# Category: Symplectic topology

Gromov's compactness theorem (topology)
In the mathematical field of symplectic topology, Gromov's compactness theorem states that a sequence of pseudoholomorphic curves in an almost complex manifold with a uniform energy bound must have a
Quantum cohomology
In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two v
Gromov–Witten invariant
In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting presc
Quasi-Frobenius Lie algebra
In mathematics, a quasi-Frobenius Lie algebra over a field is a Lie algebra equipped with a nondegenerate skew-symmetric bilinear form , which is a Lie algebra 2- of with values in . In other words, f
Stable map
In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given sympl
Lee Hwa Chung theorem
The Lee Hwa Chung theorem is a theorem in symplectic topology. The statement is as follows. Let M be a symplectic manifold with symplectic form ω. Let be a differential k-form on M which is invariant
Relative contact homology
In mathematics, in the area of symplectic topology, relative contact homology is an invariant of spaces together with a chosen subspace. Namely, it is associated to a contact manifold and one of its L
Frobenius manifold
In the mathematical field of differential geometry, a Frobenius manifold, introduced by Dubrovin, is a flat Riemannian manifold with a certain compatible multiplicative structure on the tangent space.
Taubes's Gromov invariant
In mathematics, the Gromov invariant of Clifford Taubes counts embedded (possibly disconnected) pseudoholomorphic curves in a symplectic 4-manifold, where the curves are holomorphic with respect to an
Symplectic sum
In mathematics, specifically in symplectic geometry, the symplectic sum is a geometric modification on symplectic manifolds, which glues two given manifolds into a single new one. It is a symplectic v
Pseudoholomorphic curve
In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or J-holomorphic curve) is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauc
Trivial cylinder
In geometry and topology, trivial cylinders are certain pseudoholomorphic curves appearing in certain cylindrical manifolds. In Floer homology and its variants, chain complexes or differential graded
Symplectic cut
In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose a given manifold into two pieces. There is an in
Floer homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-di
Symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space
Arnold–Givental conjecture
The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, is a statement on Lagrangian submanifolds. It gives a lower bound in terms of the Betti numbers of a Lagrangian subm
Poincaré–Birkhoff theorem
In symplectic topology and dynamical systems, Poincaré–Birkhoff theorem (also known as Poincaré–Birkhoff fixed point theorem and Poincaré's last geometric theorem) states that every area-preserving, o
Equivariant cohomology
In mathematics, equivariant cohomology (or Borel cohomology) is a cohomology theory from algebraic topology which applies to topological spaces with a group action. It can be viewed as a common genera