symplectic manifold and a Lagrangian submanifold) that count curves with boundary on the Lagrangian submanifold. Tropical geometry provides important tools for computing these invariants.
• Symplectic Field Theory (SFT) is a large framework for analogous counts in the setting when either the symplectic or Lagrangian submanifolds are non-compact with cylindrical ends. There are neck-stretching formulae that relate Gromov-Witten invariants to SFT counts.
• Techniques for understanding deformations, degenerations, and wall-crossing phenomena of (open) Gromov-Witten invariants, which occur when the symplectic or complex data changes, can be approached by using both SFT and tropical techniques. Understanding such deformations is essential for computation of the invariants.
• Various types of refinements of the above invariants have recently been suggested, e.g. via enumeration of tropical curves, upgrades to values in the Skein-module from low-dimensional topology, and counts defined for singular manifolds by using SFT.

The main aim of the workshop is to enable researchers that study these invariants from the sides of SFT and tropical geometry to gather and present the latest developments, including applications in homological mirror symmetry and low-dimensional topology. The purpose is to provide a forum that would facilitate new perspectives, scopes, and connections between these theories and their applications.