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Geometric space whose points represent algebro-geometric objects of some fixed kind
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent
Moduli_space
Geometric space
In algebraic geometry, a moduli space of curves is a space whose points correspond to isomorphism classes of algebraic curves. The term "modulus" was
Moduli_of_algebraic_curves
Partial differential equations whose solutions are instantons
anti-self-dual finite-action solutions are called instantons. The Yang–Mills moduli space was used by Simon Donaldson to prove Donaldson's theorem. In their foundational
Yang–Mills_equations
Moduli space of the Yang–Mills equations
Yang–Mills moduli space (short YM moduli space, also instanton moduli space) is the moduli space of the Yang–Mills equations, hence the space of its solutions
Yang–Mills_moduli_space
Iranian mathematician (1977–2017)
her work in "the dynamics and geometry of Riemann surfaces and their moduli spaces". Mirzakhani was considered a leading force in the fields of hyperbolic
Maryam_Mirzakhani
Space of vacuum states
more specifically, moduli space is borrowed from algebraic geometry), where it is used synonymously with "parameter". The word moduli (Moduln in German)
Moduli_(physics)
Moduli space of the Seiberg–Witten equations
Seiberg–Witten moduli space (short SW moduli space, also monopole moduli space) is the moduli space of the Seiberg–Witten equations, hence the space of its solutions
Seiberg–Witten_moduli_space
Type of smooth complex surface of kodaira dimension 0
−2.) The moduli space of quasi-polarized K3 surfaces of genus g is still irreducible of dimension 19 (containing the previous moduli space as an open
K3_surface
Correspondsnce between Higgs bundles and fundamental group representations
moduli spaces will be not just topological spaces, but have some additional structure. For example, the Dolbeault moduli space and Betti moduli space
Nonabelian Hodge correspondence
Nonabelian_Hodge_correspondence
Theory in supersymmetric gauge theory
of which the kinetic part coincides with the Kähler potential of the moduli space of vacua. Before taking the low-energy effective action, the theory is
Seiberg–Witten_theory
American mathematician (born 1937)
varieties of moduli, that is, varieties whose points parametrize isomorphism classes of some type of geometric object. The moduli space of curves of a
David_Mumford
Mathematical set with some added structure
Quadratic space Quotient space (disambiguation) Riemann's Moduli space Sample space Sequence space Sierpiński space Sobolev space Standard space State space Stone
Space_(mathematics)
{Z} )} , there is a moduli space of principally polarised abelian varieties given as a stacky quotient of Siegel upper half-space by the symplectic group
Moduli_of_abelian_varieties
Simplicial set constructed from the objects and morphisms of a small category
often used to construct topological versions of moduli spaces. If X is an object of C, its moduli space should somehow encode all objects isomorphic to
Nerve_(category_theory)
monopole moduli space is a space parametrizing monopoles (solutions of the Bogomolny equations). Atiyah and Hitchin (1988) studied the moduli space for 2
Monopole_moduli_space
Result in algebraic geometry
Grothendieck–Riemann–Roch can be used in proving that a coarse moduli space M {\displaystyle M} , such as the moduli space of pointed algebraic curves M g , n {\displaystyle
Grothendieck–Riemann–Roch theorem
Grothendieck–Riemann–Roch_theorem
Generalisation of a sheaf; a fibered category that admits effective descent
constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist. Descent theory is concerned with generalisations
Stack_(mathematics)
Symbol representing a mathematical object
parabola. That is, they specify coordinates on the 'space of parabolas': this is known as a moduli space of parabolas. Lambda calculus Observable variable
Variable_(mathematics)
Polygon with an infinite number of sides
corresponds to an isometry of the images of the mapping. Generally, the moduli space of a faithful realization of an abstract polytope is a convex cone of
Apeirogon
British-Lebanese mathematician (1929–2019)
topology of the moduli space of SU(2) instantons over a 4-sphere. They showed that the natural map from this moduli space to the space of all connections
Michael_Atiyah
Space of complex matrices with positive definite imaginary part
{M}}_{g}\to {\mathcal {A}}_{g},} from the moduli space of smooth curves of genus g {\displaystyle g} to the moduli space of principally polarised abelian varieties
Siegel_upper_half-space
Type of object in algebraic geometry
1969 paper on the irreducibility of the moduli space of algebraic curves, where they showed that the moduli stack of stable curves of fixed arithmetic
Deligne–Mumford_stack
Partial differential equation with nonlinear terms
studying the tangent space of a point of the moduli space of all solutions. Ideally one would like to describe the (moduli) space of all solutions explicitly
Nonlinear partial differential equation
Nonlinear_partial_differential_equation
Algebraic stack in mathematics
{\displaystyle {\mathcal {M}}_{1,1}} to the affine line, the coarse moduli space of elliptic curves, given by the j-invariant of an elliptic curve. It
Moduli stack of elliptic curves
Moduli_stack_of_elliptic_curves
Discrete dynamical system on polygons in the projective plane and on their moduli space
This is a projectively equivariant procedure, hence it descends to the moduli space of polygons and defines another dynamical system (which is also referred
Pentagram_map
4-manifold invariants
work with than Donaldson invariants; for example, the Seiberg–Witten moduli space of solutions of the Seiberg–Witten equations up to gauge tends to be
Seiberg–Witten_invariants
Branch of mathematics
deformation theory of the first order should equate the Zariski tangent space with a moduli space. The phenomena turn out to be rather subtle, though, in the general
Deformation_(mathematics)
Three-holed sphere
pants are used to construct the Fenchel-Nielsen coordinates on Teichmüller space, and in topological quantum field theory where they are the simplest non-trivial
Pair_of_pants_(mathematics)
Parametrizes complex structures on a surface
this way Teichmüller space can be viewed as the universal covering orbifold of the Riemann moduli space. The Teichmüller space has a canonical complex
Teichmüller_space
Type of vector bundle
(that is, the fiber is a 2-dimensional vector space). The rank 2 vector bundle arises as the solution space to Hitchin's equations for a principal SU(2)-bundle
Higgs_bundle
Algebraic variety that is a moduli space for principally polarized abelian varieties
In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed
Siegel_modular_variety
Doughnut-shaped surface of revolution
"moduli space" of the torus to contain one point for each conformal equivalence class, with the appropriate topology. It turns out that this moduli space
Torus
Riemannian manifold with SU(n) holonomy
04879. doi:10.1112/blms/bdp106. S2CID 1070427. Reid, Miles (1987). "The Moduli space of 3-folds with K = 0 may nevertheless be irreducible". Mathematische
Calabi–Yau_manifold
Study of vector bundles, principal bundles, and fibre bundles
manifold in four dimensions. In this work the moduli space of self-dual connections (instantons) on Euclidean space was studied, and shown to be of dimension
Gauge_theory_(mathematics)
One-dimensional complex manifold
the torus). To obtain the analytic moduli space (forgetting the marking) one takes the quotient of Teichmüller space by the mapping class group. In this
Riemann_surface
On when a definite intersection form of a smooth 4-manifold is diagonalizable
his Fields Medal in 1986. Donaldson's proof utilizes the Yang–Mills moduli space M P {\displaystyle {\mathcal {M}}_{P}} of solutions to the anti-self-duality
Donaldson's_theorem
American theoretical physicist (born 1963)
(1995) 109–120. P.S. Aspinwall, B.R. Greene, D.R. Morrison, "Calabi–Yau Moduli Space, Mirror Manifolds and Spacetime Topology Change in String Theory". Nuclear
Brian_Greene
Topics referred to by the same term
arithmetic Similarly, the modulus of a Dirichlet character Moduli space, in mathematics a geometric space whose points represent algebro-geometric objects Conformal
Modulus
Theory in physics
theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant
Donaldson–Thomas_theory
Study of complex manifolds and several complex variables
varieties through the minimal model program and the construction of moduli spaces sets the field apart from differential geometry, where the classification
Complex_geometry
Generalization of a scheme
several natural constructions that are used in the construction of moduli spaces but are not always possible in the smaller category of schemes, such
Algebraic_space
where it is the correct stability condition to allow the formation of moduli spaces, and where it precisely characterises the existence of Kähler–Einstein
K-stability_of_Fano_varieties
most sources work with the right action. The quotient space Mn = Xn/Out(Fn) is the moduli space which consists of isometry types of finite connected graphs
Outer_space_(mathematics)
Moduli space in the Grothendieck category of schemes
In algebraic geometry, a moduli scheme is a moduli space that exists in the category of schemes developed by French mathematician Alexander Grothendieck
Moduli_scheme
Generalization of algebraic variety
a given type can itself be viewed as a variety or scheme, known as a moduli space. For some of the detailed definitions in the theory of schemes, see the
Scheme_(mathematics)
Moduli spaces of ramified covers
algebraic geometry, Hurwitz spaces are moduli spaces of ramified covers of the projective line, and they are related to the moduli of curves. Their rational
Hurwitz_space
Russian and French mathematician (born 1964)
Gauss linking number. In topological field theory, he introduced the moduli space of stable maps, which may be considered a mathematically rigorous formulation
Maxim_Kontsevich
Algebraic curve
3, the number of moduli of a curve of genus g, unless g is 2. Much more is known about the hyperelliptic locus in the moduli space of curves or abelian
Hyperelliptic_curve
26-dimensional string theory
with an integration over the space of all possible complex structures modulo diffeomorphisms, which is simply the moduli space of the given topological surface
Bosonic_string_theory
invariants are parametrised over a well-behaved moduli space (isomorphic to the Picard variety). This space is, in particular, a proper and separated scheme
Stable_vector_bundle
Construct in mathematics
. It has a coarse moduli space M r , d s {\displaystyle M_{r,d}^{s}} , which is a quasiprojective variety. These two moduli problems parametrize the same
Gerbe
Study in mathematical gauge theory
Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons. It was started by Simon Donaldson (1983)
Donaldson_theory
the moduli stack of principal bundles (PDF) (Thesis). University of California, Berkeley. Edidin, Dan. "Notes on the construction of the moduli space of
Quotient_stack
Professor of mathematics (born 1969)
geometry. His particular interests concern moduli spaces, enumerative invariants associated to moduli spaces, such as Gromov–Witten invariants and Donaldson–Thomas
Rahul_Pandharipande
Solitons in Euclidean spacetime
Donaldson, for which he was later awarded the Fields Medal, used the moduli space of instantons over a given four-dimensional differentiable manifold as
Instanton
Concept in mathematics
points is instead an orbifold. A configuration space is a type of classifying space or (fine) moduli space. In particular, there is a universal bundle π
Configuration space (mathematics)
Configuration_space_(mathematics)
Generalization of algebraic spaces or schemes
vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques
Algebraic_stack
Algebraic surface defined by a cubic polynomial
(GIT) gives a moduli space of smooth cubic surfaces, with one point for each isomorphism class of smooth cubic surfaces. This moduli space has dimension
Cubic_surface
System of partial differential equations used in Higgs field theory
Higgs bundle moduli space, and to the moduli space of holomorphic connections. Using the metric structure on the Higgs bundle moduli space afforded by
Hitchin's_equations
Mathematics award
2022) – "For advances in Brill-Noether theory and the geometry of the moduli space of curves." Laura Monk, University of Bristol (PhD University of Strasbourg
Breakthrough Prize in Mathematics
Breakthrough_Prize_in_Mathematics
Theory of subatomic structure
"Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and SL(2,Z) invariance in string theory". Physics Letters B. 329 (2):
String_theory
^{*}\omega '=\omega } ). Let M g {\displaystyle {\mathcal {M}}_{g}} be the moduli space of Riemann surfaces of genus g {\displaystyle g} ; there is a natural
Translation_surface
Mathematic theorem about Riemann surfaces
In mathematics, the Narasimhan–Seshadri theorem, proved by Narasimhan and Seshadri (1965), says that a holomorphic vector bundle over a compact Riemann
Narasimhan–Seshadri_theorem
French mathematician (1928–2014)
cohomology – Concept in algebraic geometry Nakai conjecture Moduli scheme – Moduli space in the Grothendieck category of schemes Motive (algebraic geometry) –
Alexander_Grothendieck
{\displaystyle |X|} is a point. When X is a moduli stack, the quotient space | X | {\displaystyle |X|} is called the moduli space of X. If f : X → Y {\displaystyle
Quotient space of an algebraic stack
Quotient_space_of_an_algebraic_stack
Compactness theorem in Yang–Mills theory
used in the compactification of the anti self-dual Yang–Mills moduli space (ASDYM moduli space), which is central to the construction of Donaldson invariants
Uhlenbeck's compactness theorem
Uhlenbeck's_compactness_theorem
Belgian mathematician
He also collaborated with David Mumford on a new description of the moduli spaces for curves. Their work came to be seen as an introduction to one form
Pierre_Deligne
Orientation-preserving mapping class group of the torus
transformations. The name "modular group" comes from the relation to moduli spaces, and not from modular arithmetic. The modular group Γ is the group of
Modular_group
applications, a level structure is used in the construction of moduli spaces; a moduli space is often constructed as a quotient. The presence of automorphisms
Level structure (algebraic geometry)
Level_structure_(algebraic_geometry)
American mathematician
S2CID 125716869. Harer, J.; Zagier, D. (1986). "The Euler characteristic of the moduli space of curves". Inventiones Mathematicae. 85 (3). Springer Science and Business
Don_Zagier
American mathematician
Rahul Pandharipande; his dissertation was The tautological ring of the moduli space of curves. Pixton was appointed as a Clay Research Fellow for a term
Aaron_Pixton
Theory in differential topology
equal to the dimension of the moduli space of gradient flows between those points. Thus there is a one-dimensional moduli space of flows between a critical
Morse_homology
Brazilian mathematician (born 1979)
that the non-trivial Lyapunov exponents of the Teichmüller flow on the moduli space of Abelian differentials on compact Riemann surfaces are all distinct
Artur_Avila
Type of Riemannian manifold
instanton moduli spaces, monopole moduli spaces, spaces of solutions to Nigel Hitchin's self-duality equations on Riemann surfaces, space of solutions
Hyperkähler_manifold
Five-pointed star polygon
Discrete dynamical system on polygons in the projective plane and on their moduli space Pentalpha – Puzzle involving stones and a pentagram Petersen graph –
Pentagram
Mathematical manifold theory
topological spaces can have the structure of a smooth complex projective variety. Second, Hodge theory gives information about the moduli space of smooth
Hodge_theory
Generalized manifold
singularities. This limit represents a point on the boundary of the moduli space of K3 surfaces and corresponds to the orbifold T 4 / Z 2 {\displaystyle
Orbifold
geometry; for now see also ind-scheme). moduli See for example moduli space. While much of the early work on moduli, especially since [Mum65], put the emphasis
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Type of algebraic equation
algebraic equation satisfied by moduli, in the sense of moduli problems. That is, given a number of functions on a moduli space, a modular equation is an equation
Modular_equation
Theory of strings with supersymmetry
discover supersymmetry at CERN in the foreseeable future. Our physical space is observed to have three large spatial dimensions and, along with time
Superstring_theory
construction of elliptic cohomology (the precursor to tmf) as some kind of moduli space of conformal field theories, and these ideas have been continued and
Topological_modular_forms
Yang–Mills theory in two dimensions with a well-defined measure
function yields the volume of the moduli space of flat connections with respect to a natural volume measure on the moduli space. This volume measure is associated
Two-dimensional Yang–Mills theory
Two-dimensional_Yang–Mills_theory
Curve defined as zeros of polynomials
rational plane curve of degree d {\displaystyle d} . There is an associated moduli space M = M ¯ 0 , 0 ( P 2 , d ⋅ [ H ] ) {\displaystyle {\mathcal {M}}={\overline
Algebraic_curve
Field theory involving topological effects in physics
theory of four-manifolds, and algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have
Topological quantum field theory
Topological_quantum_field_theory
Supersymmetric generalization of quantum chromodynamics
hadrons, and the moduli space of vacua of the theory may be parametrized by their vacuum expectation values. On most of the moduli space the Higgs mechanism
Super_QCD
Conjecture in algebraic geometry
is a conjecture about intersection numbers of stable classes on the moduli space of curves, introduced by Edward Witten in the paper Witten (1991), and
Witten_conjecture
Formalism in string theory
together, These vertices and propagators produce a single cover of the moduli space of n {\displaystyle n} -point closed string scattering amplitudes so
String_field_theory
Mathematical conjecture
the moduli space of Calabi-Yau manifolds. Because of the Bogomolev-Tian-Todorov theorem, all such deformations are unobstructed, so the smooth space U smooth
Mirror_symmetry_conjecture
Framework of superstring theory
"Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and SL(2,Z) invariance in string theory". Physics Letters B. 329 (2):
M-theory
Chinese mathematician (born 1958)
a one-parameter family of Calabi-Yau metrics; this proves that the "moduli space" of Calabi-Yau metrics on the given manifold has the structure of a smooth
Tian_Gang
Concept in string theory
the moduli spaces of stable maps, which can be thought of as spaces parametrizing curves in X {\displaystyle X} . However, as these moduli spaces can
Gromov–Witten_invariant
Compact astronomical body
1950s, this solution began to be interpreted physically as a region of space from which nothing can escape. Black holes were long considered a mathematical
Black_hole
Algebraic variety
important role in arithmetic geometry. The level N modular curve X(N) is the moduli space for elliptic curves with a basis for the N-torsion. For X0(N) and X1(N)
Modular_curve
American mathematician (born 1965)
the SL ( 2 , R ) {\displaystyle {\text{SL}}(2,\mathbb {R} )} action on moduli space". Annals of Mathematics. 182 (2): 673–721. arXiv:1305.3015. doi:10.4007/annals
Alex_Eskin
Poset representing certain properties of a polytope
finitely many rotations, and possibly trivial reflection. Generally, the moduli space of realizations of an abstract polytope is a convex cone of infinite
Abstract_polytope
In physics and geometry: conjectured relation between pairs of Calabi–Yau manifolds
ISBN 978-0-9650888-0-0. Greene, Brian; Plesser, Ronen (1990). "Duality in Calabi–Yau moduli space". Nuclear Physics B. 338 (1): 15–37. Bibcode:1990NuPhB.338...15G. doi:10
Mirror symmetry (string theory)
Mirror_symmetry_(string_theory)
British mathematician
interpretation as a moduli space of semistable Higgs bundles over a compact Riemann surface or algebraic curve. This moduli space has emerged as a focal
Nigel_Hitchin
Vector bundles theorem
Nigel Hitchin, who independently conjectured in the 1980s that the moduli spaces of stable vector bundles and Einstein–Hermitian vector bundles over
Kobayashi–Hitchin correspondence
Kobayashi–Hitchin_correspondence
Relation between genus, degree, and dimension of function spaces over surfaces
useful in the construction of the moduli space of algebraic curves because it can be used as the projective space to construct the Hilbert scheme with
Riemann–Roch_theorem
Embedding a topological space into a compact space as a dense subset
compactification of the moduli space of algebraic curves. In the study of discrete subgroups of Lie groups, the quotient space of cosets is often a candidate
Compactification (mathematics)
Compactification_(mathematics)
MODULI SPACE
MODULI SPACE
Boy/Male
Christian, Hindu, Indian
Special Smile; Sweet Little Attitude
Boy/Male
Hindu
Female
English
Variant spelling of Middle English Mauld, MOULD means "mighty in battle."
Girl/Female
Indian, Sanskrit
A Bud
Girl/Female
Hindu, Indian
Good Music
Girl/Female
Hindu
Boy/Male
Hindu, Indian, Telugu
Lord Shiva; Wearing
Girl/Female
Hindu
Melodious, A musical Raag
Boy/Male
Hindu
Name of Lord Shiva
Boy/Male
Hindu
Precious, Valuable
Boy/Male
Indian, Malayalam
Cartoon Character
Girl/Female
Hindu
Happy, Cheerful
Girl/Female
Indian
Name of Godeess Durga
Boy/Male
Arabic, Assamese, Indian, Muslim
Main; New
Girl/Female
Indian
Valley of Flowers
Girl/Female
Hindu
Delightful
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Telugu
Valuable; Original
Girl/Female
Hindu, Indian, Marathi
Sacred Thread; Mother
Boy/Male
Indian, Sanskrit
Logician
Girl/Female
Bengali, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
A Small Rain Cloud
MODULI SPACE
MODULI SPACE
Girl/Female
Hindi Indian Sanskrit
A princess.
Female
Spanish
Pet form of Spanish Candelaria, CANDELAS means "candle."
Boy/Male
American, Arabic, Chinese, Hindu, Indian, Muslim
Round; Gentle; Cask; Companion; Friend
Boy/Male
Assamese, Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Sindhi, Tamil, Telugu
Gold; Lord Shiva; Silver; Closeness
Boy/Male
Hindu, Indian
Garland of Flowers
Girl/Female
Finnish, French, German, Swedish
Industrious; Striving; Work; Rival; Laborious; Eager; Beloved
Boy/Male
Hindu
Part of the sacred knowledge
Boy/Male
Hindu
Name of Lord Shiva
Boy/Male
Bengali, Hindu, Indian, Kannada, Marathi, Sanskrit, Telugu
Conquering Millions
Boy/Male
Indian, Sanskrit
Golden
MODULI SPACE
MODULI SPACE
MODULI SPACE
MODULI SPACE
MODULI SPACE
v. t.
To plan or form after a pattern; to form in model; to form a model or pattern for; to shape; to mold; to fashion; as, to model a house or a government; to model an edifice according to the plan delineated.
n.
To model; also, to modulate.
n.
A fixed compensation or equivalent given instead of payment of tithes in kind, expressed in full by the phrase modus decimandi.
a.
Of or pertaining to mode, modulation, module, or modius; as, modular arrangement; modular accent; modular measure.
v. i.
To make a copy or a pattern; to design or imitate forms; as, to model in wax.
n.
A fixed part of a module. See Module.
n.
Anything which serves, or may serve, as an example for imitation; as, a government formed on the model of the American constitution; a model of eloquence, virtue, or behavior.
n.
Something intended to serve, or that may serve, as a pattern of something to be made; a material representation or embodiment of an ideal; sometimes, a drawing; a plan; as, the clay model of a sculpture; the inventor's model of a machine.
pl.
of Modulus
p. pr. & vb. n.
of Moult
imp. & p. p.
of Moult
pl.
of Modus
n.
A model or measure.
pl.
of Morula
a.
Suitable to be taken as a model or pattern; as, a model house; a model husband.
n.
The size of some one part, as the diameter of semi-diameter of the base of a shaft, taken as a unit of measure by which the proportions of the other parts of the composition are regulated. Generally, for columns, the semi-diameter is taken, and divided into a certain number of parts, called minutes (see Minute), though often the diameter is taken, and any dimension is said to be so many modules and minutes in height, breadth, or projection.
v. t.
To form into a particular shape; to shape; to model; to fashion.