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In mathematics, rank-finiteness for fusion categories is a collection of related theorems and conjectures about structures related to fusion categories
Rank-finiteness
Linear operator in functional analysis
mathematics, a finite-rank operator is a bounded linear operator between Banach spaces whose range is finite-dimensional. Finite-rank operators are matrices
Finite-rank_operator
Mathematics award
Bruillard, Paul; Ng, Siu-Hung; Rowell, Eric C.; Wang, Zhenghan (2016). "Rank-finiteness for modular categories" (PDF). Journal of the American Mathematical
Alexanderson_Award
In algebra, module with a finite generating set
quotient of a free module of finite rank. If a set S generates a module that is finitely generated, then there is a finite generating set that is included
Finitely_generated_module
Dimension of the column space of a matrix
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal number
Rank_(linear_algebra)
Concept in model theory
groups of finite Morley rank (see below). A group of finite Morley rank is an abstract group G such that the formula x = x has finite Morley rank for the
Stable_group
Mathematical property
infinite groups. Special cases of groups with finiteness properties are finitely generated and finitely presented groups. Given an integer n ≥ 1, a group
Finiteness properties of groups
Finiteness_properties_of_groups
Theorem classifying finite simple groups
for completing the classification of finite simple groups, consisting of the following 16 steps: Groups of low 2-rank. This was essentially done by Gorenstein
Classification of finite simple groups
Classification_of_finite_simple_groups
In linear algebra, relation between 3 dimensions
{\displaystyle V} is finite dimensional. Then rank ( T ) + nullity ( T ) = dim V , {\displaystyle \operatorname {rank} (T)~+~\operatorname
Rank–nullity_theorem
Notion in abstract algebra
{\displaystyle M} has finite uniform dimension (=finite rank) n {\displaystyle n} if and only if the injective hull of M {\displaystyle M} is a finite direct sum
Injective_hull
Number of independent rational basis points with infinite order
then some point in a finite basis must have infinite order. The number of independent basis points with infinite order is the rank of the curve. In mathematical
Rank_of_an_elliptic_curve
Type of monoidal category
representation, the Bruguières modularity theorem, the Verlinde formula, the rank-finiteness theorem, the Schauenburg-Ng theorem, and Müger's theorem. A modular
Modular_tensor_category
Unproved conjecture in mathematics
known to be well-defined and the finiteness of Ш ( E ) {\displaystyle (E)} is known when additionally the analytic rank is at most 1; i.e., if L ( E , s
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Commutative group where every element is the sum of elements from one finite subset
where n ≥ 0 is the rank, and the numbers q1, ..., qt are powers of (not necessarily distinct) prime numbers. In particular, G is finite if and only if n
Finitely generated abelian group
Finitely_generated_abelian_group
Finite collection of distinct objects
numerical concept of finiteness.) Ia-finite. For every partition of S {\displaystyle S} into two sets, at least one of the two sets is I-finite. (A set with this
Finite_set
Smallest cardinality of a generating set for a group
{\displaystyle \operatorname {rank} (G)=\min\{|X|:X\subseteq G,\langle X\rangle =G\}.} If G is a finitely generated group, then the rank of G is a non-negative
Rank_of_a_group
Chinese-American mathematician
Paul; Ng, Siu-Hung; Rowell, Eric C.; Wang, Zhenghan (2015-07-21). "Rank-finiteness for modular categories". Journal of the American Mathematical Society
Zhenghan_Wang
Abstraction of linear independence of vectors
bases or circuits, rank functions, closure operators, and closed sets or flats. In the language of partially ordered sets, a finite simple matroid is equivalent
Matroid
Abelian group with no non-trivial torsion elements
(2003), "The classification problem for torsion-free abelian groups of finite rank", J. Am. Math. Soc., 16 (1): 233–258, doi:10.1090/S0894-0347-02-00409-5
Torsion-free_abelian_group
Discrete analog of a derivative
A finite difference is a mathematical expression of the form f(x + b) − f(x + a). Finite differences (or the associated difference quotients) are often
Finite_difference
Group type in algebra
inherits some finiteness property of a space. Geometric group theory studies the connections between algebraic properties of finitely generated groups
Finitely_generated_group
Commutative group (mathematics)
research: Amongst torsion-free abelian groups of finite rank, only the finitely generated case and the rank 1 case are well understood. There are many unsolved
Abelian_group
Direct summand of a free module (mathematics)
M has constant rank (as defined below). Let A be a commutative ring. If B is a (possibly non-commutative) A-algebra that is a finitely generated projective
Projective_module
Set theory concept
of all members of the set. In particular, the rank of the empty set is zero, and every ordinal has a rank equal to itself. The sets in V are divided into
Von_Neumann_universe
Group whose Cayley graph is an initially subamenable graph
subamenable graph, or equivalently a subgroup of an ultraproduct of finite-rank symmetric groups such that every two distinct elements of the group have
Sofic_group
Coordinate-free definition of a tensor
each product is of n vectors from a finite-dimensional vector space of dimension d. In indices, a tensor of rank 1 is a tensor of the form T i j … k ℓ
Tensor_(intrinsic_definition)
of sectional 2-rank at most 4. It is part of the classification of finite simple groups. Finite simple groups of section 2 with rank at least 5 have
Gorenstein–Harada_theorem
South Korean mathematician (born 1963)
arithmetic homotopy theory to the study of Diophantine problems, especially to finiteness theorems of the Faltings–Siegel type. His work was featured in 2017 in
Minhyong_Kim
Join-meet algebra on matroid flats
lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumption of finiteness. Geometric
Geometric_lattice
The group of K-rational points of an abelian variety is a finitely-generated abelian group
the finiteness of this group is a necessary condition for E ( Q ) {\displaystyle E(\mathbb {Q} )} to be finitely generated; and it shows that the rank is
Mordell–Weil_theorem
In mathematical finite group theory, a rank 3 permutation group acts transitively on a set such that the stabilizer of a point has 3 orbits. The study
Rank_3_permutation_group
Connectivity measure in graph theory
height of a regular language L equals the minimum cycle rank among all nondeterministic finite automata with ε-moves accepting L. Proofs of this theorem
Cycle_rank
In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of
List_of_finite_simple_groups
cyclic. Informally, these are the groups that resemble rank 1 groups of Lie type over a finite field of characteristic 2. Janko (1972) defined thin groups
Thin group (finite group theory)
Thin_group_(finite_group_theory)
Disproven mathematical theory concerning Banach-Tarski and amenable groups
satisfies the property F ∞ {\displaystyle F_{\infty }} , which is a stronger finiteness property. Adian, Sergei (1982), "Random walks on free periodic groups"
Von_Neumann_conjecture
Number of elements in a subset of a commutative group
dimension rank A. For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion
Rank_of_an_abelian_group
Topological structure in number theory
first occurs. If the rank, the μ-invariant, and the λ-invariant of a finitely generated module all vanish, the module is finite (and conversely); in other
Iwasawa_algebra
Use of machine learning to rank items
Learning to rank (LTR) or machine-learned ranking (MLR) is the application of machine learning, often supervised, semi-supervised or reinforcement learning
Learning_to_rank
Maximum size of an independent set of the matroid
mathematical theory of matroids, the rank of a matroid is the maximum size of an independent set in the matroid. The rank of a subset S of elements of the
Matroid_rank
groups with finite Prüfer rank are more amenable to analysis. Specifically in the case of finitely generated pro-p groups, having finite Prüfer rank is equivalent
Prüfer_rank
Three groups
(see Whitehead torsion) or the projective class group of F (see Wall's finiteness obstruction) is trivial, though it easily shown that F satisfies the strong
Thompson_groups
Set of isolated points in the spectrum of an operator with finite-rank Riesz projectors
of isolated points of its spectrum such that the rank of the corresponding Riesz projector is finite. The discrete spectrum can also be defined as the
Discrete spectrum (mathematics)
Discrete_spectrum_(mathematics)
Basis of generic programming
B. (1 January 1999). "Principality and decidable type inference for finite-rank intersection types". Proceedings of the 26th ACM SIGPLAN-SIGACT Symposium
Parametric_polymorphism
Theorem in Lie theory in mathematics
local rigidity and finite generation of lattices the Kazhdan-Margulis theorem is an important ingredient in the proof of Wang's finiteness theorem. If G {\displaystyle
Kazhdan–Margulis_theorem
Template that specifies one or more axioms
is an instance of the schema. Axiom schemata are commonly used to give finite descriptions of theories whose axioms include infinitely many formulas.
Axiom_schema
Algebraic curve in mathematics
combination of a finite number of fixed points. The theorem however doesn't provide a method to determine any representatives of E(Q)/mE(Q). The rank of E(Q) is
Elliptic_curve
remarkable feature is that for every Nichols algebra (under sufficient finiteness conditions) there exists a generalized root system with a set of roots
Nichols_algebra
Statement in abstract algebra
{\displaystyle M/tM} is a finitely generated torsion-free module, and such a module over a commutative PID is a free module of finite rank. Thus M / t M {\displaystyle
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
Branch of mathematics that studies the properties of groups
finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite)
Group_theory
property if all finitely generated free modules over R have a well-defined rank. In the case of fields, the IBN property is the fact that finite-dimensional
Invariant_basis_number
Partially ordered set in Mathematics
In mathematics, a ranked poset is a partially ordered set in which one of the following (non-equivalent) conditions hold: it is a graded poset, or a poset
Ranked_poset
Concept in mathematics
In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections:
Complex_reflection_group
Error-correcting code
erasures. A rank code is an algebraic linear code over the finite field G F ( q N ) {\displaystyle GF(q^{N})} similar to Reed–Solomon code. The rank of the
Rank_error-correcting_code
Depth of nesting of quantifiers in a formula
quantifier rank of a formula is the depth of nesting of its quantifiers. It plays an essential role in model theory. The quantifier rank is a property
Quantifier_rank
In mathematics, a module that has a basis
the rank of the free module M {\displaystyle M} . If this cardinality is finite, the free module is said to be free of finite rank, or free of rank n if
Free_module
Generalized alphabetical order
used in combinatorics, orders subsets of a given finite set by assigning a total order to the finite set, and converting subsets into increasing sequences
Lexicographic_order
Collection of mathematical objects
{\displaystyle \emptyset } ) and the latter has no elements at all. A set is finite if there exists a natural number n {\displaystyle n} such that the first
Set_(mathematics)
there exist only finitely many pro-p groups of coclass r. This finiteness result is fundamental for the classification of finite p-groups by means of
Pro-p_group
Concept in linear algebra
where r = rank A {\displaystyle r=\operatorname {rank} A} is the rank of A {\displaystyle A} . Every finite-dimensional matrix has a rank decomposition:
Rank_factorization
This table shows a summary of regular polytope counts by rank. There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its
List_of_regular_polytopes
Specific algebraic group
isogenous, non-isomorphic tori. Over the finite field F q {\displaystyle \mathbb {F} _{q}} there are two rank-1 tori: the split one, of cardinality q −
Algebraic_torus
Algebraic structure
In mathematics, a quasithin group is a finite simple group that resembles a group of Lie type of rank at most 2 over a field of characteristic 2. The classification
Quasithin_group
If G is a finitely generated group with exponent n, is G necessarily finite?
finitely generated group with exponent n, is G necessarily finite? It turns out that this problem can be restated as a question about the finiteness of
Burnside_problem
Concept in geometry
singleton or the whole l. Finite polar spaces (where P is a finite set) are also studied as combinatorial objects. A polar space of rank two is a generalized
Polar_space
Branch of logic
interpretations (semantics). Finite model theory is a restriction of model theory to interpretations on finite structures, which have a finite universe. Since many
Finite_model_theory
Graph width parameter used in graph theory
matrix; for the purposes of rank-width, this matrix is defined over the finite field GF(2) rather than using real numbers. The rank-width of a graph is the
Rank-width
American mathematician
Griggs Thompson with the thesis Characterizations of Some Finite Simple Groups with Small 2-Rank. From 1972 to 2017, he was a professor at Rutgers University
Richard_Lyons_(mathematician)
Theorem in group theory
be finitely generated groups and let A∗B be the free product of A and B. Then rank(A∗B) = rank(A) + rank(B). It is obvious that rank(A∗B) ≤ rank(A) +
Grushko_theorem
Mathematics concept
group of countable rank (given by 1 plus the Euler characteristic of the quotient graph). The Cayley graph of a free group of finite rank, with respect to
Free_group
Theorem in mathematics
be seen as a special case of the constant rank theorem, which states that a smooth map with constant rank near a point can be put in a particular normal
Inverse_function_theorem
Type of continuous linear operator
{\displaystyle K(X,Y)} . Every finite-rank operator is compact. Indeed, if the range of T : X → Y {\displaystyle T:X\to Y} is finite-dimensional, then the image
Compact_operator
Computation model defining an abstract machine
into discrete cells, each of which can hold a single symbol drawn from a finite set of symbols called the alphabet of the machine. It has a "head" that
Turing_machine
Natural number
4-dimensions. There are also 5 compact hyperbolic Coxeter groups, or 4-prisms, of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations
5
Theorem in geometric group theory
Gk/Gk+1 is a finitely generated abelian group. The Bass–Guivarc'h formula states that the order of polynomial growth of G is d ( G ) = ∑ k ≥ 1 k rank ( G k
Gromov's theorem on groups of polynomial growth
Gromov's_theorem_on_groups_of_polynomial_growth
Finite sets whose elements are all hereditarily finite sets
can be built by a finite number of applications of these two rules are hereditarily finite. This class of sets is naturally ranked by the number of bracket
Hereditarily_finite_set
Number of vectors in any basis of the vector space
{\displaystyle F} -vector space. An important result about dimensions is given by the rank–nullity theorem for linear maps. If F / K {\displaystyle F/K} is a field
Dimension_(vector_space)
Set of all limit points of a set
{\displaystyle X^{\alpha +1}=X^{\alpha }} is called the Cantor–Bendixson rank of X . {\displaystyle X.} This investigation into the derivation process
Derived_set_(mathematics)
Matroid associated with a group
paper (1973b) Dowling gave an intrinsic definition of the rank-n Dowling lattice of any finite group G. Let S be the set {1,...,n}. A G-labelled set (T
Dowling_geometry
Pictorial representation of symmetry
{\displaystyle i=1,2,3...} for each rank. Very-extended groups are Lorentz groups, defined by adding three nodes to the finite groups. The E8, E7, E6, F4, and
Dynkin_diagram
Filling in missing entries of a matrix
or is low-rank. For example, one may assume the matrix has low-rank structure, and then seek to find the lowest rank matrix or, if the rank of the completed
Matrix_completion
involutions resemble semisimple elements. Groups of characteristic 2 type and rank at least 3 are classified by the trichotomy theorem. A group is said to be
Characteristic_2_type
President of the United States (2017–2021; since 2025)
ranked him third-worst. He was ranked near the bottom in all categories except for luck, willingness to take risks, and party leadership, and ranked last
Donald_Trump
Random process independent of past history
there is a unique stationary distribution π. In this case Pk converges to a rank-one matrix in which each row is the stationary distribution π: lim k → ∞
Markov_chain
Mathematical theorem
the ring of invariants of a finite group acting linearly on a complex vector space is Cohen-Macaulay, so it is a finite-rank free module over a polynomial
Chevalley–Shephard–Todd theorem
Chevalley–Shephard–Todd_theorem
Group that admits a formal description in terms of reflections
1016/0022-4049(94)90019-1. Brink, Brigitte; Howlett, Robert B. (1993). "A finiteness property and an automatic structure for Coxeter groups". Mathematische
Coxeter_group
Theorem in group theory
divides the finite simple groups of characteristic 2 type and rank at least 3 into three classes. It was proved by Michael Aschbacher for rank 3, and by
Trichotomy_theorem
varieties with volume bounded below by V {\displaystyle V} . The Higher Rank Finite Generation is established by Liu-Xu-Zhuang. As a consequence, X n , V
K-stability_of_Fano_varieties
Limitative results in mathematical logic
formal systems that can express basic arithmetic, a complete and consistent finite list of axioms can never be created: each time an additional, consistent
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Discrete group type in group theory
important class arises from Riemannian symmetric spaces of rank 1: the n-sphere Sn, corresponding to finite reflection groups, the Euclidean space Rn, corresponding
Reflection_group
Natural number
Vinberg polytopes of rank n + 4 mirrors, where there is one unique figure with eleven facets. On the other hand, such figures with rank n + 3 mirrors exist
7
Mathematical-logic system
re-creation until the next call makes its existence possible by having two finite lambda-terms E inside it re-create it on the fly later as needed. This self-applicational
Lambda_calculus
Regulations on arms and ammunition
society outside of the military, and gun ownership and gun-related deaths rank among the lowest in the world. South Korea has strict gun policies. Hunting
Overview of gun laws by nation
Overview_of_gun_laws_by_nation
Ability to make choices voluntarily
independent senses of will. In addition, one of the most important ("first rank") diagnostic symptoms of schizophrenia is the patient's delusion of being
Free_will
Compact operator for which a finite trace can be defined
dt=\sum _{i}\lambda _{i}.} Every finite-rank operator is a trace-class operator. Furthermore, the space of all finite-rank operators is a dense subspace
Trace_class
J(P)} to be the subgroup generated by the abelian subgroups of P of maximal rank. More often the Thompson subgroup J ( P ) {\displaystyle J(P)} is defined
Thompson_subgroup
the question of the p-rank of the Jacobian variety J of C; the p-rank is bounded by the rank of H, specifically it is the rank of the Frobenius mapping
Hasse–Witt_matrix
Curves of genus > 1 over the rationals have only finitely many rational points
field with good reduction outside a fixed finite set of places. Aleksei Parshin showed that Shafarevich's finiteness conjecture would imply the Mordell conjecture
Faltings'_theorem
Polygon with an infinite number of sides
polygon is a polygon with an infinite number of sides. Apeirogons are the rank-2 case of infinite polytopes. In some literature, the term "apeirogon" may
Apeirogon
Distinction between nominal, ordinal, interval and ratio variables
of some rule. Rank orders represent ordinal scales and are frequently used in research relating to qualitative phenomena. A student's rank in his graduation
Level_of_measurement
Algebraic manipulation of "true" and "false"
set is cofinite, while the union of two finite sets is finite. Intersection behaves like union with "finite" and "cofinite" interchanged. This example
Boolean_algebra
travel, tourism, insurance
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