6,052 articles
Modular surface
The modular surface is the quotient of the hyperbolic upper half-plane by the modular group PSL(2,Z), a finite-area orbifold that classifies elliptic curves and hosts modular forms.
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.
Modulus of continuity
The modulus of continuity of a function measures the largest amount it can change over pairs of points no farther apart than a given distance.
Modus ponens
Modus ponens, also called the law of detachment, is a valid rule of inference in propositional logic: from if A then B, and A, infer B.
Modus tollens
Modus tollens, or denying the consequent, is a valid rule of inference in propositional logic: from a conditional p → q and the denial ¬q, one infers ¬p.
Moment (mathematics)
A moment is a numerical characteristic of a probability distribution, the expected value of a power of a random variable; the first moment is the mean.
Moment closure
Moment closure is a mathematical technique approximating the infinite hierarchy of moment equations of a stochastic process by a finite system, expressing higher-order moments as functions of lower-order ones.
Moment determinacy and indeterminate distributions
Moment determinacy is the property of a probability distribution being uniquely determined by its moments. Carleman's condition implies determinacy, and the lognormal distribution is the standard indeterminate example.
Moment map
A moment map, also written momentum map, is a tool in symplectic geometry that packages the conserved quantities generated by a Lie group's symmetry action into a single map.
Moment problem
The moment problem, or problem of moments, asks whether a probability distribution is uniquely determined by its moments, a question introduced by Stieltjes in 1894 and 1895.
Moment-generating function
The moment-generating function (MGF) of a random variable X is M_X(t)=E(e^{tX}); differentiating at 0 yields every moment, and an MGF that exists uniquely determines the distribution.
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.
Monad (functional programming)
In functional programming, a monad is an abstraction that structures computations, letting pure languages represent effects such as state, exceptions, and input and output behind a uniform interface.
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.
Monadic second-order logic
Monadic second-order logic (MSO logic) extends first-order logic by quantifying over sets of elements, and over words, trees, and bounded graphs it has the power of finite automata.
Monge–Ampère equation
The Monge–Ampère equation is a fully nonlinear second-order partial differential equation that prescribes the determinant of the Hessian of a convex function, central to optimal transport.
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.
Monodromy theorem
The monodromy theorem is a result of complex analysis: an analytic function continued along every path in a simply connected domain yields a single-valued branch.
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.
Monomial order
A monomial order is a total order on the monomials of a polynomial ring, compatible with multiplication, that fixes the leading terms used in Gröbner basis computation.
Monotone class theorem
The monotone class theorem is a measure theory result stating that a monotone class containing an algebra also contains the σ-algebra it generates, attributed to Paul Halmos.