AI & ChatGPT searches , social queries for STABLE VECTOR-BUNDLE
Search references for STABLE VECTOR-BUNDLE. Phrases containing STABLE VECTOR-BUNDLE
See searches and references containing STABLE VECTOR-BUNDLE!
Search references for STABLE VECTOR-BUNDLE. Phrases containing STABLE VECTOR-BUNDLE
See searches and references containing STABLE VECTOR-BUNDLE!STABLE VECTOR-BUNDLE
stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle
Stable_vector_bundle
Mathematical parametrization of vector spaces by another space
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space
Vector_bundle
algebraic geometry, a stable principal bundle is a generalisation of the notion of a stable vector bundle to the setting of principal bundles. The concept of
Stable_principal_bundle
Type of vector bundle
In mathematics, a Higgs bundle is a pair ( E , φ ) {\displaystyle (E,\varphi )} consisting of a holomorphic vector bundle E and a Higgs field φ {\displaystyle
Higgs_bundle
Correspondsnce between Higgs bundles and fundamental group representations
Narasimhan–Seshadri theorem which defines a correspondence between stable vector bundles and unitary representations of the fundamental group of a compact
Nonabelian Hodge correspondence
Nonabelian_Hodge_correspondence
Tangent spaces of a manifold
tangent bundle of a differentiable manifold M {\displaystyle M} is a manifold T M {\displaystyle TM} which assembles all the tangent vectors in M {\displaystyle
Tangent_bundle
Vector bundles theorem
Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The correspondence is named after
Kobayashi–Hitchin correspondence
Kobayashi–Hitchin_correspondence
Mathematic theorem about Riemann surfaces
Narasimhan and Seshadri (1965), says that a holomorphic vector bundle over a compact Riemann surface is stable if and only if it comes from an irreducible projective
Narasimhan–Seshadri_theorem
Generalization of vector bundles
information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under
Coherent_sheaf
Fiber bundle whose fibers are projective spaces
projective bundle is of the form P ( E ) {\displaystyle \mathbb {P} (E)} for some vector bundle (locally free sheaf) E. Every vector bundle over a variety
Projective_bundle
Study of vector bundles, principal bundles, and fibre bundles
gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused
Gauge_theory_(mathematics)
a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal (dually, tangential) data
Stable_normal_bundle
Way to create new manifolds out of disk bundles
_{M_{B}^{4k}}\rightarrow \xi } is a bundle map from the stable normal bundle of the Milnor manifold to a certain stable vector bundle. A crucial theorem for the
Plumbing_(mathematics)
Concept in mathematics
a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or
Normal_bundle
Concept in algebraic geometry
the group PGL5g–5. Example: A vector bundle W over an algebraic curve (or over a Riemann surface) is a stable vector bundle if and only if deg ( V ) rank
Geometric_invariant_theory
Indian mathematician (1932–2021)
Narasimhan–Seshadri theorem which proved the necessary conditions for stable vector bundles on a Riemann surface. He was a recipient of the Padma Bhushan, India's
M._S._Narasimhan
Conjecture in symplectic geometry
especially the Kobayashi–Hitchin correspondence relating slope stable vector bundles to Hermitian Yang–Mills metrics. The conjecture is intimately related
Thomas–Yau_conjecture
Construct in mathematics
objects, but the stacky version remembers automorphisms of vector bundles. For any stable vector bundle E {\displaystyle E} the automorphism group A u t ( E
Gerbe
Mathematical object studied in the field of algebraic geometry
d)} of isomorphism classes of stable vector bundles of rank n and degree d as an open subset. Since a line bundle is stable, such a moduli is a generalization
Algebraic_variety
Indian mathematician (1932–2020)
Narasimhan–Seshadri theorem which proved the necessary conditions for stable vector bundles on a Riemann surface.He also introduced and named the concept called
C._S._Seshadri
Moduli space in the Grothendieck category of schemes
moduli space of curves. Using the notion of stable vector bundle, coarse moduli schemes for the vector bundles on any smooth complex variety have been shown
Moduli_scheme
Type of group in mathematics
clutching construction, homotopy groups of the stable space O are identified with stable vector bundles on spheres (up to isomorphism), with a dimension
Orthogonal_group
Mathematical theorem
algebraic geometry, Lange's conjecture is a theorem about stability of vector bundles over curves, introduced by Herbet Lange [de] and proved by Montserrat
Lange's_conjecture
American mathematician (born 1937)
embedding, which proves the stability of this point. The concept of stable vector bundle from moduli theory has been consequential in mathematical physics:
David_Mumford
Techniques in topology used to produce one finite-dimensional manifold from another
and only if the Spivak normal fibration of X has a reduction to a stable vector bundle. If normal maps of degree one to X exist, their bordism classes (called
Surgery_theory
Algebro-geometric stability condition
Donaldson, the theorem states that a holomorphic vector bundle over a compact Riemann surface is stable if and only if it corresponds to an irreducible
K-stability
Geometric space whose points represent algebro-geometric objects of some fixed kind
physics, the number of moduli of vector bundles and the closely related problem of the number of moduli of principal G-bundles has been found to be significant
Moduli_space
Type of differentiable manifold
of the normal bundle, and also for an abstract (that is, non-embedded) manifold with a given stable trivialisation of the tangent bundle. A related notion
Parallelizable_manifold
Chinese-American mathematician (born 1949)
dimension two, a holomorphic vector bundle admits a hermitian Yang–Mills connection if and only if the bundle is stable. A result of Yau and Karen Uhlenbeck
Shing-Tung_Yau
Hodge bundle The Hodge bundle on the moduli space of curves (of fixed genus) is roughly a vector bundle whose fiber over a curve C is the vector space
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Topological space associated to a vector bundle
topology and differential topology is a topological space associated to a vector bundle, over any paracompact space. One way to construct this space is as follows
Thom_space
British-Lebanese mathematician (1929–2019)
S. Narasimhan described the cohomology of the moduli spaces of stable vector bundles over Riemann surfaces by counting the number of points of the moduli
Michael_Atiyah
English mathematician (born 1957)
This contrasts with the situation in higher dimensions. A stable holomorphic vector bundle over a non-singular projective algebraic variety admits a Hermitian–Einstein
Simon_Donaldson
System of partial differential equations used in Higgs field theory
differential equations for a connection and Higgs field on a vector bundle or principal bundle over a Riemann surface, written down by Nigel Hitchin in 1987
Hitchin's_equations
Concept in geometric topology
the classifying space for stable spherical fibrations, B O {\displaystyle BO} the classifying space for stable vector bundles and the map J : B O → B G
Normal_invariant
correspond to k = 2 instantons on S4. Hartshorne, Robin (1978), "Stable vector bundles and instantons", Communications in Mathematical Physics, 59 (1):
Hartshorne_ellipse
Concept in algebraic geometry
Noetherian, such as the Moduli of algebraic curves and Moduli of stable vector bundles. Also, this property can be used to show many schemes considered
Noetherian_scheme
Public research institute in Mumbai, India
differential operators. Narasimhan and Seshadri wrote a seminal paper on stable vector bundles, work which has been recognised as one of the most influential articles
Tata Institute of Fundamental Research
Tata_Institute_of_Fundamental_Research
Characteristic class of oriented, real vector bundles
oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth
Euler_class
Spanish mathematician
− 1 ) {\displaystyle 0<s\leq n'(n-n')(g-1)} , then there exist stable vector bundles with s n ′ ( E ) = s {\displaystyle s_{n'}(E)=s} ." They also clarified
Montserrat_Teixidor_i_Bigas
Association of cohomology classes to principal bundles
whenever there was a vector bundle involved. The prime mechanism then appeared to be this: Given a space X carrying a vector bundle, that implied in the
Characteristic_class
Branch of algebraic topology
K-theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general)
Topological_K-theory
= ηx4 = 0, and x42 = 4v14. KO0(X) is the ring of stable equivalence classes of real vector bundles over X. Bott periodicity implies that the K-groups
List_of_cohomology_theories
Describes a periodicity in the homotopy groups of classical groups
much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can
Bott_periodicity_theorem
of semistable vector bundles on an algebraic curve. Let X be a projective curve over an algebraically closed field k. A vector bundle on X can be considered
S-equivalence
Characteristic class for real vector bundles
classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Given a real vector bundle E {\displaystyle
Pontryagin_class
_{\mathcal {F}}(\lambda )=n\lambda +d+n(1-g)} Then, the locus of semi-stable vector bundles is contained in Q u o t O C ⊕ N / C / Z Φ F , L {\displaystyle {\mathcal
Quot_scheme
Branch of mathematics
mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology
K-theory
Short exact sequence of sheaves on projective space
The Euler sequence generalizes to that of a projective bundle as well as a Grassmann bundle (see the latter article for this generalization.) Let P A
Euler_sequence
Metric on a determinant line bundle
the ample line bundle over the moduli space of vector bundles on a compact Riemann surface, known as the Quillen determinant line bundle. It can be seen
Quillen_metric
Branch of geometry
produces a transport of unit-length tangent vectors, and thus a vector flow field on the unit tangent bundle U T ( M ) {\displaystyle UT(M)} . This is the
Contact_geometry
Study of spaces with group actions
one can replace the bundle by a homotopy quotient where G {\displaystyle G} acts freely and is bundle homotopic to the induced bundle on X {\displaystyle
Equivariant_topology
American mathematician
ample vector bundle". Inventiones Mathematicae. 12 (2): 112–117. doi:10.1007/BF01404655. S2CID 122235253. Gieseker, D. (1977). "On the moduli of vector bundles
David_Gieseker
California, Berkeley under the direction of Robin Hartshorne with thesis Stable vector bundles on projective spaces in char p > 0 (which was published in Mathematische
Lawrence_Ein
{\displaystyle X} is called convex if the pullback of the tangent bundle to a stable rational curve f : C → X {\displaystyle f:C\to X} has globally generated
Convexity (algebraic geometry)
Convexity_(algebraic_geometry)
American mathematician (born 1942)
(1986). "On the existence of Hermitian-Yang-Mills connections in stable vector bundles". Communications on Pure and Applied Mathematics. 39: S257–S293
Karen_Uhlenbeck
Neighborhood of a submanifold
produce analogs to the normal bundle, or rather to the stable normal bundle, which are replacements for the tangent bundle (which does not admit a direct
Tubular_neighborhood
Diffeomorphism that has a hyperbolic structure on the tangent bundle
unit-length vectors on H. Note that a bundle of unit-length vectors on a surface is the principal bundle of a complex line bundle. One starts by noting that T
Anosov_diffeomorphism
Operation in differential geometry
theorem, which he used in his study of stable mappings. Suppose that E is a finite-dimensional smooth vector bundle over a manifold M, with projection π
Jet_(mathematics)
Partial differential equations whose solutions are instantons
system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations of
Yang–Mills_equations
characteristic class used in the study of cones, a generalization of vector bundles. For vector bundles the total Segre class is inverse to the total Chern class
Segre_class
Concept in differential geometry
adjoint bundle. A = Ω Ad 1 ( E , g ) {\displaystyle {\mathcal {A}}=\Omega _{\operatorname {Ad} }^{1}(E,{\mathfrak {g}})} , an affine vector space (not
Stable_Yang–Mills_connection
Yang–Mills coupled to a Higgs field
connection, and a Higgs field, given by a section of a vector bundle (specifically, the adjoint bundle). These equations are D A ∗ F A + ∗ [ Φ , D A Φ ] =
Yang–Mills–Higgs_equations
pair ( A , Z ) {\displaystyle (A,Z)} including a bundle homomorphism (i.e. a homomorphism of vector bundles) or the ( r + 1 {\displaystyle r+1} )-tuple (
Stochastic analysis on manifolds
Stochastic_analysis_on_manifolds
Generalisation of a sheaf; a fibered category that admits effective descent
situations where isomorphic, compatible geometrical objects (such as vector bundles on topological spaces) can be "glued together" within a restriction
Stack_(mathematics)
Subbundle of the tangent bundle
vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle
Distribution (differential geometry)
Distribution_(differential_geometry)
Branch of mathematics
considerable interest in physics. The apparatus of vector bundles, principal bundles, and connections on bundles plays an extraordinarily important role in modern
Differential_geometry
Type of integrable system
system is a partial compactification of the cotangent bundle to the moduli space of stable G-bundles for some reductive group G, on some compact algebraic
Hitchin_system
Type of wave
which concerns a high-order phase-locked vector soliton in SMFs, a stable high-order phase-locked vector soliton has recently been created in a fiber
Vector_soliton
Differential geometry topic
function that maps each point in the surface to its normal direction, a unit vector that is orthogonal to the surface at that point. Namely, given a surface
Gauss_map
Result in algebraic geometry
for line bundles on compact Riemann surfaces. Riemann–Roch type theorems relate Euler characteristics of the cohomology of a vector bundle with their
Grothendieck–Riemann–Roch theorem
Grothendieck–Riemann–Roch_theorem
Special tangential structure
of Real Vector Bundles". arXiv:2310.05061 [math.AT]. Lawson & Michelson 90, Definition D.3 Albanese & Milivojević 2021, Definition 3.1 Stable complex
Spinc_structure
Concerned with the notion of stability in model theory
model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof
Stable_theory
Restriction of electrical impulse flow in the heart's bundle branches
A bundle branch block is a partial or complete interruption in the flow of electrical impulses in either of the bundle branches of the heart's electrical
Bundle_branch_block
Differentiable function whose derivative is everywhere injective
normal bundle ν of the immersion i, which has dimension n − m, for there to be a codimension k immersion of M, there must be a vector bundle of dimension
Immersion_(mathematics)
Mathematical result in differential geometry
symbol of a differential operator between two vector bundles E and F is a section of the pullback of the bundle Hom(E, F) to the cotangent space of X. The
Atiyah–Singer_index_theorem
stability. Let C be a connected smooth curve. A rank-2 vector bundle E on C is said to be stable if for every line subbundle L of E, deg L < 1 2 deg
Complete_algebraic_curve
Algebraic geometry analog of a principal bundle in algebraic topology
In algebraic geometry, a torsor or a principal bundle is an analogue of a principal bundle in algebraic topology. Because there are few open sets in Zariski
Torsor_(algebraic_geometry)
Exact homotopy case
families of n vectors in Ck and let Gn(Ck) be the Grassmannian of n-dimensional subvector spaces of Ck. The total space of the universal bundle can be taken
Classifying_space_for_U(n)
Differential form on a manifold which is permitted to have complex coefficients
Because this direct sum decomposition is stable under holomorphic coordinate changes, it also determines a vector bundle decomposition. In particular, for each
Complex_differential_form
and David Mumford, characterizes the semistable and stable points of a group action on a vector space in terms of eigenvalues of 1-parameter subgroups
Hilbert–Mumford_criterion
Every vector bundle E → X {\displaystyle E\to X} over a scheme is a smooth morphism. For example, it can be shown that the associated vector bundle of O
Smooth_morphism
Differentiable manifold
The holomorphic tangent bundle of C 2 {\displaystyle \mathbb {C} ^{2}} consists of all linear combinations of the vectors ∂ ∂ z , ∂ ∂ w . {\displaystyle
CR_manifold
Invariant of a quadratic form over a field of characteristic 2
we now have a trivial normal 3-plane vector bundle. Trivialise it using the trivial framing of the normal bundle to the embedding M ↪ D 4 {\displaystyle
Arf_invariant
when a geometric object, for example a point, an algebraic variety, a vector bundle, or a sheaf, has some desirable properties for the purpose of classifying
Stability (algebraic geometry)
Stability_(algebraic_geometry)
Concept in differential geometry
adjoint bundle. A = Ω Ad 1 ( E , g ) {\displaystyle {\mathcal {A}}=\Omega _{\operatorname {Ad} }^{1}(E,{\mathfrak {g}})} , an affine vector space (not
Stable_Yang–Mills–Higgs_pair
Topologically stable solution of a partial differential equation
for a vector field. (A three-vector, its direction plus length, can be thought of as specifying a point on a 3-sphere. The orientation of the vector specifies
Topological_defect
Commutative algebra theorem
In the geometric setting it is a statement about the triviality of vector bundles on affine space. The theorem states that every finitely generated projective
Quillen–Suslin_theorem
Hypothetical particle with one magnetic pole
can include (but are not limited to) spin-0 monopoles or spin-1 massive vector mesons. The term "magnetic monopole" only refers to the nature of the particle
Magnetic_monopole
Mathematical theory
equips these Riemann surfaces with Hermitian metrics on holomorphic vector bundles over X(C), the complex points of X {\displaystyle X} . This extra Hermitian
Arakelov_theory
Subject area in mathematics
says that these are equal. When Y is a point, a vector bundle is a vector space, the class of a vector space is its dimension, and the Grothendieck–Riemann–Roch
Algebraic_K-theory
Mathematical concept
\mathbf {CP} ^{\infty }} is the associated vector bundle of the principal U ( 1 ) {\displaystyle U(1)} -bundle S ∞ → C P ∞ {\displaystyle S^{\infty }\to
Complex_projective_space
Spacetime modeled by four pointwise-orthonormal vector fields
notational conventions for sections of a tangent bundle. Alternative notations for the coordinate basis vector fields in common use are ∂ / ∂ x μ ≡ ∂ x μ ≡
Frame fields in general relativity
Frame_fields_in_general_relativity
Economic concept proposed by Erik Lindahl
a single price-vector for all agents, but each agent has a different bundle In a Lindahl equilibrium, there is a personal price-vector for each agent
Lindahl_tax
Chinese mathematician
"for his contributions to the study of moduli spaces of vector bundles and to the theory of stable maps and invariants of Calabi-Yau manifolds." "Jun Li
Jun_Li_(mathematician)
Algebraic topology uses abstract algebra to study topological spaces
Eilenberg–MacLane space Fibre bundle Möbius strip Line bundle Canonical line bundle Vector bundle Associated bundle Fibration Hopf bundle Classifying space Cofibration
List of algebraic topology topics
List_of_algebraic_topology_topics
Topological space that locally resembles Euclidean space
what a tangent vector might be, and there is no intrinsic notion of a normal bundle, but instead there is an intrinsic stable normal bundle. The n-sphere
Manifold
Mathematics textbook
with complex geometry, which includes Kähler manifolds and (stable) holomorphic vector bundles. Simon Donaldson and Peter Kronheimer both obtained their
The Geometry of Four-Manifolds
The_Geometry_of_Four-Manifolds
Economic equilibrium concept
current bundle as long as it is in the demand-set for price vector P ϵ x {\displaystyle P_{\epsilon }^{x}} . This makes the equilibrium more stable. The
Competitive_equilibrium
Matrix representing a Euclidean rotation
with standard coordinates v = (x, y), it should be written as a column vector, and multiplied by the matrix R: R v = [ cos θ − sin θ sin θ cos
Rotation_matrix
STABLE VECTOR-BUNDLE
STABLE VECTOR-BUNDLE
STABLE VECTOR-BUNDLE
STABLE VECTOR-BUNDLE
STABLE VECTOR-BUNDLE
STABLE VECTOR-BUNDLE
STABLE VECTOR-BUNDLE
STABLE VECTOR-BUNDLE
STABLE VECTOR-BUNDLE