We are working towards a formula to that gives some lowerbound on the number of pseudo-holomorphic sections of a Lefschetz fibration over the disk, while keeping track of their relative homology class. To achieve this, we developed a particular local coefficient system and gave a fully explicit and geometric proof of the exactness of Seidel's triangle in Lagrangian and Fixed Point Floer homology.
Draft available upon request.
I gave an explicit description of the Floer cohomology of a family of Dehn twists about disjoint Lagrangian spheres in a \(w^+\)-monotone rational symplectic manifold. This is a generalization of a classic result by P. Seidel (1996) and it is based on a neck-stretching argument and some delicate reasoning with an energy filtration for $CF(\tau_V)$ which show that certain ``bad trajectories" do not count towards the differentials for $CF(\tau_V)$, proving that the chain complex can be naturally identified with the one for Morse relative cohomology of $(M,V)$. As a byproduct of our framework, in a monotone symplectic manifold, I defined a class in the fixed point Floer cohomology of a Dehn twist by counting half strips bound to the given Lagrangian sphere and prove it must vanish. In subsequent work, joint with T. Perutz, we plan on using this vanishing result to give a new geometric proof of Seidel's long exact sequence.
Journal of Symplectic Geometry 2024 vol 22 issue #3. arXiv.
This is my master thesis project, where I (almost) completely classified four-manifold with prescribed fundamental group up to stable diffeomorphism. As a corollary I got some restriction on the divisibility of the signature of such manifolds under some additional assumptions. A. Debray communicated me he was able to figure out the classification in the missing case.
If you are interested in the pdf copy of my thesis e-mail me. I will try to make it more broadly available in the future