Symplectic geometry

Symplectic geometry
Gargan · CC BY-SA 3.0 · source
Name	Symplectic geometry
Field	Mathematics
Subdisciplines	Differential geometry

Symplectic geometry is a branch of differential geometry concerned with the study of smooth manifolds equipped with a closed nondegenerate 2-form that defines a geometry appropriate for classical and modern physical systems. It provides the natural mathematical language for Hamiltonian mechanics, connects to complex and algebraic geometry, and drives developments in topology and dynamical systems.

Introduction

Symplectic geometry originated in the formalism of Isaac Newton-era mechanics and acquired modern form through contributions related to William Rowan Hamilton, Carl Gustav Jacobi, Joseph-Louis Lagrange, Sophus Lie, and later reformulations by Hermann Weyl, Élie Cartan, Marston Morse, and Andrei Kolmogorov. It bridges ideas from Pierre-Simon Laplace's celestial mechanics, the phase-space approach of Ludwig Boltzmann, and modern mathematical frameworks developed in contexts such as the International Congress of Mathematicians and institutions like the Institute for Advanced Study and the École Normale Supérieure.

Basic Definitions and Examples

A symplectic manifold is a smooth manifold M with a symplectic form ω: a closed (dω = 0), nondegenerate 2-form. Classic examples include the standard form on R^{2n} related to Joseph-Louis Lagrange's canonical coordinates, cotangent bundles T^*Q of configuration spaces Q used since William Rowan Hamilton's work, and complex projective space CP^n with the Fubini–Study form studied by researchers at institutions like Princeton University and University of Cambridge. Other examples arise in moduli spaces studied by teams associated with Clay Mathematics Institute problems and centers such as Mathematical Sciences Research Institute.

Symplectic Topology and Global Properties

Symplectic topology studies global invariants and rigidity phenomena not present in Riemannian geometry, including non-squeezing theorems first proved by Mikhail Gromov and foundational compactness results related to pseudoholomorphic curves introduced by Gromov and further developed by researchers working at Institut des Hautes Études Scientifiques and Stanford University. Global questions link to celebrated conjectures and theorems associated with names such as Ludwig Faddeev, Michael Atiyah, Raoul Bott, Simon Donaldson, and institutions like Harvard University and University of Chicago where deep relationships with gauge theory and four-manifold topology have been pursued.

Hamiltonian Dynamics and Moment Maps

Hamiltonian vector fields and flows on symplectic manifolds encode classical dynamics studied since William Rowan Hamilton and formalized in modern monographs associated with scholars at Oxford University and Massachusetts Institute of Technology. Moment maps tie group actions by Lie groups like Élie Cartan's circle actions and groups such as SU(2), SO(3), and U(n) to conserved quantities; foundational results include convexity theorems by Atiyah and Victor Guillemin and constructions used in geometric quantization programs related to work at University of California, Berkeley and Yale University.

Invariants and Techniques (Floer Homology, Gromov–Witten)

Floer homology, introduced by Andreas Floer and expanded by many collaborators at centers including Max Planck Institute for Mathematics and ETH Zurich, provides infinite-dimensional Morse-theoretic invariants for Hamiltonian diffeomorphisms and Lagrangian intersections. Gromov–Witten invariants, arising from pseudoholomorphic curve theory of Mikhail Gromov and developed further by researchers affiliated with Courant Institute of Mathematical Sciences and University of Bonn, connect to quantum cohomology and enumerative geometry explored in collaboration with scholars from Institut Henri Poincaré and Kolmogorov Institute-associated programs.

Applications and Connections

Symplectic methods are central in classical mechanics from William Rowan Hamilton to contemporary studies of celestial mechanics at observatories like Royal Observatory, Greenwich and modern physics areas including quantum field theory developments at CERN and string theory work at Caltech and Perimeter Institute for Theoretical Physics. Links to algebraic geometry involve mirror symmetry conjectures championed by research groups at University of Cambridge and Princeton University; interactions with low-dimensional topology appear in work by Simon Donaldson and Richard S. Hamilton across institutions like Columbia University and Duke University.

Historical Development and Key Results

The subject's modern form crystallized through contributions by William Rowan Hamilton in the 19th century, consolidation by Élie Cartan and Hermann Weyl in the early 20th century, breakthroughs by Mikhail Gromov in the 1980s, and deep structural advances by Andreas Floer, Simon Donaldson, Maxim Kontsevich, and Kenji Fukaya in the late 20th and early 21st centuries. Landmark results include Gromov's non-squeezing theorem, Arnold conjectures on fixed points studied by Vladimir Arnold and contributors at venues like Steklov Institute of Mathematics, and the development of homological mirror symmetry proposed by Maxim Kontsevich and elaborated by collaborative teams across the European Research Council-funded projects.

Category:Differential geometry