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Circulant matrix
A circulant matrix is an n × n matrix in which each row is the row above shifted right with wrap-around; the discrete Fourier transform diagonalizes it, enabling fast multiplication and solving by FFT.
Class field theory
Class field theory is the branch of algebraic number theory that describes all abelian extensions of local and global fields, centered on the reciprocity map.
Class formation
A class formation is a cohomological datum, introduced by Emil Artin and John Tate, from which the main theorems of class field theory can be derived abstractly.
Class number
The class number h_K of a number field K is the order of its ideal class group, and h_K = 1 exactly when K has unique factorization.
Class number formula
The class number formula expresses the residue at s = 1 of a number field's Dedekind zeta function through its class number, regulator, discriminant, and roots of unity.
Class number problem
The class number problem, or class number conjecture, asks which imaginary quadratic fields have each class number; posed by Gauss in 1801 and settled for N = 100.
Classification of finite simple groups
The classification of finite simple groups (CFSG) is a theorem listing every finite simple group as cyclic, alternating, of Lie type, or one of 26 sporadic groups.
Classification of Kac–Moody algebras
The classification of Kac–Moody algebras divides their generalized Cartan matrices into finite, affine, and indefinite types; finite recovers the nine classical semisimple families, and hyperbolic matrices are too numerous to classify.
Clifford algebra
A Clifford algebra is an associative algebra in which vectors square to scalars given by a quadratic form, generalizing complex numbers and quaternions and underlying spinors.
Closed-form expression
A closed-form expression, also called an analytical solution, evaluates a quantity using a finite number of standard operations such as roots, exponentials, and logarithms, without limits or infinite sums.
Closest vector problem
The closest vector problem (CVP) asks, given a basis for a lattice and a target vector, for the closest lattice vector; it is NP-hard and underpins lattice-based cryptography.
Closure (mathematics)
Closure in mathematics is the property that applying an operation to elements of a set produces results that stay within that set; examples include topological and transitive closure.
Closure operator
A closure operator is a function c on subsets of a set that is extensive, monotone, and idempotent, so c(X) is the smallest closed set containing X.
Cluster algebra
Cluster algebra is a commutative algebra whose generators, called cluster variables, are produced from an initial seed by iterative mutation, created by Fomin and Zelevinsky in 2000.
Coadjoint orbit
A coadjoint orbit is the orbit of a Lie group in the dual of its Lie algebra, carrying a canonical symplectic structure central to Kirillov's orbit method.
Coalgebra
Coalgebra is a mathematical structure dual to an algebra, using comultiplication instead of multiplication, and foundational to Hopf algebras and state-based systems in computer science.
Coding theory
Coding theory, also called code theory, is the branch of mathematics and engineering that designs error-correcting codes; its methods appear in 5G, QR codes, and deep space communication.
Coefficient
A coefficient, or multiplicative factor, is the number multiplying a variable in an algebraic term, such as the 3 in 3x or the leading term of a polynomial.
Cofinality
Cofinality, written cf(α), is a set theory concept measuring how hard an ordinal is to approach from below: the least order type of a cofinal subset of it.
Cohen structure theorem
The Cohen structure theorem is a 1946 result in commutative algebra describing every complete local ring as a quotient of a formal power series ring.
Cohen–Macaulay ring
A Cohen–Macaulay ring, also called a Macaulay ring, is a Noetherian commutative ring in which depth equals Krull dimension locally, a standard measure of how well behaved a ring is.
Coherent sheaf
A coherent sheaf is a sheaf of modules over a ringed space that is locally finitely generated with finitely generated relations, introduced by Cartan in 1944 and Serre in 1955.
Coherent sheaf cohomology
Coherent sheaf cohomology, or coherent cohomology, studies coherent sheaves on schemes via the right derived functors of the global-section functor; it was founded in Jean-Pierre Serre's 1955 paper FAC.
Cokernel
The cokernel of a homomorphism f: M → N between modules is the quotient N/im(f), the dual of the kernel, and is zero exactly when f is surjective.
Colimit (category theory)
A colimit in category theory is a universal construction producing, from a diagram, a new object receiving maps from every object in the most economical way; it is the dual of a limit.
Collatz conjecture
The Collatz conjecture, also called the 3x + 1 problem, is an unsolved number theory problem stating that iterating 3n + 1 for odd numbers and n/2 for even numbers carries every positive integer to 1.
Combinatorial representation theory
Combinatorial representation theory describes representations of groups and algebras by explicit combinatorial objects such as tableaux and permutations, so that dimensions and character values become counts and weighted sums.
Combinatorics on words
Combinatorics on words, also known as word combinatorics, is the branch of discrete mathematics that studies sequences of symbols from a finite alphabet, including their patterns, repetitions, and equations.
Communication-avoiding algorithms
Communication-avoiding algorithms are numerical linear algebra algorithms restructured to minimize data movement between memory levels and processors, because communication is the bottleneck; examples include CALU and TSQR.
Commutation theorem for traces
The commutation theorem for traces is a result in von Neumann algebra theory stating that an algebra with a semifinite trace is the commutant of its opposite algebra.