289 articles
Conformal algebra
A conformal algebra is the Lie algebra of infinitesimal conformal transformations of a geometry or field theory, central to conformal field theory and the operator product expansion.
Connes classification of type III factors
The Connes classification of type III factors is the 1973 partition of type III von Neumann factors into III₀, IIIλ, and III₁, defined by Alain Connes via the S and T invariants from modular theory.
Connes embedding problem
The Connes embedding problem, also called Connes' embedding conjecture, asked whether every tracial von Neumann algebra embeds into an ultrapower of the hyperfinite II1 factor; it was refuted in 2020.
Construction and structure of Kac–Moody algebras
Kac–Moody algebras are generally infinite-dimensional Lie algebras built from a generalized Cartan matrix by generators and relations, introduced independently by Victor Kac and Robert Moody in the late 1960s.
Cotangent complex
The cotangent complex is a homological invariant of a ring map or scheme morphism that replaces Kähler differentials for nonsmooth maps and controls first-order deformation theory.
Coxeter group
A Coxeter group, or Coxeter system, is a group generated by involutions satisfying relations (r_i r_j)^{n_ij} = 1, introduced by H. S. M. Coxeter in 1934 through reflection groups.
Crossed product of von Neumann algebras
The crossed product of von Neumann algebras, or Verschränktes Produkt, is a von Neumann algebra M ⋊ G built from M and a group action, analogous to the semidirect product.
Crystal base
A crystal base is a combinatorial structure attached to a representation of a quantum group, introduced in the early 1990s by Masaki Kashiwara and George Lusztig.
Cyclic homology
Cyclic homology is a homological invariant of associative algebras, created by Connes and by Feigin and Tsygan in the 1980s, whose periodic variant recovers algebraic de Rham cohomology.
Cylindric algebra
A cylindric algebra is a Boolean algebra equipped with cylindrifications and diagonal elements, introduced by Alfred Tarski in 1948–1952 as an algebraic counterpart of first-order predicate logic.
De Morgan's laws
De Morgan's laws, also called De Morgan's theorem, are a pair of rules stating that the negation of AND is the OR of negations, and vice versa.
Decomposition theorem
The decomposition theorem is a 1981 result in algebraic geometry, proved by Beilinson, Bernstein, Deligne, and Gabber, stating that derived direct images of intersection cohomology complexes split into direct sums.
Deformation quantization
Deformation quantization is a mathematical technique for constructing quantum theories by deforming the algebra of functions on a classical phase space into a noncommutative star product.
Deligne–Mumford stack
A Deligne–Mumford stack is an algebraic stack locally modeled on schemes by an étale surjective covering, introduced by Deligne and Mumford in 1969 for the moduli of algebraic curves.
Derived algebraic geometry
Derived algebraic geometry, or DAG, is a branch of mathematics studying derived schemes and stacks, classical schemes enhanced with homotopical, higher categorical data, with foundations laid in the 2000s by Toën, Vezzosi, and Lurie.
Derived category
A derived category is a mathematical construction that replaces an abelian category by a category of chain complexes, invented by Grothendieck and Verdier in the early 1960s.
Derived functor
A derived functor, in left and right variants, is a homological algebra construction measuring how far a functor fails to be exact, computed by applying it to a projective or injective resolution and taking homology.
Descent theory
Descent theory is a set of techniques in algebraic geometry and category theory for recovering an object from its pullback along a covering morphism, presented in general form by Alexander Grothendieck in 1960.
Direct integral
A direct integral assembles a family of Hilbert spaces, von Neumann algebras, or operators indexed by a measure space, generalizing the direct sum of Hilbert spaces.
Dirichlet algebra
A Dirichlet algebra is a uniform algebra of continuous functions on a compact space whose real parts are dense in all real continuous functions, defined by Gleason in 1958.
Divisor (algebraic geometry)
In algebraic geometry, a divisor is a formal object on an algebraic variety or scheme recording codimension-1 subvarieties with integer multiplicities, tied to line bundles through the Picard group.
Drinfeld center
The Drinfeld center of a monoidal category is a braided monoidal category whose objects are pairs with half-braidings, the categorical counterpart of Drinfeld's quantum double of a Hopf algebra.
Drinfeld double
The Drinfeld double, also called the quantum double, is a Hopf algebra built from a Hopf algebra and its dual, introduced by Vladimir Drinfeld in 1986.
Du Val singularity
A Du Val singularity, also called a rational double point or Kleinian singularity, is a surface singularity classified by the Dynkin diagrams A, D, and E, named after Patrick Du Val.
Dual representation
The dual representation, or contragredient representation, is the representation a group or Lie algebra induces on the dual space of its representation space, preserving the natural pairing.
Dualizing sheaf
The dualizing sheaf is a coherent sheaf on a Noetherian scheme representing duality on coherent sheaves, generalizing the canonical bundle to singular spaces and underlying Serre and Grothendieck duality.
Dustin Clausen
Dustin Clausen is an American mathematician known for algebraic K-theory and for developing condensed mathematics with Peter Scholze; he has been a permanent professor at IHES since 2023.
Dynkin diagram
A Dynkin diagram is a graph encoding a root system, with one vertex per simple root, introduced by Evgeny Dynkin in 1947 to classify semisimple Lie algebras.
E7 (mathematics)
E7 is an exceptional simple Lie group and Lie algebra, one of five exceptional algebras, of dimension 133 and rank 7, distinguished by a 56-dimensional fundamental representation.
E8 (mathematics)
E8 (Lie group E8, E8 Lie algebra) is the largest of the five exceptional simple Lie algebras, a 248-dimensional structure of continuous symmetry with 240 roots.