Linking of Lagrangian Tori and Embedding Obstructions in Symplectic 4-Manifolds

We classify weakly exact, rational Lagrangian tori in $T^* \mathbb{T}^2- 0_{\mathbb{T}^2}$ up to Hamiltonian isotopy. This result is related to the classification theory of closed $1$-forms on $\mathbb{T}^n$ and also has applications to symplectic topology. As a 1st corollary, we strengthen a result due independently to Eliashberg–Polterovich and to Giroux describing Lagrangian tori in $T^* \mathbb{T}^2-0_{\mathbb{T}^2}$, which are homologous to the zero section. As a 2nd corollary, we exhibit pairs of disjoint totally real tori $K_1, K_2 \subset T^*\mathbb{T}^2$, each of which is isotopic through totally real tori to the zero section, but such that the union $K_1 \cup K_2$ is not even smoothly isotopic to a Lagrangian. In the 2nd part of the paper, we study linking of Lagrangian tori in $({\mathbb{R}}^4, \omega )$ and in rational symplectic $4$-manifolds. We prove that the linking properties of such tori are determined by purely algebro-topological data, which can often be deduced from enumerative disk counts in the monotone case. We also use this result to describe certain Lagrangian embedding obstructions.

From Pseudo-Rotations to Holomorphic Curves via Quantum Steenrod Squares

In the context of symplectic dynamics, pseudo-rotations are Hamiltonian diffeomorphisms with finite and minimal possible number of periodic orbits. These maps are of interest in both dynamics and symplectic topology. We show that a closed, monotone symplectic manifold, which admits a nondegenerate pseudo-rotation, must have a deformed quantum Steenrod square of the top degree element and hence nontrivial holomorphic spheres. This result (partially) generalizes a recent work by Shelukhin and complements the results by the authors on nonvanishing Gromov–Witten invariants of manifolds admitting pseudo-rotations.

A brief survey of the spectral numbers in floer homology

We give a brief introduction and a partial survey of some of the results about spectral numbers in symplectic topology that we are aware of. Without attempting to be comprehensive, we will select just some of the constructions and ideas that, according to our personal taste and our point of view, give a flavor of this fast developing theory.