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We show that for singular hypersurfaces, a version of their genus-zero Gromov-Witten theory may be described in terms of a direct limit of fixed point Floer cohomology groups, a construction which is more amenable to computation and easier…

A self-dual harmonic 2-form on a 4-dimensional Riemannian manifold is symplectic where it does not vanish. Furthermore, away from the form's zero set, the metric with the 2-form give a compatible almost complex structure and thus…

In this note we extend to non trivial Hamiltonian fibrations over symplectically uniruled manifolds a result of Lu's, \cite{Lu}, stating that any trivial symplectic product of two closed symplectic manifolds with one of them being…

The first part of this work constructs positive-genus real Gromov-Witten invariants of real-orientable symplectic manifolds of odd "complex" dimensions; the present part focuses on their properties that are essential for actually working…

We compute, with Symplectic Field Theory techniques, the Gromov-Witten theory of the complex projective line with orbifold points. A natural subclass of these orbifolds, the ones with polynomial quantum cohomology, gives rise to a family of…

We construct Gromov-Witten invariants of general symplectic manifolds.

In this paper the Gromov-Witten invariants on a class of noncompact symplectic manifolds are defined by combining Ruan-Tian's method with that of McDuff-Salamon. The main point of the arguments is to introduce a method dealing with the…

In this note we prove that if a closed monotone symplectic manifold admits a Hamiltonian pseudo-rotation, which may be degenerate, then the quantum Steenrod square of the cohomology class Poincar\'{e} dual to the point must be deformed.…

We describe a formalism based on quantization of quadratic hamiltonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures about Gromov-Witten invariants of compact…

In this paper, using pseudo-holomorphic curve method, one proves the Weinstein conjecture in the product $P_1\times P_2$ of two strongly geometrically bounded symplectic manifolds under some conditions with $P_1$. In particular, if $N$ is a…

We consider 6-manifolds endowed with a symplectic half-flat SU(3)-structure and acted on by a transitive Lie group G of automorphisms. We review a classical result allowing to show the non-existence of compact non-flat examples. In the…

In this paper we introduce invariants of semi-free Hamiltonian actions of $S\sp 1$ on compact symplectic manifolds (which satisfy some technical conditions related to positivity) using the space of solutions to certain gauge theoretical…

We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for…

Taubes has recently defined Gromov invariants for symplectic four-manifolds and related them to the Seiberg-Witten invariants. Independently, Ruan and Tian defined symplectic invariants based on ideas of Witten. In this note, we show that…

We describe a natural isomorphism between the set of equivalence classes of pseudocycles and the integral homology groups of a smooth manifold. Our arguments generalize to settings well-suited for applications in enumerative algebraic…

In this note we describe the physics of equivalence of the Seiberg-Witten invariants of 4-manifolds and certain Gromov-Witten invariants defined by pseudo-holomorphic curves. We show that physics of the pseudo-holomorphic curves should be…

In this paper, we explicitly prove that statistical manifolds, related to exponential families and with flat structure connection have a Frobenius manifold structure. This latter object, at the interplay of beautiful interactions between…

A smooth counterexample to the Hamiltonian Seifert conjecture for six-dimensional symplectic manifolds is found. In particular, we construct a smooth proper function on the symplectic 2n-dimensional vector space, 2n > 4, such that one of…

In 2002 Polterovich has notably established that on closed aspherical symplectic manifolds, Hamiltonian diffeomorphisms of finite order, which we call Hamiltonian torsion, must in fact be trivial. In this paper we prove the first…

We show that, on a closed semipositive symplectic manifold with semisimple quantum homology, any Hamiltonian diffeomorphism possessing more contractible fixed points, counted homologically, than the total Betti number of the manifold, must…