We present a new and simple proof of Eliashberg-Gromov's theorem based on the notion of $C^0$-commutativity introduced by Cardin and Viterbo.

We show that there exists a Legendrian knot with maximal Thurston–Bennequin invariant whose contact homology is trivial. We also provide another Legendrian knot which has the same knot type and classical invariants but nonvanishing contact homology.

In Towards Generalizing Schubert Calculus in the Symplectic Category, Goldin and Tolman extend some ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. The main goal of this paper is to build on this work by finding more effective formulas. More explicitly, given a generic component of the moment map, they define a canonical class $\alpha_p$ in the equivariant cohomology of the manifold $M$ for each fixed point $p \in M$. When they exist, canonical classes form a natural basis of the equivariant cohomology of $M$. In particular, when $M$ is a flag variety, these classes are the equivariant Schubert classes. It is a long-standing problem in combinatorics to find positive integral formulas for the equivariant structure constants associated to this basis. Since computing the restriction of the canonical classes to the fixed points determines these structure constants, it is important to find effective formulas for these restrictions. In this paper, we introduce new techniques for calculating the restrictions of a canonical class $\alpha_p$ to a fixed point $q$. Our formulas are nearly always simpler, in the sense that they count the contributions over fewer paths. Moreover, our formula is manifestly positive and integral in certain important special cases.

There are two purposes of the present paper. One is to correct an error in the proof of Theorem 6.1.25 in Lagrangian intersection Floer theory anomaly and obstruction, Part I, and Part II, from which Theorem J follows. In the course of doing so, we also obtain a new lower bound of the displacement energy of polydisks in general dimension.

We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic four-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes. As a corollary, we give explicit values for when an ellipsoid of type $E(a, b)$, with $\frac{b}{a} \in \mathbb{N}$, embeds in a polydisc $P(s,t)$. Under this integrality assumption, we also give an alternative proof of a recent result of M. Hutchings showing that the embedded contact homology capacities give sharp inequalities for embedding ellipsoids into polydisks.