symplectic topology

symplectic topology
Name	Symplectic topology
Field	Differential topology, Geometry
Key concepts	Symplectic manifold, Lagrangian submanifold, J-holomorphic curve, Floer homology
Notable figures	Vladimir Arnold, Mikhail Gromov, Yakov Eliashberg, Andreas Floer, Dusa McDuff
Related areas	Hamiltonian mechanics, Mirror symmetry, String theory, Contact geometry

symplectic topology is a branch of differential topology and geometry that studies the global properties of symplectic manifolds. It emerged from the mathematical formalization of classical mechanics, particularly the work of William Rowan Hamilton and Carl Gustav Jacob Jacobi, and has since developed deep connections with many areas of mathematics and theoretical physics. The field investigates the rigidity and flexibility of symplectic structures, leading to profound results about the existence of periodic orbits and the classification of manifolds.

Introduction

The field originated from the study of Hamiltonian mechanics, where the phase space of a physical system naturally carries a symplectic structure. Foundational work by Henri Poincaré on the three-body problem hinted at the topological richness of these spaces. A major breakthrough came with Vladimir Arnold's conjecture on the number of fixed points of certain maps, which spurred the development of modern techniques. Key objects of study include Lagrangian submanifolds, which generalize the configuration space of a mechanical system, and the behavior of Hamiltonian diffeomorphisms.

Symplectic manifolds

A central object is the symplectic manifold, a smooth manifold equipped with a closed, non-degenerate 2-form. The prototypical example is Euclidean space $\backslash mathbb\{R\}^\{2n\}$ with its standard symplectic form, which serves as the local model by Darboux's theorem. Important classes include Kähler manifolds from complex geometry and cotangent bundles, which are fundamental in analytical mechanics. The work of Mikhail Gromov introduced revolutionary methods for studying these spaces, demonstrating their distinct nature from merely Riemannian manifolds.

Symplectic invariants

Unlike in Riemannian geometry, local invariants are trivial; the interest lies in global invariants that distinguish symplectic structures. Gromov-Witten invariants, defined via counts of J-holomorphic curves, are powerful tools originating from Gromov's nonsqueezing theorem. Floer homology, pioneered by Andreas Floer to prove the Arnold conjecture, provides an infinite-dimensional homology theory for Lagrangian submanifolds and symplectomorphisms. Other significant invariants include symplectic capacity, introduced by Helmut Hofer and Eduard Zehnder, and various forms of quantum cohomology.

Symplectic techniques and results

A hallmark technique is the analysis of J-holomorphic curves, a non-linear version of Cauchy-Riemann equations adapted to almost complex structures compatible with the symplectic form. This led to Gromov's celebrated compactness theorem and the founding of symplectic field theory by Yakov Eliashberg and collaborators. Major results include the Arnold conjecture, various rigidity theorems like the Eliashberg-Gromov C^0-rigidity, and the existence of pseudoholomorphic curves. The study of Lagrangian intersection theory, advanced by Fukaya categories, is another central theme.

Relations to other fields

The field has extensive and deep interactions with Hamiltonian dynamics, providing existence proofs for periodic orbits as in the Weinstein conjecture. Through mirror symmetry, a conjecture from string theory, it connects intimately with algebraic geometry and derived categories. It also shares borders with contact geometry, the odd-dimensional counterpart studied by Emmanuel Giroux, and with low-dimensional topology via symplectic 4-manifold theory and gauge theory, notably the work of Simon Donaldson and Clifford Taubes. Applications appear in topological quantum field theory and the study of integrable systems.

History and development

Early roots lie in the 19th-century work of Joseph-Louis Lagrange, Hamilton, and Jacobi on analytical mechanics. The modern formulation began with Hermann Weyl and was solidified by Jean-Marie Souriau. The field was transformed in the 1960s and 70s by Vladimir Arnold and his school, leading to the seminal conjectures. The 1980s brought the revolutionary methods of Mikhail Gromov, particularly the introduction of J-holomorphic curves. The subsequent decades saw the development of Floer homology by Andreas Floer and its expansion by Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, alongside significant contributions from Dusa McDuff, Leonid Polterovich, and many others, establishing it as a major area of modern mathematics.

Category:Differential topology Category:Symplectic geometry