A branch of mathematics, classically studying zeros of polynomial equations. It is based on techniques of abstract algebra, with the language and the problems of geometry.

Roughly, a set that can be constructed from open or closed sets by repeatedly taking countable unions and intersections.

The set constructed by taking the interval [0,1], removing the open middle third, removing the middle third of each of the two remaining pieces, and continuing ad infinitum.

A map between manifolds which is differentiable and has a differentiable inverse.

A branch of mathematics that studies dynamical systems with an invariant measure and related problems.

A function between categories which maps objects to objects and morphisms to morphisms.

The theorem that x^y is transcendental if x is algebraic and not equal to 0 or 1, and y is algebraic and irrational.

An inner product space that is a complete metric space under the metric induced by the inner product.

A bijective map between metric spaces that preserves distances.

The theorem that any simple closed curve separates the plane into an 'inside' and an 'outside.'

A complex manifold for which the exterior derivative of the fundamental form associated with the given Hermitian metric vanishes.

A ring which has a unique maximal ideal.

A holomorphic function on the upper half-plane satisfying a certain kind of functional equation with respect to the action of SL(2,Z), and also satisfying a growth condition.

A ring which does not have an infinitely ascending chain of ideals.

The quotient Aut(G)/Inn(G).

The first space filling curve to be discovered.

A four-dimensional noncommutative division algebra over the real numbers, where elements are typically written in the form a + b i + c j + d k.

A conjecture that all the nontrivial zeros of the Riemann-zeta function have real part 1/2.

A smooth closed 2-form on a manifold which is nondegenerate such that at every point p of the manifold, the alternating bilinear form on the tangent space at p is nondegenerate.

A space that parametrizes complex structures on a surface up to the action of homeomorphisms that are isotopic to the identity homeomorphism.

The smallest number of times a knot must pass through itself to become untied.

A matrix where each row is of the form [1, x, x^2, x^3, ... , x^n]

A knot property, also called the twist number, defined as the sum of signed crossings of a link.

A variant of the Riemann zeta function which was defined so as to have a particularly simple functional equation.

The conjecture that, for any triangle, 8 w^3 < ABC where A, B, and C are the vertex angles of the triangle and w is the Brocard angle.

The statement that if S is any nonempty partially ordered set in which every chain has an upper bound, then S has a maximal element.

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