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See searches and references containing TOPOLOGICAL K-THEORY!TOPOLOGICAL K-THEORY
Branch of algebraic topology
In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas
Topological_K-theory
Branch of mathematics
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology
K-theory
Field theory involving topological effects in physics
under any deformation of spacetime and are therefore topological invariants. Topological field theories are not very interesting on flat Minkowski spacetime
Topological quantum field theory
Topological_quantum_field_theory
Topological quantum field theory
The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type. It was discovered first by mathematical physicist Albert Schwarz
Chern–Simons_theory
denoted ψk for natural numbers k, is a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic
Adams_operation
British-Lebanese mathematician (1929–2019)
contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the Fields Medal in 1966 and the Abel Prize in 2004
Michael_Atiyah
Describes a periodicity in the homotopy groups of classical groups
the theory associated to the unitary group. See for example topological K-theory. There are corresponding period-8 phenomena for the matching theories, (real)
Bott_periodicity_theorem
k-quasi-planar topological graph is n log O ( log k ) n {\displaystyle n\log ^{O(\log k)}n} . This implies that every complete topological graph with
Topological_graph
Branch of the mathematical field of graph theory
In mathematics, topological graph theory is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and
Topological_graph_theory
operator K-theory is a noncommutative analogue of topological K-theory for Banach algebras with most applications used for C*-algebras. Operator K-theory resembles
Operator_K-theory
Subject area in mathematics
define topological K-theory. Topological K-theory was one of the first examples of an extraordinary cohomology theory: It associates to each topological space
Algebraic_K-theory
Hirzebruch (1961) in the special case of topological K-theory. For a CW complex X {\displaystyle X} and a generalized cohomology theory E ∙ {\displaystyle E^{\bullet
Atiyah–Hirzebruch spectral sequence
Atiyah–Hirzebruch_spectral_sequence
Theory in theoretical physics
In theoretical physics, topological string theory is a version of string theory. Topological string theory appeared in papers by theoretical physicists
Topological_string_theory
How many linearly independent smooth nowhere-zero vector fields can be on an n-sphere
{\displaystyle \rho (n)-1} such fields. Adams applied homotopy theory and topological K-theory to prove that no more independent vector fields could be found
Vector_fields_on_spheres
combinatorics. Topological degree theory Topological graph theory Topological K-theory Topos theory Toric geometry Transcendental number theory a branch of
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Topics referred to by the same term
equivariant K-theory refers to either equivariant algebraic K-theory, an equivariant analog of algebraic K-theory equivariant topological K-theory, an equivariant
Equivariant_K-theory
Application of K-theory in string theory
condensed matter physics K-theory has also found important applications, specially in the topological classification of topological insulators, superconductors
K-theory_(physics)
Algebraic structure used in topology
specifically in homology theory and algebraic topology, cohomology is a way of attaching algebraic invariants to a topological space or other mathematical
Cohomology
French mathematician (1928–2014)
this theorem started the study of algebraic and topological K-theory, which explores the topological properties of objects by associating them with rings
Alexander_Grothendieck
Exact homotopy case
U(1), but need not have a chosen identification, one writes BT. The topological K-theory K0(BT) is given by numerical polynomials; more details below. Let
Classifying_space_for_U(n)
Mathematics concept
In mathematics, KR-theory is a variant of topological K-theory defined for spaces with an involution. It was introduced by Atiyah (1966), motivated by
KR-theory
Topics referred to by the same term
K-group or K group may refer to: A group in algebraic K-theory A group in topological K-theory A complemented group K-Groups (Germany), small Communist
K-group
Result on the topology of operators on an infinite-dimensional, complex Hilbert space
its homotopy groups are trivial. This result has important uses in topological K-theory. For finite dimensional H, this group would be a complex general
Kuiper's_theorem
State of matter with insulating bulk but conductive boundary
{Z} _{2}} topological order has also been used to describe the topological order with emergent Z 2 {\displaystyle \mathbb {Z} _{2}} gauge theory discovered
Topological_insulator
German mathematician (1927–2012)
'new methods' of sheaf theory, in complex algebraic geometry. He went on to write the foundational papers on topological K-theory with Michael Atiyah, and
Friedrich_Hirzebruch
Mathematical subject
The mathematical discipline of topological combinatorics is the application of topological and algebro-topological methods to solve problems in combinatorics
Topological_combinatorics
Number representing system complexity
In mathematics, the topological entropy of a topological dynamical system is a nonnegative extended real number that is a measure of the complexity of
Topological_entropy
Mathematical theories
diffeomorphism. In both cases there are two obstructions for n>9, a primary topological K-theory obstruction to the existence of a vector bundle: if this vanishes
Obstruction_theory
Mathematical object
As a second important example, consider topological K-theory. At least for X compact, K 0 ( X ) {\displaystyle K^{0}(X)} is defined to be the Grothendieck
Spectrum_(topology)
Group that is a topological space with continuous group operations
many results from the theory of topological groups can be applied to functional analysis. A topological group, G, is a topological space that is also a
Topological_group
Subject in mathematics
In mathematics, measure theory in topological vector spaces refers to the extension of measure theory to topological vector spaces. Such spaces are often
Measure theory in topological vector spaces
Measure_theory_in_topological_vector_spaces
Mathematical parametrization of vector spaces by another space
on real vector bundles in the category of topological spaces. A real vector bundle consists of: topological spaces X {\displaystyle X} (base space) and
Vector_bundle
Properties that homology theories of topological spaces have in common
are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular
Eilenberg–Steenrod_axioms
Type of order at absolute zero
"topological order". The name "topological order" is motivated by the low energy effective theory of the chiral spin states which is a topological quantum
Topological_order
Mathematical space with a notion of closeness
defined. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, topological spaces are fundamental
Topological_space
Generalization of category theory
enriched models like topologically enriched categories. Topologically enriched categories (sometimes simply called topological categories) are categories
Higher_category_theory
compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually invariant under isomorphism. Topological indices
Topological_index
Analysis of datasets using techniques from topology
geometry) Size theory Algebraic topology Topological deep learning Epstein, Charles; Carlsson, Gunnar; Edelsbrunner, Herbert (2011-12-01). "Topological data analysis"
Topological_data_analysis
topological K-theory to noncommutative C*-algebras in the form of operator K-theory. A further development in this is a bivariant version of K-theory
Noncommutative_topology
Topologically stable solution of a partial differential equation
In mathematics and physics, solitons, topological solitons and topological defects are three closely related ideas, all of which signify structures in
Topological_defect
Topological field
The BF model or BF theory is a topological field, which when quantized, becomes a topological quantum field theory. BF stands for background field B and
BF_model
Mathematical manifold theory
different roles played by Hodge theory in complex algebraic geometry. First, Hodge theory gives restrictions on which topological spaces can have the structure
Hodge_theory
Type of topological order in condensed matter physics
entanglement see topological order, which is not related to the famous EPR paradox). Since short-range entangled states have only trivial topological orders we
Symmetry-protected topological order
Symmetry-protected_topological_order
Classification of topological quantum field theories
and James Dolan, concerns the classification of extended topological quantum field theories (TQFTs). In 2008, Jacob Lurie outlined a proof of the cobordism
Cobordism_hypothesis
Type of quantum computer
claiming partial evidence of topological behaviour. Topological order Symmetry-protected topological order Ginzburg–Landau theory Husimi Q representation Random
Topological_quantum_computer
Algebraic structure associated with a topological space
homology of a topological space. For sufficiently nice topological spaces and compatible choices of coefficient rings, any homology theory satisfying the
Homology_(mathematics)
Vector space with a notion of nearness
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures
Topological_vector_space
D. dissertation, "A new cohomology theory". Spectrum: ku for connective K-theory, ko for connective real K-theory. Coefficient ring: For ku, the coefficient
List_of_cohomology_theories
Design theory
C-K design theory or concept-knowledge theory is both a design theory and a theory of reasoning in design. It defines design reasoning as a logic of expansion
C-K_theory
Indication of topological symmetry groups to topological condensed matter
The periodic table of topological insulators and topological superconductors, also called tenfold classification of topological insulators and superconductors
Periodic table of topological insulators and topological superconductors
Periodic_table_of_topological_insulators_and_topological_superconductors
Directed graph with no directed cycles
a topological ordering is acyclic. Conversely, every directed acyclic graph has at least one topological ordering. The existence of a topological ordering
Directed_acyclic_graph
Research field in deep learning
graphs, or general topological spaces like simplicial complexes and CW complexes. TDL addresses this by incorporating topological concepts to process
Topological_deep_learning
Type of monoidal category
the algebraic theory of topological quantum information, as they are used to store the algebraic data describing anyons in topological quantum phases
Modular_tensor_category
Fewest edge crossings in drawing of a graph
subgraphs. Theory and Practice of Combinatorics. North-Holland Mathematics Studies. Vol. 60. pp. 9–12. MR 0806962. Ackerman, Eyal (2013). "On topological graphs
Crossing number (graph theory)
Crossing_number_(graph_theory)
American mathematician (1944–2025)
one-dimensional formal group laws over a field, which generalize classical topological K-theory. From a modern point of view (i.e., since Ethan Devinatz, Michael
Jack_Morava
Tool to track locally defined data attached to the open sets of a topological space
framework for a very general cohomology theory, which encompasses also the "usual" topological cohomology theories such as singular cohomology. Especially
Sheaf_(mathematics)
Conjecture that states that certain assembly maps are isomorphisms
statement, for the topological K-theory of reduced group C ∗ {\displaystyle C^{*}} -algebras K n t o p ( C ∗ r ( G ) ) {\displaystyle K_{n}^{top}(C_{*}^{r}(G))}
Farrell–Jones_conjecture
Homotopy invariant of maps between n-spheres
Adams, and subsequently by Adams and Michael Atiyah with methods of topological K-theory, that these are the only maps with Hopf invariant 1. J. H. C. Whitehead
Hopf_invariant
Abelian group extending a commutative monoid
Field of fractions Localization Topological K-theory Atiyah–Hirzebruch spectral sequence for computing topological K-theory Bruns, Winfried; Gubeladze, Joseph
Grothendieck_group
Journal publisher
California, Berkeley. Algebra & Number Theory Algebraic & Geometric Topology Analysis & PDE Annals of K-Theory Communications in Applied Mathematics and
Mathematical Sciences Publishers
Mathematical_Sciences_Publishers
Characteristic classes of vector bundles
degree 2 k {\displaystyle 2k} is a spin manifold. Chern classes can be used to construct a homomorphism of rings from the topological K-theory of a space
Chern_class
The analogous result holds between the K-theory of the symplectic quotient and the equivariant topological K-theory of M {\displaystyle M} . Kirwan, F.C
Kirwan_map
in a specified sense. A topological algebra A {\displaystyle A} over a topological field K {\displaystyle K} is a topological vector space together with
Topological_algebra
Mathematical result in differential geometry
dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other theorems, such as
Atiyah–Singer_index_theorem
Field of mathematics and science based on non-linear systems and initial conditions
such that f k ( U ) ∩ V ≠ ∅ {\displaystyle f^{k}(U)\cap V\neq \emptyset } . Topological transitivity is a weaker version of topological mixing. Intuitively
Chaos_theory
In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G.
Bohr_compactification
Algebraic tool for computing topological spaces' invariants
topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces. The result is
Mayer–Vietoris_sequence
Concept in algebraic topology
can be applied to all topological spaces, and so singular homology is expressible as a functor from the category of topological spaces to the category
Singular_homology
a lattice gauge theory, and it has applications in many fields, like topological quantum computation, topological order, topological quantum memory, quantum
Quantum_double_model
Concept in mathematics
provide a set with group structure (a group) or a topological space with group structure (a topological group), supplying appropriate names to the generic
Lawvere_theory
Class of compact connected topological spaces
a class of topological groups. For the wrapped loop of wire, see Solenoid. In mathematics, a solenoid is a compact connected topological space (i.e.
Solenoid_(mathematics)
In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer
Topological_modular_forms
Type of mathematical measure
(specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is
Radon_measure
Topological space
a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the
Cantor_space
Type of topological group in mathematics
targets Topological group – Group that is a topological space with continuous group operations Topological module Topological ring Topological semigroup
Locally_compact_group
Algebraic structure formed from a collection of algebraic structures
{B} \end{bmatrix}}.} A topological vector space (TVS) X , {\displaystyle X,} such as a Banach space, is said to be a topological direct sum of two vector
Direct_sum
Particle
entirely on braiding and performing topological charge measurements, and hence form a natural setting for topological quantum computing. This is in contrast
Fibonacci_anyons
Branch of topology
generally topological spaces, of four or fewer dimensions. Representative topics are the theory of 3-manifolds and 4-manifolds, knot theory, and braid
Low-dimensional_topology
Space with topology generated by convex sets
of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize
Locally convex topological vector space
Locally_convex_topological_vector_space
Branch of mathematics
geometry (e.g., KK-theory and Noncommutative topology) In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order
Homotopy_theory
Branch of mathematics that studies the properties of groups
between infinite abstract groups and topological groups: whenever a group Γ can be realized as a lattice in a topological group G, the geometry and analysis
Group_theory
Study of mathematical knots
mathematical theory of knots was first developed in 1771 by Alexandre-Théophile Vandermonde who explicitly noted the importance of topological features when
Knot_theory
American theoretical physicist
theoretical physicist known for his contributions to string theory, topological quantum field theory, general relativity and various areas of mathematics. He
Edward_Witten
Algebraic topology uses abstract algebra to study topological spaces
from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though
List of algebraic topology topics
List_of_algebraic_topology_topics
{\displaystyle [V]} to its associated character. Topological K-theory, the topological equivariant K-theory Charles A. Weibel, Robert W. Thomason (1952–1995)
Equivariant algebraic K-theory
Equivariant_algebraic_K-theory
Topological invariants concerning BPS states
S2CID 13824856 Gopakumar, Rajesh; Vafa, Cumrun (1998d), "Topological Gravity as Large N Topological Gauge Theory", Adv. Theor. Math. Phys., 2 (2): 413–442, arXiv:hep-th/9802016
Gopakumar–Vafa_invariant
\ker D^{+})-\operatorname {tr} (g\mid \ker D^{-}).} Equivariant topological K-theory Kawasaki's Riemann–Roch formula Berline, Nicole; Getzler, E.; Vergne
Equivariant_index_theorem
Mathematical set with some added structure
requires a theory capable of assigning extra data to degenerate situations. One of the building blocks of a scheme is a topological space. Topological spaces
Space_(mathematics)
Topological group structure arising in Fourier analysis
groups. A topological group is called locally compact if the underlying topological space is locally compact and Hausdorff; the topological group is called
Locally_compact_abelian_group
Right inverse of a morphism
Lane the founder of category theory, and (as the earliest publications on category theory concerned various topological spaces) one might have expected
Section_(category_theory)
Study of sudden qualitative behavior changes caused by small parameter changes
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral
Bifurcation_theory
Sheaf cohomology on the étale site
finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct
Étale_cohomology
Mathematical approach to quantum physics
E k 1 n E k 2 k 3 2 E k 2 k 4 − V k 2 k 3 V k 3 k 4 V k 4 n V k 1 k 2 E k 1 n E k 2 n E n k 3 E n k 4 + V k 1 k 2 E k 1 n ( | V k 2 k 3 | 2 V k 2 k 2
Perturbation theory (quantum mechanics)
Perturbation_theory_(quantum_mechanics)
Combinatorial approach of studying the topology of a manifold
configuration spaces, homology computation, denoising, mesh compression, and topological data analysis. Let X {\displaystyle X} be a CW complex and denote by
Discrete_Morse_theory
Mathematical concept
mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only
Finite_topological_space
Class of mathematical sets
In mathematics, the Borel sets of a topological space are a particular class of "well-behaved" subsets of that space. For example, whereas an arbitrary
Borel_set
Conjecture linking two mathematical areas
continuous functions on the circle. So the right hand side is the topological K-theory of the circle. One can then show that the assembly map is KK-theoretic
Baum–Connes_conjecture
Concept in topology
functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct
Topological_conjugacy
Number of "holes" of a surface
a sphere with n cross-caps or on a sphere with n/2 handles. In topological graph theory there are several definitions of the genus of a group. Arthur T
Genus_(mathematics)
Topological group with compact topology
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space. Compact groups are a natural
Compact_group
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TOPOLOGICAL K-THEORY
TOPOLOGICAL K-THEORY
Girl/Female
American, British, English
Sparkling; K from the Greek Spelling of Krystallos
Male
Czechoslovakian
, butcher.
Male
Icelandic
Icelandic form of German Ludwig, LÚÃVÃK means "famous warrior."
Male
Czechoslovakian
, famous war.
Male
Hungarian
Hungarian form of Old High German Berhtram, BERTÓK means "bright raven."
Girl/Female
American, British, English
A Combination of Initials K and C; Alert; Vigorous
Boy/Male
Hindu, Indian
K for Krishna, S for Shiv and G for Ganesh
Male
Polish
Polish form of Russian Svyatopolk, ÅšWIĘTOPEÅK means "blessed people."
Girl/Female
American, British, English
Sparkling; K from the Greek Spelling of Krystallos
Girl/Female
English Greek
Sparkling. 'K' from the Greek spelling of krystallos.
Male
Hungarian
Hungarian form of Greek Isaák, IZSÃK means "he will laugh."Â
Girl/Female
American, British, English, Gaelic, Irish
A Combination of Initials K and C; Alert; Watchful; Vigorous
Male
Egyptian
, the name of a mystical deity.
Girl/Female
American, British, English, Gaelic, Irish
A Combination of Initials K and C; Alert; Vigorous; Watchful
Girl/Female
English Greek
Sparkling. 'K' from the Greek spelling of krystallos.
Girl/Female
American, British, English, Polish
Sparkling; K from the Greek Spelling of Krystallos; Crystal Ice
Girl/Female
British, English, Greek
Sparkling; K from the Greek Spelling of Krystallos
Male
Greek
(Ἰσαάκ) Greek form of Hebrew Yitzchak, ISAÃK means "he will laugh."Â
Girl/Female
English Greek
Sparkling. 'K' from the Greek spelling of krystallos.
Girl/Female
English Greek
Sparkling. 'K' from the Greek spelling of krystallos.
TOPOLOGICAL K-THEORY
TOPOLOGICAL K-THEORY
TOPOLOGICAL K-THEORY
TOPOLOGICAL K-THEORY
TOPOLOGICAL K-THEORY
TOPOLOGICAL K-THEORY
TOPOLOGICAL K-THEORY
a.
Of or pertaining tootology.
a.
Pertaining to posology.
a.
Of or pertaining to oology.
a.
Of or pertaining to nosology.
a.
Characterized by tropes; varied by tropes; tropical.
a.
Of or pertaining to pomology.
a.
Applied to certain mute consonants, as p, k, and t (or Gr. /, /, /).
n.
A student in a theological seminary.
a.
Alt. of Posological
a.
Of or pertaining to theology, or the science of God and of divine things; as, a theological treatise.
n. pl.
A class of levelers in the time of K. Henry I.
a.
Theological.
a.
Alt. of Tropological
a.
Pertaining to doxology; giving praise to God.
a.
Of or pertaining to zoology, or the science of animals.
a.
Relating to a horologe, or to horology.
a.
Of or pertaining to noology.
a.
Of or pertaining to orology.
a.
Pertaining to homology; having a structural affinity proceeding from, or base upon, that kind of relation termed homology.
v. i.
To introduce innovations in doctrine, esp. in theological doctrine.
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