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Branch of topology
String topology, a branch of mathematics, is the study of algebraic structures on the homology of free loop spaces. The field was started by Moira Chas
String_topology
American mathematician (born 1941)
is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at
Dennis_Sullivan
American mathematician
1952) is an American mathematician, specializing in algebraic topology and differential topology. Cohen received his bachelor's degree from the University
Ralph_Louis_Cohen
Sequence of characters, data type
same topology. Isomorphisms between string representations of topologies can be found by normalizing according to the lexicographically minimal string rotation
String_(computer_science)
Branch of mathematics
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants
Algebraic_topology
Branch of mathematics studying (smooth) functions of manifolds
geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. Geometric topology as an area
Geometric_topology
Branch of mathematics
Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric
Topology
Topological variant of the loop
the advent of string topology, i.e. the study of new algebraic structures on the homology of the free loop space. Loop space Loop (topology) Quasigroup
Free_loop
American theoretical physicist (born 1963)
form of topology change, and the conifold transition, a more severe transformation of space, showing that topology can smoothly change in string theory
Brian_Greene
Vector space with a notion of nearness
Hausdorff TVS is metrizable if and only if its topology can be induced by a single topological string. A vector space is an abelian group with respect
Topological_vector_space
Manifold modelled on Hilbert spaces
it suited to the study of algebraic topology of the free loop space, especially in the field of string topology. We can do an analogous Sobolev construction
Hilbert_manifold
Russian-American mathematician
Hochschild cohomology, cohomology of vertex operator algebras, and string topology (see cactus operad). He is a Fellow of the American Mathematical Society
Alexander_A._Voronov
Colombian mathematician
mathematician. Uribe's research deals with algebraic geometry and topology with string theory applications. Uribe graduated from secondary school in Bogotá
Bernardo_Uribe
Topological quantum field theory Topological quantum number Topological string theory Topology of the universe Milnor–Thurston kneading theory Topological conjugacy
List_of_topology_topics
German mathematician
to the invention of stringy K-theory. Kaufmann has also worked on string topology, invented by Moira Chas and Dennis Sullivan, and operad theory. Here
Ralph_Kaufmann
Speculative feature of the early universe
topological defect: a cosmic string) Cosmic string loop stabilised by a fermionic supercurrent: vorton Kibble, Tom W K (1976). "Topology of cosmic domains and
Cosmic_string
Local and global geometry of the universe
the shape of the universe refers to both its local geometry and cosmic topology. Local geometry is defined primarily by its curvature, general relativity
Shape_of_the_universe
Topics referred to by the same term
Germany String theory landscape, the large number of possible false vacua in string theory Knot theory, a branch of mathematical topology This disambiguation
String theory (disambiguation)
String_theory_(disambiguation)
Branch of topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions
Low-dimensional_topology
Topological model
(two geometries in two-dimensions, R2), in geometry, point-set topology, geospatial topology, and fields related to computer spatial analysis. The spatial
DE-9IM
Simple mechanical puzzles using topology
which the string is threaded. One can distinguish three subgroups of wire-and-string puzzles: Closed string subgroup: The pieces of string consist of
Disentanglement_puzzle
JSON subset for geospatial data
containing all metadata, Polygon, LineString, Point elements, arcs and properties is defined as follows: { "type": "Topology", "transform": { "scale": [1, 1]
GeoJSON
American theoretical physicist
1951) is an American theoretical physicist known for his contributions to string theory, topological quantum field theory, general relativity and various
Edward_Witten
Mathematics timeline
matter of manifolds is a strand common to algebraic topology, differential topology and geometric topology. Terminology: By this period manifolds are generally
Timeline_of_manifolds
Branch of mathematics
'topology is rubber-sheet geometry'. Subfields of topology include geometric topology, differential topology, algebraic topology and general topology.
Geometry
Symplectic topology tool
on the Floer homology of a cotangent bundle that correspond to the string topology operations on the homology of the loop space of the underlying manifold
Floer_homology
Aspect of theoretical physics
charged) and F1 string and NS5 brane among other objects. The mathematical treatment of type IIA string theory belongs to symplectic topology and algebraic
Type_II_string_theory
In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or J-holomorphic curve) is a smooth map, from a Riemann surface into
Pseudoholomorphic_curve
Concept of universality in physical science
consistent quantum theory of gravity should include topology change as a dynamical process. String theory is usually formulated with perturbation theory
Background_independence
Topologically stable solution of a partial differential equation
for line (string) defects in liquid crystals that can cross each other without entanglement. It was a non-trivial application of topology that first
Topological_defect
26-dimensional string theory
Bosonic string theory is the original version of string theory, developed in the late 1960s. It is so called because it contains only bosons in the spectrum
Bosonic_string_theory
Number of times a curve wraps around a point in the plane
objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and
Winding_number
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
Collection of knots that do not intersect, but may be linked
it an "ℓ-component string link". A string link need not be a braid – it may double back on itself, such as a two-component string link that features an
Link_(knot_theory)
Mathematical concept
Strings are further classified into open and closed. The topology of the worldsheet of an open string is R × I {\displaystyle \mathbb {R} \times I} , where
Worldsheet
Low energy theories not compatible with string theory
This is in contrast with the so-called "string theory landscape" that are known to be compatible with string theory, which is hypothesized to be a consistent
Swampland_(physics)
History of maths
Homotopical algebra; Topology using categories, including algebraic topology, categorical topology, quantum topology, low-dimensional topology; Categorical logic
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Theorem in topology
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. Brouwer. It states that for any continuous function f {\displaystyle
Brouwer_fixed-point_theorem
Mathematical object
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable
Spectrum_(topology)
Infinite-dimensional group in topology
In topology, a branch of mathematics, a string group is an infinite-dimensional group String ( n ) {\displaystyle \operatorname {String} (n)} introduced
String_group
Study of mathematical knots
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a
Knot_theory
Continuous deformation between two continuous functions
In topology, two continuous functions from one topological space to another are called homotopic (from Ancient Greek: ὁμός homós 'same, similar' and τόπος
Homotopy
Topological invariant in mathematics
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré
Euler_characteristic
Characteristic classes of vector bundles
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated
Chern_class
each instance. There are two deployment topologies for Caching: Dedicated Co-located In the dedicated topology, you define a worker role that is dedicated
Windows_Azure_Caching
cohomology which consist of 1-form connections and 2-form curvatures. The topology of a U ( 1 ) {\displaystyle U(1)} bundle is classified by its Chern class
Bundle_gerbe
Concept in string theory
previously indistinguishable. They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten. The rigorous
Gromov–Witten_invariant
Graphical representation of a morphism
between category theory and low-dimensional topology, a combinatorial definition is necessary to formalise string diagrams in computer algebra systems and
String_diagram
Generalization of a differentiable manifold
In differential topology, a branch of mathematics, a stratifold is a generalization of a differentiable manifold where certain kinds of singularities are
Stratifold
Hypothetical physical concept
(1994). "Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory". Nuclear Physics B. 416 (2): 414. arXiv:hep-th/9309097
Theory_of_everything
Set of mathematical concepts in quantum gravity
fashion. String theory uses quantum geometry to describe exotic phenomena such as T-duality and other geometric dualities, mirror symmetry, topology-changing
Quantum_geometry
Finite or infinite ordered list of elements
\mathbb {N} })=x_{i}} . Then the product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all
Sequence
Mathematician
Abouzaid, Mohammed; Seidel, Paul (2010). "An open string analogue of Viterbo functoriality". Geometry & Topology. 14 (2): 627–718. arXiv:0712.3177. doi:10.2140/gt
Mohammed_Abouzaid
of topology topics List of general topology topics Glossary of general topology List of topologies Topological property List of algebraic topology topics
Lists_of_mathematics_topics
Field theory involving topological effects in physics
Topological quantum number Topological quantum computer Topological string theory Arithmetic topology Cobordism hypothesis Atiyah, Michael (1988a). "New invariants
Topological quantum field theory
Topological_quantum_field_theory
Intersection graph for curves in the plane
"Recognizing string graphs in NP", Journal of Computer and System Sciences, 67 (2): 365–380, doi:10.1016/S0022-0000(03)00045-X. Sinden, F. W. (1966), "Topology of
String_graph
System for dealing with email spam
the client has to concatenate a random number with a string several times and hash this new string. It then has to do so over and over until a hash beginning
Hashcash
Ten-dimensional supergravity
important role in modern physics since it is the low-energy limit of type IIB string theory. After supergravity was discovered in 1976, there was a concentrated
Type_IIB_supergravity
Secondary characteristic classes of 3-manifolds
Bosonic string theory Superstring theory Type I string Type II string Type IIA string Type IIB string Heterotic string N=2 superstring F-theory String field
Chern–Simons_form
applications in enumerative geometry, random matrix theory, mathematical physics, string theory, knot theory. The topological recursion is a construction in algebraic
Topological_recursion
commutative algebra in statistics. Algebraic topology a branch that uses tools from abstract algebra for topology to study topological spaces. Algorithmic
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Property of a mathematical space
also the dimension of the tangent vector space at any point. In geometric topology, the theory of manifolds is characterized by the way dimensions 1 and 2
Dimension
Condensed matter phenomenon; vortex-like magnetic quasiparticle
interpretations with subtle differences. Most descriptions include the notion of topology – a categorization of shapes and the way in which an object is laid out
Magnetic_skyrmion
Branch of mathematics
is closely related to, and is sometimes taken to include, differential topology, which concerns itself with properties of differentiable manifolds that
Differential_geometry
Belgian mathematician
vanishing cycles, central extensions of reductive groups, geometry and topology of braid groups, providing the modern axiomatic definition of Shimura varieties
Pierre_Deligne
PSTN network topology is the switching network topology of a telephone network connected to the public switched telephone network (PSTN). In the United
PSTN_network_topology
structures on the Hochschild complex. It is of importance in relation with string theory. Some sources cite Tamarkin's proof as the first real proof. Piecewise
Deligne's conjecture on Hochschild cohomology
Deligne's_conjecture_on_Hochschild_cohomology
Set of points on a line segment with certain topological properties
generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). The Cantor
Cantor_set
Unique knot with a crossing number of four
William Thurston (March 2002), "7. Computation of volume", The Geometry and Topology of Three-Manifolds, p. 165, archived from the original (PDF) on 2020-07-27
Figure-eight knot (mathematics)
Figure-eight_knot_(mathematics)
Converts output of a photovoltaic panel into a utility frequency alternating current
microinverter developers, who have introduced a variety of conversion topologies with lowered storage requirements, some using the much less capable but
Solar_inverter
Property holding for typical examples
}} has a topology with basic open sets [ σ ] = { f : σ ≼ f } {\displaystyle [\sigma ]=\{f:\sigma \preccurlyeq f\}} for every finite string of natural
Generic_property
Framework of superstring theory
Quantum Topology. 3 (1): 1–137. arXiv:1101.3216. doi:10.4171/QT/26. S2CID 119248828. Woit, Peter (2006). Not Even Wrong: The Failure of String Theory and
M-theory
Topological space with only one nontrivial homotopy group
In mathematics, specifically algebraic topology, an Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group. Let G be a
Eilenberg–MacLane_space
Method of fastening or securing linear material
practical uses, as well as their topological intricacy, studied in the area of topology known as knot theory. Knots and knotting have been used and studied throughout
Knot
Russian physicist
Nakajima and other mathematicians to explore hidden algebraic structures in topology and in quantum field theory. Interview at MIPT (in Russian)：https://vk
Sergei_Gukov
Generalized alphabetical order
Lexicographic order topology on the unit square Lexicographic ordering in tensor abstract index notation Lexicographically minimal string rotation Leximin
Lexicographic_order
Russian-American mathematician (born 1939)
in the representation theory of infinite-dimensional Lie groups and in topology. He was born in Kazan, Soviet Tatarstan. Fuchs received in 1964 his Russian
Dmitry_Fuchs
Additional mathematical object
with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the
Mathematical_structure
similarities at certain string positions. Schemata are a special case of cylinder sets, forming a basis for a product topology on strings. In other words
Schema_(genetic_algorithms)
Russian American mathematician
contributions have been in symplectic topology and singularity theory, as well as their relation to topological string theories. Givental graduated from the
Alexander_Givental
invariants, which find application in enumerative geometry and type IIA string theory. The idea of stable maps was proposed by Maxim Kontsevich around
Stable_map
Construct in mathematics
gerbe (/dʒɜːrb/; French: [ʒɛʁb]) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud (Giraud 1971) following ideas of
Gerbe
Neil Strickland, the Witten genus can be lifted to topology. That is, there is a map from the string bordism spectrum to tmf, the Ando–Hopkins–Rezk orientation
Topological_modular_forms
Concept in algebraic geometry
In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of
Quantum_cohomology
Mathematical conjecture
02632 [math.DG]. McDuff, Dusa (2012). J-holomorphic curves and symplectic topology. Salamon, D. (Dietmar) (2nd ed.). Providence, R.I.: American Mathematical
Mirror_symmetry_conjecture
Theories of higher-dimensional general relativity
Calabi–Yau manifolds in string theory, reduce the apparent number of dimensions to four at observable scales. The geometry and topology of the compactified
Higher-dimensional Einstein gravity
Higher-dimensional_Einstein_gravity
Objects of certain abelian categories associated to topological spaces
fundamental mathematical objects at the crossroads of algebraic geometry, topology, analysis and differential equations. They also play an important role
Perverse_sheaf
Mathematical space with a notion of distance
metric, such balls form a basis for a topology on X, but this topology need not be metrizable. For example, the topology induced by the quasimetric on the
Metric_space
Study of categorified structures
of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra. A first step towards defining higher
Higher-dimensional_algebra
Generalization of a manifold
Greene, Morrison & Strominger (1995), this provides the string-theoretic description of the topology-change via the conifold transition originally described
Conifold
plane. The Cayley complex is an infinite string of spheres. Hatcher, Allen (2001-12-03). Algebraic Topology (1st ed.). Cambridge: Cambridge University
Presentation_complex
Conjectured duality combining S-duality and T-duality
This is the union of all the S-duality and T-duality available in that topology. The narrow meaning of the word "U-duality" is one of those dualities that
U-duality
Generalized manifold
explaining the origin of the word "orbifold" In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of
Orbifold
Graphical language for quantum processes
for reasoning about linear maps between qubits, which are represented as string diagrams called ZX-diagrams. A ZX-diagram consists of a set of generators
ZX-calculus
American mathematician
April 18, 1958) is an American mathematician known for work in algebraic topology. He received his PhD from Northwestern University in 1984 under the direction
Michael_J._Hopkins
Transforming a function in such a way that it only takes a single argument
correspondence to quantum mechanics, to cobordisms in algebraic topology, and to string theory. The linear type system, and linear logic are useful for
Currying
Instruction for x86 microprocessors
ID string as well as the highest calling parameter that the CPU implements. .intel_syntax noprefix .text .m0: .string "CPUID: %x\n" .m1: .string "Largest
CPUID
American mathematician and billionaire (1938–2024)
contributed to the development of string theory by providing a theoretical framework to combine geometry and topology with quantum field theory. In 1994
Jim_Simons
Computational problems no algorithm can solve
1016/S0167-6911(00)00049-9. ISSN 0167-6911. Stillwell, John (1993), Classical Topology and Combinatorial Group Theory, Graduate Texts in Mathematics, vol. 72
List_of_undecidable_problems
In string theory, high-energy scattering amplitudes describe the interactions of strings at extreme energy scales, such as the Planck scale. Unlike point-particle
High-energy string scattering amplitudes
High-energy_string_scattering_amplitudes
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