Quantum topology and contact geometry – Quantact

Modern invariants of low-dimensional manifolds and knots come from frictions between symplectic geometry, mathematical physics, geometric topology and dynamical systems. When representing particles in String theory, a lot of algebraic structures emerge from knot invariants, such as Topological Quantum Field theories (TQFT) and A-infinity-operations. A new source of such invariants comes from symplectic and contact geometry via the use of holomorphic curves and related Floer type homologies.

In recent years, it was understood how these interact with classical algebraico-topological methods. Heegaard-Floer homology has revolutionized the landscape of knot invariants as a categorification of the Alexander polynomial. Khovanov Homology, originally described algebraically, categorifies the Jones polynomial and now also has its geometric version through a dedicated Lagrangian Floer homology: symplectic Khovanov homology by Seidel and Smith. Legendrian contact homology also provides new invariants of smooth knots through the conormal construction, surprisingly again related to Khovanov homology. Cobordisms maps are obtained by looking at Lagrangian cobordisms between Legendrians. In these theories, the differential counts pseudo-holomorphic curves in a symplectic manifold. Topological recursion of Eynard and Orantin provides formulas for this, and the hope is to use it to categorify the Reshetikin-Turaev invariants.

The goal of the project is to interpret by means of symplectic geometry some of the main invariants of quantum topology and reciprocally to systematically study algebraic properties of new invariants arising from symplectic and contact constructions.
A typical example is the study of a cylindrical version of symplectic Khovanov homology which we would like to extend to links in arbitrary 3-manifolds and for which crossed skills and new methods are required.

The project gathers active young members of symplectic and contact geometry, low-dimensional topology/knot theory, dynamical systems and mathematical physics communities who are willing to learn from each others. It has been voluntarily kept at a small size to make it very effective. It intends to create new small groups of collaborators of 2-4 people.

One success would be that each participant manages to extend its expertise and publishes a paper outside of its current research area.