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Topics referred to by the same term
algebra Symplectic integrator Symplectic manifold Symplectic matrix Symplectic representation Symplectic vector space, a vector space with a symplectic bilinear
Symplectic
Branch of differential geometry and differential topology
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds
Symplectic_geometry
Mathematical concept
mathematics, particularly in representation theory, a symplectic resolution is a morphism that combines symplectic geometry and resolution of singularities. Let
Symplectic_resolution
Mathematical group
In mathematics, the symplectic group is the group of linear transformations that preserve the geometric structure of phase space, the space of position
Symplectic_group
Type of manifold in differential geometry
\omega } , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally
Symplectic_manifold
Topics referred to by the same term
Symplectic form refers to a type of bilinear form or a type of 2-form. See: Symplectic vector space, a vector space with a symplectic bilinear form Symplectic
Symplectic_form
Mathematical concept
In mathematics, a symplectic vector space is a vector space V {\displaystyle V} over a field F {\displaystyle F} (for example the real numbers R {\displaystyle
Symplectic_vector_space
Mathematical concept
In mathematics, a symplectic matrix is a 2 n × 2 n {\displaystyle 2n\times 2n} matrix M {\displaystyle M} with real entries that satisfies the condition
Symplectic_matrix
In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations
Symplectic_category
Mathematical structure in differential geometry
Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics
Poisson_manifold
Numerical integration scheme for Hamiltonian systems
In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric
Symplectic_integrator
algebra, a standard symplectic basis is a basis e i , f i {\displaystyle {\mathbf {e} }_{i},{\mathbf {f} }_{i}} of a symplectic vector space, which is
Symplectic_basis
In mathematics, specifically finite group theory, a p-group of symplectic type is a p-group such that all characteristic abelian subgroups are cyclic.
Group_of_symplectic_type
Branch of mathematics
example, in Riemannian geometry distances and angles are specified, in symplectic geometry volumes may be computed, in conformal geometry only angles are
Differential_geometry
properties and concepts in symplectic geometry in mathematics. The terms listed here cover the occurrences of symplectic geometry both in topology as
Glossary of symplectic geometry
Glossary_of_symplectic_geometry
Tool in symplectic geometry
In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action
Momentum_map
mathematics, a symplectic vector field is one whose flow preserves a symplectic form. That is, if ( M , ω ) {\displaystyle (M,\omega )} is a symplectic manifold
Symplectic_vector_field
Topics referred to by the same term
A symplectic space may refer to: Symplectic manifold Symplectic vector space This disambiguation page lists mathematics articles associated with the same
Symplectic_space
Canonical subbundle
In symplectic geometry, the symplectic frame bundle of a given symplectic manifold ( M , ω ) {\displaystyle (M,\omega )\,} is the canonical principal S
Symplectic_frame_bundle
theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (V, ω) which preserves the symplectic form
Symplectic_representation
Symplectic topology tool
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is an invariant that arises as an
Floer_homology
Cobordism W between X and the empty set
by a symplectic structure. Let ξ denote the kernel of the contact form α. A weak symplectic filling of a contact manifold (X,ξ) is a symplectic manifold
Symplectic_filling
In differential geometry, an almost symplectic structure on a differentiable manifold M {\displaystyle M} is a non-degenerate two-form ω {\displaystyle
Almost_symplectic_manifold
In mathematics, specifically in symplectic geometry, the symplectic sum is a geometric modification on symplectic manifolds, which glues two given manifolds
Symplectic_sum
In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose
Symplectic_cut
Modification of the Euler method for solving Hamilton's equations
In mathematics, the semi-implicit Euler method, also called symplectic Euler, semi-explicit Euler, Euler–Cromer, and Newton–Størmer–Verlet (NSV), is a
Semi-implicit_Euler_method
Isomorphism of symplectic manifolds
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism
Symplectomorphism
2 n {\displaystyle 2n} -dimensional symplectic manifold ( M , ω ) , {\displaystyle (M,\omega ),\,} the symplectic spinor bundle is the Hilbert space bundle
Symplectic_spinor_bundle
Area of differential geometry and topology
important generalisation is the theory of symplectic Dirac operators in symplectic spin geometry and symplectic topology, which have become important fields
Spin_geometry
Austrian mathematician and mathematical physicist
first to prove that Mikhail Gromov's symplectic non-squeezing theorem (also called the Principle of "the Symplectic Camel") allowed the derivation of a
Maurice_A._de_Gosson
symplectic geometry. It was first proven in 1985 by Mikhail Gromov. The theorem states that one cannot embed a ball into a cylinder via a symplectic map
Non-squeezing_theorem
American mathematician
He works at the Massachusetts Institute of Technology in the field of symplectic geometry, and he has also made contributions to the fields of microlocal
Victor_Guillemin
British mathematician (born 1973)
Ivan Smith (born 1973) FRS is a British mathematician who deals with symplectic manifolds and their interaction with algebraic geometry, low-dimensional
Ivan_Smith_(mathematician)
French mathematician (b. 1977)
from Paris-Sud University with a thesis on Seiberg-Witten invariants of symplectic manifolds. In 1999, he received his doctorate from the École polytechnique
Denis_Auroux
Russian-American mathematician
Stanford University. His research interests are differential topology, symplectic topology, and contact topology. He was awarded many prizes for his work
Yakov_Eliashberg
Formulation of classical mechanics using momenta
Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical
Hamiltonian_mechanics
Japanese mathematician
薫, Ono Kaoru, born 1962) is a Japanese mathematician, specializing in symplectic geometry. He is a professor at the Research Institute for Mathematical
Kaoru_Ono
Mathematical matrix
forms a Lie algebra (the symplectic Lie algebra); its associated Lie group is the symplectic group, whose elements are the symplectic matrices. Suppose that
Hamiltonian_matrix
Random matrix with gaussian entries
three main examples are the Gaussian orthogonal (GOE), unitary (GUE), and symplectic (GSE) ensembles. These are classified by the Dyson index β, which takes
Gaussian_ensemble
Vector field defined for any energy function
In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named
Hamiltonian_vector_field
Portuguese mathematician (born 1968)
Cannas da Silva (born 1968) is a Portuguese mathematician specializing in symplectic geometry and geometric topology. She works in Switzerland as a professor
Ana_Cannas_da_Silva
Bony structure that forms the head in vertebrates
the maxilla itself located further back, and an additional bone, the symplectic, linking the jaw to the rest of the cranium. Although the skulls of fossil
Skull
English mathematician
CorrFRSE (born 18 October 1945) is an English mathematician who works on symplectic geometry. She was the first recipient of the Ruth Lyttle Satter Prize
Dusa_McDuff
Matrix-valued random variable
with IID samples from the standard normal distribution. The Gaussian symplectic ensemble GSE ( n ) {\displaystyle {\text{GSE}}(n)} is described by the
Random_matrix
Branch of geometry
odd-dimensional counterpart of symplectic geometry, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by
Contact_geometry
Israeli and Canadian mathematician
mathematician who has been described as "one of Canada's leading experts in symplectic geometry". She works as a professor at the University of Toronto Mississauga
Yael_Karshon
Group in mathematical representation theory
of classical mechanics act in quantum mechanics. More precisely, the symplectic group consists of the linear changes of position and momentum that preserve
Metaplectic_group
Key result in Hamiltonian mechanics and statistical mechanics
and momentum coordinates is available in the mathematical setting of symplectic geometry. Liouville's theorem ignores the possibility of chemical reactions
Liouville's theorem (Hamiltonian)
Liouville's_theorem_(Hamiltonian)
American mathematician (born 1943)
to many new developments in symplectic and contact geometry. In 1981 he formulated a general principle, called symplectic creed, stating that "everything
Alan_Weinstein
Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis. It is associated most classically
Maslov_index
Vector bundle of cotangent spaces at every point in a manifold
algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent
Cotangent_bundle
the symplectic group with abelian radical, given by the matrices of the symplectic group whose lower left quadrant is 0 (for the standard symplectic form)
Siegel_parabolic_subgroup
Differential algebra
{\displaystyle V} (of dimension 2 n {\displaystyle 2n} ) equipped with a symplectic form ω {\displaystyle \omega } . Define the Weyl algebra W ( V ) {\displaystyle
Weyl_algebra
Russian mathematician (1937–2010)
systems, algebra, catastrophe theory, topology, real algebraic geometry, symplectic geometry, differential equations, classical mechanics, differential-geometric
Vladimir_Arnold
circular unitary ensemble (CUE) on unitary matrices, and the circular symplectic ensemble (CSE) on self dual unitary quaternionic matrices. The distribution
Circular_ensemble
Canonical differential form
derivative of this form defines a symplectic form, giving T ∗ Q {\displaystyle T^{*}Q} the structure of a symplectic manifold. The tautological one-form
Tautological_one-form
intersects the interior of the Weyl group invariant cone. For the real symplectic group, the maximal and minimal cone coincide, so there is only one invariant
Invariant_convex_cone
Associative algebra together with a Lie bracket that satisfies Leibniz's law
Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson–Lie groups are a special case. The algebra is
Poisson_algebra
Russian-French mathematician
for the existence of exact Lagrangian immersions and similar objects in symplectic and contact geometry. His well-known book Partial Differential Relations
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
Closed degenerate differential 2-form of constant rank
differentiable manifolds. It is a generalization of symplectic form. Given a differentiable manifold, a symplectic form over it is differential 2-form that is
Presymplectic_form
Supergeometric generalization of a manifold
{\displaystyle \xi _{i}} odd coordinates. (An odd symplectic form should not be confused with a Grassmann-even symplectic form on a supermanifold. In contrast, the
Supermanifold
Mathematician
(Arabic: محمد أبوزيد; born 1981) is a Moroccan mathematician working in symplectic topology. He is a professor at Stanford University. He obtained his PhD
Mohammed_Abouzaid
Trick relating differential forms
when two volume forms are equivalent, but its main applications are in symplectic geometry. It is the standard argument for the modern proof of Darboux's
Moser's_trick
Japanese mathematician
Kenji, born in 1959) is a Japanese mathematician known for his work in symplectic geometry and Riemannian geometry. His many fundamental contributions to
Kenji_Fukaya
American mathematician (1936–2024)
an American mathematician known for his work in geometry, particularly symplectic geometry and Lie theory. He also wrote some well-known textbooks.[which
Shlomo_Sternberg
German mathematician (1953–2025)
2018. Salamon's field of research is symplectic topology and related fields such as symplectic geometry. Symplectic topology is a relatively new field of
Dietmar_Salamon
Foundational result in symplectic geometry
symplectic manifold can be made to look locally like the linear symplectic space C n {\displaystyle \mathbb {C} ^{n}} with its canonical symplectic form
Darboux's_theorem
More unsolved problems in mathematics In mathematics, more specifically symplectic topology, the nearby Lagrangian conjecture, is an open mathematical problem
Nearby_Lagrangian_conjecture
Mathematician
Ailsa Macgregor Keating is a mathematician specialising in symplectic geometry and homological mirror symmetry. She is a professor in the Department of
Ailsa_Keating
Concept in string theory
In mathematics, specifically in symplectic geometry and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations
Gromov–Witten_invariant
Differential form
generally, the n {\displaystyle n} th exterior power of the symplectic form on a symplectic manifold is a volume form. Many classes of manifolds have canonical
Volume_form
manifold is a symplectic manifold with a compatible torsion-free connection, that is, a triple (M, ω, ∇), where (M, ω) is a symplectic manifold (that
Fedosov_manifold
Operation in Hamiltonian mechanics
of two symplectic vector fields is a Hamiltonian vector field and hence is also symplectic. In the language of abstract algebra, the symplectic vector
Poisson_bracket
bundle L satisfying the quantization condition on the symplectic quotient of a compact symplectic manifold is the space of invariant sections[vague] of
Quantization commutes with reduction
Quantization_commutes_with_reduction
double cover of the symplectic group) there is a corresponding quadratic Fourier transform. Gosson, Maurice A. de (2011). Symplectic Methods in Harmonic
Quadratic_Fourier_transform
American mathematician and professor (born 1976)
"Lenny" Ng (born 1976) is an American mathematician, working primarily on symplectic geometry. Ng is a professor of mathematics at Duke University. Lenhard
Lenhard_Ng
Representation theory of the symplectic group
oscillator representation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David Shale, and André Weil
Oscillator_representation
American mathematician
Information Technology on the St. George campus. Murphy works in the area of symplectic topology, contact geometry and geometric topology. Murphy graduated from
Emmy_Murphy
Theorem stating that smooth algebraic curve has minimum genus its homology class
as the symplectic Thom conjecture (which is now a theorem, as proved for example by Peter Ozsváth and Szabó in 2000). It states that a symplectic surface
Thom_conjecture
In mathematics, in the area of symplectic topology, relative contact homology is an invariant of spaces together with a chosen subspace. Namely, it is
Relative_contact_homology
Comprehensive physical model
representation of O(16). Symplectic gauge groups could also be considered. For example, Sp(8) (which is called Sp(4) in the article symplectic group) has a representation
Grand_Unified_Theory
South Korean mathematician
and Physics located on that campus. His fields of study have been on symplectic topology, Floer homology, Hamiltonian mechanics, and mirror symmetry He
Yong-Geun_Oh
Type of group in mathematics
quaternionic general linear, special linear, orthogonal, unitary, and symplectic groups, together with their indefinite analogues. In the language of linear
Classical_group
Describes a periodicity in the homotopy groups of classical groups
KSp-theory, associated to the real orthogonal group and the quaternionic symplectic group, respectively. The J-homomorphism is a homomorphism from the homotopy
Bott_periodicity_theorem
Spanish mathematician
2025) was a Spanish mathematician specializing in differential geometry, symplectic geometry, and G2-structures. She was professor of geometry and topology
Marisa_Fernández
German mathematician
May 1991) was a German mathematician who made seminal contributions to symplectic topology, and mathematical physics, in particular the invention of Floer
Andreas_Floer
symplectic manifolds, the meaning of "aspherical" is a little bit different. Specifically, we say that a symplectic manifold (M,ω) is symplectically aspherical
Aspherical_space
Topics referred to by the same term
between layers of a cake Satiety after eating food Dental restoration Symplectic filling, a kind of cobordism in mathematics Part of the leather crusting
Filling
Type of vector space in mathematics
Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V. Its dimension is 1/2n(n + 1) (where the dimension of
Lagrangian_Grassmannian
Property of a differential manifold that includes complex structures
differential manifold that includes as special cases a complex structure and a symplectic structure. Generalized complex structures were introduced by Nigel Hitchin
Generalized_complex_structure
British mathematician (born 1959)
University of Oxford. Her fields of specialisation are algebraic and symplectic geometry. Kirwan was educated at Oxford High School, and studied maths
Frances_Kirwan
Mathematical space with two coordinates
signed areas can be meaningfully compared, as they can in a more general symplectic surface. The projective plane does away with both distance and parallelism
Two-dimensional_space
Mathematical conjecture
symplectic geometry, a branch of differential geometry. Let ( M , ω ) {\displaystyle (M,\omega )} be a closed (compact without boundary) symplectic manifold
Arnold_conjecture
German-American mathematician
is a German-American mathematician, one of the founders of the area of symplectic topology. He is a member of the National Academy of Sciences, and the
Helmut_Hofer
Method in contact geometry
{\displaystyle (P,\alpha )} as principal S 1 {\displaystyle S^{1}} -bundles over symplectic manifolds ( M , ω ) {\displaystyle (M,\omega )} where ω {\displaystyle
Boothby-Wang_construction
Rwandan-born American mathematician (born 1947)
is a Rwandan-born American mathematician whose research fields include symplectic topology and contact geometry. He is currently a professor of mathematics
Augustin_Banyaga
Mathematicians in 2022. with Yakov Eliashberg, From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds (Colloquium Publications, Band 59)
Kai_Cieliebak
Concept in mathematics
{\displaystyle X} is embedded in to a symplectic manifold ( M , ω ) {\displaystyle (M,\omega )} , such that the pullback of the symplectic form has constant rank on
Normal_bundle
Smooth manifold
complex manifolds. Almost complex structures have important applications in symplectic geometry. The concept is due to Charles Ehresmann and Heinz Hopf in the
Almost_complex_manifold
Canadian mathematician
(born 17 September 1955) is a Canadian mathematician specializing in symplectic geometry and topology. Lalonde received a bachelor's degree in physics
François_Lalonde
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