1,733 articles
Minimal model program
The minimal model program, also called the Mori program, is a procedure in birational algebraic geometry that classifies projective varieties by replacing them with simpler birational models.
Minimal polynomial (linear algebra)
In linear algebra, the minimal polynomial of a matrix or operator is the monic polynomial of least degree that annihilates it, and it divides the characteristic polynomial.
Minkowski's theorem
Minkowski's theorem, or Minkowski's convex body theorem, is a geometry of numbers result stating that any sufficiently large convex body symmetric about the origin contains a lattice point.
Minor (linear algebra)
A minor in linear algebra is the determinant of a square submatrix formed by choosing certain rows and columns of a matrix, also called a minor determinant.
Mixed radix
Mixed radix, also called mixed base, is a positional numeral system in which each digit position has its own radix, unlike the single fixed base of decimal.
Möbius function
The Möbius function, μ(n), is an arithmetic function taking values −1, 0, and 1: (−1)^k for products of k distinct primes, and 0 on non-squarefree integers.
Möbius inversion formula
The Möbius inversion formula, also called Möbius inversion, is a number theory theorem stating that if f(n) is the divisor sum of g(n), then g is recovered using the Möbius function.
Modal algebra
A modal algebra, also called a Magari algebra, is a Boolean algebra equipped with a normal operator representing a modal connective, providing the algebraic semantics of propositional modal logic.
Modular arithmetic
Modular arithmetic, also called clock arithmetic, treats integers as congruent when they leave the same remainder; Gauss formalized it in 1801, and it underlies RSA cryptography.
Modular curve
A modular curve is an algebraic curve whose points parameterize elliptic curves with extra level structure, such as a point of order N, central to the arithmetic of elliptic curves.
Modular exponentiation
Modular exponentiation, also called Modpow, computes the remainder when a base raised to an exponent is divided by a modulus, and is the core operation of RSA and Diffie-Hellman.
Modular form
A modular form is a holomorphic function on the complex upper half-plane that transforms in a prescribed way under a subgroup of the modular group, holomorphic at the cusps.
Modular function
A modular function is a meromorphic function on the upper half-plane, invariant under a subgroup of the modular group and meromorphic at the cusps; the j-invariant is the principal example.
Modular lattice
A modular lattice is a lattice satisfying the modular law, a weakening of distributivity formulated by Richard Dedekind around 1900; examples include subspaces of a vector space and normal subgroups of a group.
Modular multiplicative inverse
The modular multiplicative inverse, also called the modular inverse, is the analog of division in modular arithmetic: an integer b⁻¹ with b·b⁻¹ ≡ 1 (mod m). It exists when b and m are relatively prime.
Modular representation theory
Modular representation theory, also called Brauer or modular character theory, studies representations of finite groups over fields whose characteristic divides the group order, developed by Richard Brauer from 1935.
Modular tensor category
A modular tensor category is a semisimple braided tensor category with finitely many simple objects, duals, a compatible twist, and a non-degenerate braiding, meaning an invertible S-matrix.
Modularity theorem
The modularity theorem, once the Taniyama-Shimura-Weil conjecture, states that every elliptic curve over the rationals is modular; proved between 1995 and 2001, it settled Fermat's Last Theorem.
Module (mathematics)
In mathematics, a module, or module over a ring, is an algebraic structure generalizing a vector space by taking scalars from a ring rather than a field.
Module homomorphism
A module homomorphism, also called an R-linear map, is a map between modules over a ring preserving addition and scalar multiplication; over a field it is a linear map.
Module over a restricted Lie algebra
A module over a restricted Lie algebra is a representation compatible with the pth power map, organized around reduced enveloping algebras and baby Verma modules in prime characteristic.
Moduli of algebraic curves
The moduli of algebraic curves, also called the Deligne–Mumford moduli space of curves, is the geometric object parametrizing smooth projective curves of fixed genus up to isomorphism.
Moduli space
A moduli space is a geometric space whose points correspond to isomorphism classes of objects like Riemann surfaces or vector bundles, recording how they vary.
Moduli stack of elliptic curves
The moduli stack of elliptic curves, written M₁,₁ or M_ell, is the Deligne–Mumford stack that classifies elliptic curves and serves as the geometric home of modular forms.
Modulo
Modulo, also known as the mod or modulus operator, returns the remainder when one integer is divided by another; Gauss introduced it in his 1801 Disquisitiones Arithmeticae.
Monad (category theory)
A monad in category theory, formerly called a triple or standard construction, is an endofunctor with unit and multiplication satisfying the monad laws, a monoid in the category of endofunctors.
Monadic Boolean algebra
A monadic Boolean algebra, also called a monadic algebra, is a Boolean algebra with a unary operation acting like the existential quantifier, introduced by Halmos in the 1950s.
Monic polynomial
A monic polynomial is a nonzero polynomial whose leading coefficient is exactly 1. Products of monic polynomials stay monic, and division by a monic works over any ring.
Monoid
A monoid is an algebraic structure consisting of a set with an associative binary operation and a two-sided identity element, like the natural numbers under addition or multiplication.
Monoidal category
A monoidal category is a category with a tensor product, a unit object, and coherence isomorphisms making associativity hold up to isomorphism, introduced by Saunders Mac Lane in 1963.