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Concept in algebraic geometry
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli
Geometric_invariant_theory
Mathematical study of invariants under symmetries
his geometric invariant theory. In large measure due to the influence of Mumford, the subject of invariant theory is seen to encompass the theory of actions
Invariant_theory
Classification of computer problems
advanced tools in algebraic geometry and representation theory (i.e., geometric invariant theory) to prove lower bounds for problems. Currently the main
Geometric_complexity_theory
In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme X = Spec A {\displaystyle X=\operatorname
GIT_quotient
Algebro-geometric stability condition
Simon Donaldson. The definition was inspired by a comparison to geometric invariant theory (GIT) stability. In the special case of Fano varieties, K-stability
K-stability
American mathematician (born 1937)
relies on the more tractable theory of moduli of abelian varieties. In the introduction to his 1965 book Geometric Invariant Theory, Mumford described the construction
David_Mumford
rational, as proven by Katzarkov, Kontsevich, Pantev and Yu. Using geometric invariant theory (GIT), Radu Laza constructed a compactification of cubic fourfolds
Cubic_fourfold
computational geometry. Geometric function theory the study of geometric properties of analytic functions. Geometric invariant theory a method for constructing
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Concept in mathematical knot theory
invariant Arf invariant Hopf invariant Invariant theory Framed knot Chern–Simons theory Algebraic geometry Seifert surface Geometric invariant theory
Quantum_invariant
Branch of mathematics that studies abstract algebraic structures
in the form of his geometric invariant theory. The representation theory of semisimple Lie groups has its roots in invariant theory and the strong links
Representation_theory
where observations going back to the original development of geometric invariant theory show that it is necessary to restrict to a class of stable objects
K-stability_of_Fano_varieties
Asymptotically stable in the sense of geometric invariant theory
algebraic curve that is asymptotically stable in the sense of geometric invariant theory. This is equivalent to the condition that it is a complete connected
Stable_curve
(holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using
Stable_vector_bundle
Subgroup of the group of invertible n×n matrices
of geometric objects. Part of the theory of group actions is geometric invariant theory, which aims to construct a quotient variety X/G, describing the
Linear_algebraic_group
Study of space and shapes locally given by a convergent power series
Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem
Geometric_function_theory
Topics referred to by the same term
award Gastrointestinal tract Geographic information technology Geometric invariant theory Geoscientist In Training, a professional designation Git, Iran
Git_(disambiguation)
Invariant of a quadratic form over a field of characteristic 2
perfect field. The Arf invariant is particularly applied in geometric topology, where it is primarily used to define an invariant of (4k + 2)-dimensional
Arf_invariant
Function of a knot that takes the same value for equivalent knots
In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots
Knot_invariant
Field of higher mathematics
far back as Hodge theory. More recently, it refers largely to the use of nonlinear partial differential equations to study geometric and topological properties
Geometric_analysis
from geometric invariant theory, or inspired by it. A completely general theory of stability does not exist (although one attempt to form such a theory is
Stability (algebraic geometry)
Stability_(algebraic_geometry)
Branch of mathematics studying (smooth) functions of manifolds
not homeomorphic. This was the origin of simple homotopy theory. The use of the term geometric topology to describe these seems to have originated rather
Geometric_topology
Study of mathematical knots
the knot group and invariants from homology theory such as the Alexander polynomial. This would be the main approach to knot theory until a series of breakthroughs
Knot_theory
Branch of mathematics
understood as geometric objects since Klein's Erlangen programme. Geometric group theory studies group actions on objects that are regarded as geometric (significantly
Geometry
Physical theory with fields invariant under the action of local "gauge" Lie groups
gauge theory, the usual example being the Yang–Mills theory. Many powerful theories in physics are described by Lagrangians that are invariant under some
Gauge_theory
Study of vector bundles, principal bundles, and fibre bundles
of infinite-dimensional version of the Kempf–Ness theorem from geometric invariant theory, relating critical points of the norm squared of the moment map
Gauge_theory_(mathematics)
the universal invariant or u-invariant of a field describes the structure of quadratic forms over the field. The universal invariant u(F) of a field
U-invariant
Set with associative invertible operation
equations are well-behaved. Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory. Matrix groups consist
Group_(mathematics)
Hyperspace in algebraic geometry
space of lines on a non-singular cubic 3-fold is a Fano surface. Geometric invariant theory (GIT) gives a moduli space of smooth cubic threefolds, with one
Cubic_threefold
Concept in mathematics
In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular
Invariant_measure
Each semi-simple algebraic group is geometrically reductive
edition of his book Geometric Invariant Theory. Haboush's theorem can be used to generalize results of geometric invariant theory from characteristic
Haboush's_theorem
Invariant that plays a role in algebraic geometry and singularity theory
considered both a geometric invariant and an algebraic invariant. This is why it plays an important role in algebraic geometry and singularity theory. Consider
Milnor_number
isomorphism. (Here, k is the base field.) The notion appears in geometric invariant theory. (i), (ii) say that Y is an orbit space of X in topology. (iii)
Geometric_quotient
functional of the moment map in geometric invariant theory and symplectic reduction. The Mabuchi functional appears in the theory of K-stability as an analytical
Mabuchi_functional
of Galois theory. Along with a module of covariants, the ring of invariants is a central object of study in invariant theory. Geometrically, the rings
Fixed-point_subring
Concept in string theory
IIA string theory. They are named after Mikhail Gromov and Edward Witten. The rigorous mathematical definition of Gromov–Witten invariants is lengthy
Gromov–Witten_invariant
Topological quantum field theory
used to calculate knot invariants and three-manifold invariants such as the Jones polynomial. Particularly, Chern–Simons theory is specified by a choice
Chern–Simons_theory
Math concept
Bayer, David; Morrison, Ian (1988). "Standard bases and geometric invariant theory I. Initial ideals and state polytopes". Journal of Symbolic Computation
Polyhedral_complex
Type of category in mathematics
sometimes associate to a homotopy invariant in classical topology an inverse system of invariants in topos theory. The study of the pro-simplicial set
Topos
Topics referred to by the same term
functions Stable theory, concerned with the notion of stability in model theory Stability, a property of points in geometric invariant theory K-Stability,
Stability
Geometric space whose points represent algebro-geometric objects of some fixed kind
admit a solution; however, it is addressed by the groundbreaking geometric invariant theory (GIT), developed by David Mumford in 1965, which shows that under
Moduli_space
For the formulation due to Teleman, see C. Woodward's notes. Geometric invariant theory This means that the curvature of the connection on the line bundle
Quantization commutes with reduction
Quantization_commutes_with_reduction
Homotopy invariant of maps between n-spheres
mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres. In 1931 Heinz Hopf used
Hopf_invariant
Phase of a cycle
In classical and quantum mechanics, the geometric phase (also known as the Pancharatnam–Berry phase, Pancharatnam phase, or Berry phase) is a phase difference
Geometric_phase
Area in mathematics devoted to the study of finitely generated groups
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups by exploring the connections between algebraic properties
Geometric_group_theory
traditional gauge theory places the gauge potential as a 1-form on a principal bundle over spacetime. Higher gauge theories provide geometric and category-theoretic
Higher_gauge_theory
British-Lebanese mathematician (1929–2019)
theorems. Atiyah showed that the moment map was closely related to geometric invariant theory, and this idea was later developed much further by his student
Michael_Atiyah
Mathematical transformation in engineering
transform Vector control (motor) O'Rourke, Colm J. (December 2019). "A Geometric Interpretation of Reference Frames and Transformations: dq0, Clarke, and
Alpha–beta_transformation
Quantum field theory enjoying conformal symmetry
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional
Conformal_field_theory
One of six awards by the Wolf Foundation
work on algebraic surfaces; on geometric invariant theory; and for laying the foundations of the modern algebraic theory of moduli of curves and theta
Wolf_Prize_in_Mathematics
Chinese-American mathematician (born 1949)
Almgren–Pitts min-max theory of the area functional from geometric measure theory; Li and Yau's approach depended on their new "conformal invariant", which is a
Shing-Tung_Yau
Superconformal Yang–Mills theory
N = 4 supersymmetric Yang–Mills (SYM) theory is a relativistic conformally invariant Lagrangian gauge theory describing the interactions of fermions via
N = 4 supersymmetric Yang–Mills theory
N_=_4_supersymmetric_Yang–Mills_theory
Point minimizing sum of distances to given points
data set is not in general rotation invariant, nor is it independent of the choice of coordinates. The geometric median has a breakdown point of 0.5.
Geometric_median
Modular function in mathematics
In mathematics, the j-invariant or j function is a modular function of weight zero for the special linear group SL ( 2 , Z ) {\displaystyle \operatorname
J-invariant
Branch of mathematics
characteristic classes are a basic invariant, and surgery theory is a key theory. Low-dimensional topology is strongly geometric, as reflected in the uniformization
Topology
compact Lie group G {\displaystyle G} is closely related to a geometric invariant theory quotient by the complexification of G {\displaystyle G} . Hyperkähler
Kähler_quotient
{\displaystyle \pi } . One of the main motivations for the development of geometric invariant theory was the construction of a categorical quotient for varieties or
Categorical_quotient
English mathematician (born 1957)
1090/S0273-0979-1983-15090-5. MR 0682827. ——— (1984b). "Instantons and geometric invariant theory". Comm. Math. Phys. 93 (4): 453–460. Bibcode:1984CMaPh..93..453D
Simon_Donaldson
Gauge symmetry cannot be spontaneously broken
entirely in terms of gauge invariant quantities in what is known as the Fröhlich–Morchio–Strocchi mechanism. A field theory admits different types of symmetries
Elitzur's_theorem
Algebraic variety in a projective space
special cases, are also projective schemes in their own right. Geometric invariant theory offers another approach. The classical approaches include the
Projective_variety
Field theory involving topological effects in physics
topological invariants. While TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory, the
Topological quantum field theory
Topological_quantum_field_theory
American mathematician
professor emeritus in 2022. The topics of his research include geometric invariant theory and moduli of vector bundles over algebraic curves. with Spencer
David_Gieseker
On the intersection form of a smooth, closed 4-manifold with a spin structure
is equal to half the Casson invariant mod 2. The Casson invariant is viewed as the Z-valued lift of the Rokhlin invariant of integral homology 3-sphere
Rokhlin's_theorem
Branch of mathematics that studies the properties of groups
investigations further by creating the theory of permutation groups. The second historical source for groups stems from geometrical situations. In an attempt to
Group_theory
Aspect of theoretical physics
Quantum geometry in condensed matter physics refers to gauge-invariant geometric properties of quantum states as functions of external parameters—most
Quantum geometry (condensed matter)
Quantum_geometry_(condensed_matter)
Mathematical object studied in the field of algebraic geometry
possibly reducible algebraic variety; for example, one way is to use geometric invariant theory which ensures a set of isomorphism classes has a (reducible) quasi-projective
Algebraic_variety
Theory of interwoven space and time by Albert Einstein
Electrodynamics of Moving Bodies", the theory is presented as being based on just two postulates: The laws of physics are invariant (identical) in all inertial frames
Special_relativity
Concept in abstract algebra
an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology
Cohomological_dimension
Concept in differential topology
and thus analogous to the other invariants from L-theory: the signature, a 4 k {\displaystyle 4k} -dimensional invariant (either symmetric or quadratic
Kervaire_invariant
1957 book by Emil Artin
Geometric Algebra is a book written by Emil Artin and published by Interscience Publishers, New York, in 1957. It was republished in 1988 in the Wiley
Geometric_Algebra_(book)
Vector bundles theorem
work on geometric invariant theory, with a view to constructing moduli spaces of various geometric objects. Mumford applied this new theory vector bundles
Kobayashi–Hitchin correspondence
Kobayashi–Hitchin_correspondence
Regularity in sensory qualia or abstract ideas
well as a connection with mathematics. A geometric pattern is a type of pattern formed of repeating geometric shapes and typically repeated like a wallpaper
Pattern
geometric terms. It plays an important role in the theory of Fourier integral operators, geometric quantization, Hamiltonian systems, spectral theory
Maslov_index
Branch of mathematics
Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatorial
Algebraic_graph_theory
equation J-invariant Algebraic function Algebraic form Addition theorem Invariant theory Symbolic method of invariant theory Geometric invariant theory Toric
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Application of Lagrangian mechanics to field theories
contact geometry. The field theories of physics can be developed in terms of gauge invariant fiber bundles. In Lagrangian field theory, the Lagrangian as a function
Lagrangian_(field_theory)
Moduli space in the category of schemes
of Michael Artin). Work of Grothendieck and David Mumford (see geometric invariant theory) opened up this area in the early 1960s. The more algebraic and
Moduli_scheme
Branch of mathematics
bracket between left-invariant vector fields. Beside the structure theory there is also the wide field of representation theory. Geometric analysis is a mathematical
Differential_geometry
Classical field theory describing gravitation
Poincaré gauge theory is developed and studied by many authors. The core idea of a gauge theory is that a theory's equations should remain invariant under local
Poincaré_gauge_theory
Concept in mathematics
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times
Integral_geometry
Secondary characteristic classes of 3-manifolds
The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants,"
Chern–Simons_form
Arithmetic mean is greater than or equal to geometric mean
matrix generalizations of the arithmetic geometric mean inequality apply on the level of unitarily invariant norms, since, even if the matrices A {\displaystyle
AM–GM_inequality
Area of discrete mathematics
Algebraic graph theory also studies the algebraic invariants, chromatic polynomial, Tutte polynomial of a graph, and knot invariant. A graph invariant is a property
Graph_theory
Theory in physics
specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau
Donaldson–Thomas_theory
Property of measure-preserving dynamical systems
invariant measure. This provides one of the classical families of examples in ergodic theory and is closely related to Anosov flows. Other geometric examples
Ergodicity
Property in algebraic geometry
surface, or more simply its Geometric genus is a birational invariant. A more complicated example is given by Hodge theory: in the case of an algebraic
Birational_invariant
Topics referred to by the same term
said orbits The closure of the orbit of a reductive group, in geometric invariant theory A stable electron configuration This disambiguation page lists
Orbital_Stability
Mathematics concept
group over a local field with a ring of invariants of the Weyl group. The geometric Satake equivalence is a geometric version of the Satake isomorphism, proved
Satake_isomorphism
Branch of algebra
the commutative development by building the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of functions
Ring_theory
Disproved conjecture in multilinear algebra on the rank of symmetric tensors
G-stable rank, a notion introduced by Harm Derksen and defined via geometric invariant theory. A structural obstacle to settling the conjecture for small tensors
Comon's_conjecture
is closely related to topological quantum field theory in dimension 2+1. Geometric invariant theory Horowitz, R.D. (1972). "Characters of Free Groups
Character_variety
Mod 2 invariant of (4k+1)-dimensional manifold
In geometric topology, the de Rham invariant is a mod 2 invariant of a (4k+1)-dimensional manifold, that is, an element of Z / 2 {\displaystyle \mathbf
De_Rham_invariant
Graph defined from a mathematical group
generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs make them particularly
Cayley_graph
ISBN 978-1-4008-3720-5. Mumford, David; Fogarty, J.; Kirwan, F. (1994). Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results
Level structure (algebraic geometry)
Level_structure_(algebraic_geometry)
Hypothetical physical concept
continuous geometric nature of classical spacetime in general relativity. Reconciling the background-independent, diffeomorphism-invariant formulation
Theory_of_everything
terms in invariant theory. For descriptions of particular invariant rings, see invariants of a binary form, symmetric polynomials. For geometric terms used
Glossary_of_invariant_theory
Subspace preserved by a linear mapping
proper non-trivial invariant subspace. Determining whether a given subspace W is invariant under T is ostensibly a problem of geometric nature. Matrix representation
Invariant_subspace
Number of "holes" of a surface
projective algebraic scheme X {\displaystyle X} : the arithmetic genus and the geometric genus. When X {\displaystyle X} is an algebraic curve with field of definition
Genus_(mathematics)
Algebraic structure designed for geometry
geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra
Geometric_algebra
View of mathematicians to consolidate two or more theories into a more generalized one
under which the geometric objects were invariant. This unification of geometry goes by the name of the Erlangen programme. The general theory of angle can
Unifying theories in mathematics
Unifying_theories_in_mathematics
recreational number theory topics (see number theory, recreational mathematics). Listing here is not pejorative: many famous topics in number theory have origins
List of recreational number theory topics
List_of_recreational_number_theory_topics
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GEOMETRIC INVARIANT-THEORY
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