Symplectic Geometry

We study the geometry of the twistor space of the universal hyperkaehler implosion Q for SU(n). Using the description of Q as a hyperkaehler quiver variety, we construct a holomorphic map from the twistor space Z_Q of Q to a complex vector…

We describe the extent to which Ionel-Parker's proposed refinement of the standard relative Gromov-Witten invariants sharpens the usual symplectic sum formula. The key product operation on the target spaces for the refined invariants is…

We construct Ionel-Parker's proposed refinement of the standard relative Gromov-Witten invariants in terms of abelian covers of the symplectic divisor and discuss in what sense it gives rise to invariants. We use it to obtain some vanishing…

We list the properties of contact homology, beyond purely formal, needed for the proofs of some of the recent applications of contact homology in dynamics to work. The list is put together for the AIM Transversality in Contact Homology…

This manuscript describes in detail the symplectic sum formulas in Gromov-Witten theory and related topological and analytic issues. In particular, we analyze and compare two analytic approaches to these formulas. The Ionel-Parker formula…

This is a survey about certain "almost homomorphisms" and "almost linear" functionals (called quasi-morphisms and quasi-states) in symplectic topology and their applications to Hamiltonian dynamics, functional-theoretic properties of…

We describe a (nonlinear) Fredholm theory for a new class of ambient spaces, as well as for a certain type of categories. The theory is illustrated by an application to the category of stable maps.

For n odd the Lagrangian Grassmannian of \R^{2n} is a \Gamma-manifold.

In this paper, one considers the change of orbifold Gromov-Witten invariants under weighted blow-up at smooth points. Some blow-up formula for Gromov-Witten invariants of symplectic orbifolds is proved. These results extend the results of…

Using the framework of quasi-Hamiltonian actions, we compute the obstruction to prequantization for the moduli space of flat ${\rm PU}(p)$-bundles over a compact orientable surface with prescribed holonomies around boundary components,…

In this paper we start with the applications of polyfold theory to symplectic field theory.

We show that under some topological assumptions, an exact Lagrangian cobordism $(W; L_{0}, L_{1})$ of dimension $dim(W) >5$ is a Lagrangian pseudo-isotopy. This result is a weaker form of a conjecture proposed by Biran and Cornea, which…

We review the quiver descriptions of symplectic and hyperk\"ahler implosion in the case of SU(n) actions. We give quiver descriptions of symplectic implosion for other classical groups, and discuss some of the issues involved in obtaining a…

We examine how symplectic cohomology may be used as an invariant on symplectic structures, and investigate the non-uniqueness of these structures on Liouville domains, a field which has seen much development in the past decade. Notably, we…

Let K $\subset$ G be compact connected Lie groups with common maximal torus T. Let (M, $\omega$) be a prequantisable compact connected symplectic manifold with a Hamiltonian G-action. Geometric quantisation gives a virtual representation of…

This is (mainly) a survey of recent results on the problem of the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb flows. We focus on the Conley conjecture, proved for a broad class of closed symplectic…

We use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and Maslov class with boundary on Lagrangian submanifolds of nonpositive curvature. Applications include the proof of Audin's conjecture on…

Using the structural theorems developed in [Hua13], we study the deformation theory of coisotropic submanifolds in contact manifolds, under the assumption that the characteristic foliation is nonsingular. In the "middle" dimensions, we find…

Let k > 2. We show that if a closed orientable 2k-manifold K, with Euler characteristic not equal to -2, admits an exact Lagrangian immersion into complex Euclidean 2k-space with one transverse double point and no other self-intersections,…

Let $(X,\omega)$ be a compact symplectic manifold, $L$ be a Lagrangian submanifold and $V$ be a codimension 2 symplectic submanifold of $X$, we consider the pseudoholomorphic maps from a Riemann surface with boundary…