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Inverse limit
The inverse limit, also called the projective limit, is a mathematical construction that assembles objects connected by backward-pointing maps into a single object of compatible choices, used across algebra, topology, and number theory.
Inverse semigroup
An inverse semigroup is a semigroup in which every element has a unique inverse, the algebraic structure capturing partial symmetries, defined independently by Vagner and Preston in the 1950s.
Invertible matrix
An invertible matrix, also called a non-singular matrix, is a square matrix with an inverse that multiplies it to give the identity; such an inverse exists exactly when the determinant is nonzero.
Irrational number
An irrational number is a real number that cannot be written as a ratio of two integers; its decimal expansion neither terminates nor repeats, as with √2 and π.
Irrational rotation algebra
The irrational rotation algebra, also called the noncommutative torus, is a universal C-algebra generated by two unitaries with uv = e^(2πiθ)vu; for irrational θ it is simple with a unique trace.
Irreducible element
An irreducible element, also called an atom, is a nonzero, non-unit element of an integral domain that cannot be written as a product of two non-unit elements; it generalizes prime numbers to abstract rings.
Irreducible polynomial
An irreducible polynomial is a nonconstant polynomial that cannot be written as a product of two lower-degree polynomials, the polynomial analog of a prime number, with irreducibility depending on the coefficient field.
Irreducible representation
An irreducible representation, or irrep, is a representation of a group or algebra with no smaller nontrivial representation; irreps are the building blocks of representation theory.
Isomorphism theorems
The isomorphism theorems, also known as Noether's isomorphism theorems, are three or four results in abstract algebra describing how quotients and images of algebraic structures relate.
Ivan Fesenko
Ivan Fesenko is a Russian-born mathematician in number theory, known for class field theory and adelic analysis, a distinguished professor at Westlake University since 2023.
Iwasawa theory
Iwasawa theory is a branch of number theory, founded by Kenkichi Iwasawa in the late 1950s, that studies how class groups grow in infinite towers of number fields.
j-invariant
The j-invariant is a quantity attached to an elliptic curve, taking the same value on two curves exactly when they are isomorphic over an algebraically closed field.
Jack function
The Jack function, also called the Jack polynomial, is a symmetric polynomial depending on a partition and a parameter, introduced by Henry Jack in 1970.
Jacobi method
The Jacobi method, also known as Jacobi iteration, is an iterative algorithm of numerical linear algebra that solves systems of linear equations; Jacobi described it in 1845.
Jacobi's formula
Jacobi's formula is a theorem of matrix calculus stating that the differential of a square matrix's determinant equals the trace of its adjugate times the matrix differential.
Jacobian matrix and determinant
The Jacobian matrix of a vector-valued function is the matrix of its first-order partial derivatives; when square, its Jacobian determinant measures local volume scaling and appears in change of variables.
Jacobson radical
The Jacobson radical of a ring R, written J(R), is the intersection of all maximal left ideals, equivalently of all maximal right ideals; it was introduced by Nathan Jacobson in 1945.
Jacobson ring
A Jacobson ring, also called a Hilbert ring, is a commutative ring in which every radical ideal is the intersection of the maximal ideals containing it.
Jacobsthal numbers
The Jacobsthal numbers are the sequence 0, 1, 1, 3, 5, 11, ... defined by the recurrence J(n) = J(n−1) + 2J(n−2), named for the German mathematician Ernst Erich Jacobsthal.
Janko groups
The Janko groups J₁, J₂, J₃, and J₄ are four of the 26 sporadic finite simple groups, named after mathematician Zvonimir Janko, who found them between 1964 and 1975.
Jeu de taquin
Jeu de taquin is an algorithm in algebraic combinatorics, introduced by Marcel-Paul Schützenberger in 1977, that transforms a skew Young tableau into a straight-shape tableau by sliding entries.
Joel David Hamkins
Joel David Hamkins is an American mathematician and philosopher working in logic and set theory, known for the set-theoretic multiverse view and infinite time Turing machines.
John Napier
John Napier of Merchiston (1550–1617) was a Scottish mathematician and theologian who invented logarithms, published the first table of them in 1614, and devised the calculating rods known as Napier's bones.
Jones polynomial
The Jones polynomial is a polynomial invariant of knots and links in three-dimensional space, introduced by Vaughan Jones in 1984. It distinguishes all knots of 10 crossings or fewer, though computing it is #P-hard.
Jordan algebra
A Jordan algebra is a commutative non-associative algebra satisfying the Jordan identity, introduced by Pascual Jordan in 1933 as an algebraic foundation for quantum mechanics.
Jordan block
A Jordan block is a square matrix with a single number λ on its main diagonal, 1s on the diagonal above, and 0s everywhere else.
Jordan normal form
The Jordan normal form, also called the Jordan canonical form, is a block-diagonal matrix representing a linear operator when a matrix cannot be diagonalized, with eigenvalues on the diagonal and 1s on the superdiagonal.
K-theory
K-theory is a branch of mathematics assigning algebraic invariants, the K-groups, to spaces, rings, and operator algebras; introduced by Grothendieck in 1957, it splits into algebraic and topological branches.
Kac–Moody algebra
A Kac–Moody algebra is a usually infinite-dimensional Lie algebra defined by generators and relations from a generalized Cartan matrix, studied independently by Victor Kac and Robert Moody.
Kac–Moody flag variety
A Kac–Moody flag variety is the homogeneous space of flags attached to a Kac–Moody group, built from a generalized Cartan matrix; in finite type it is the familiar projective variety G/P.