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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
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
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
Iranian mathematician (1977–2017)
geometry of moduli spaces. In collaboration with Alex Eskin and later Amir Mohammadi, she achieved breakthrough results on the dynamics of moduli spaces that
Maryam_Mirzakhani
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
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
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
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
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
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)
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
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
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
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)
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
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
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)
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)
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
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
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
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
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)
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
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)
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
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)
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
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)
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
Hyperspace in algebraic geometry
not rational. The 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
Cubic_threefold
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
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
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)
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
{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
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
Moduli space in the 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
about it, similar to the much more well-known Yang–Mills moduli space and Seiberg–Witten moduli space. The Kapustin–Witten equations are named after Anton
Kapustin–Witten_equations
about it, similar to the much more well-known Yang–Mills moduli space and Seiberg–Witten moduli space. The Vafa–Witten equations are named after Cumrun Vafa
Vafa–Witten_equations
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
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
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
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
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
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
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
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
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
of hyperplanes in the period space. Yuchen Liu further showed that the GIT compactification is isomorphic to the K-moduli of cubic fourfolds. Katzarkov
Cubic_fourfold
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
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
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)
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
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
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
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
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
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
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
^{*}\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
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
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)
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
symplectic geometry and algebraic geometry, the moduli spaces of stable maps generalise the moduli spaces of curves, allowing the study of the geometry
Stable_map
theorem. In particular, key informations are obtained from the Yang–Mills moduli space of the Yang–Mills equations on the 4-manifold, which as partial differential
Donaldson_invariant
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
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
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)
over the moduli space of pseudoholomorphic curves M ¯ g , n ( X , A ) {\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,A)} . This moduli space is roughly
Kuranishi_structure
Concept in algebraic geometry
{\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
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
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
British mathematician
(MMath 2006, DPhil 2009), where he wrote his doctoral thesis Stable moduli spaces of manifolds under the supervision of Ulrike Tillmann. Since 2012 he
Oscar_Randal-Williams
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
Mathematician
algebraic topology, who has made important contributions to the study of the moduli space of algebraic curves. She was the president of the London Mathematical
Ulrike_Tillmann
Mathematical classification of surfaces
by a moduli space. For most of the classes the moduli spaces are well understood, but for the class of surfaces of general type the moduli spaces seem
Enriques–Kodaira classification
Enriques–Kodaira_classification
French mathematician (1928–2014)
Moduli scheme – Moduli space in the category of schemes Motive (algebraic geometry) – Structure in algebraic geometry Nuclear operator Nuclear space Parafactorial
Alexander_Grothendieck
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
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
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
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
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
Cubic graph with 10 vertices and 15 edges
geometry. The cone over the Petersen graph is naturally identified with the moduli space of five-pointed rational tropical curves. The Petersen graph is the complement
Petersen_graph
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
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
Concept in algebraic geometry
quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper
Geometric_invariant_theory
and an integral over the moduli space of stable curves. Several fundamental results in the intersection theory of moduli spaces of curves can be deduced
ELSV_formula
Type of integrable system
algebraic geometry, the phase space of the system is a partial compactification of the cotangent bundle to the moduli space of stable G-bundles for some
Hitchin_system
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)
Mathematical concept
fails to be a fine moduli space due to the presence of elliptic curves with automorphisms, necessitating the construction of the Moduli stack of elliptic
J-line
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
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
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
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)
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
Equivalence in 3D quantum field theory
Seiberg in 1996. They showed that for a pair of mirror theories, the moduli space of each theory is swapped. Specifically, what is known as the Coulomb
3D_mirror_symmetry
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
Mathematical Concept
geometry, the tautological ring is the subring of the Chow ring of the moduli space of curves generated by tautological classes. These are classes obtained
Tautological_ring
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