(topology) A theorem which (in one of its simplest forms) states that a continuous function from an n-dimensional unit cube (in ℝⁿ) to itself has a fixed point.

(differential geometry) A theorem stating that, for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data).

(number theory) A theorem stating that there are infinitely many prime numbers of the form a² + b⁴.

(mathematics) A formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X.

(graph theory) A theorem stating that, when all finite subgraphs of an infinite graph can be colored using c colors, the same is true for the parent graph.

(mathematics) A result giving an explicit description of the universal enveloping algebra of a Lie algebra.

(mathematics, complex analysis) The theorem that if a complex number is a root for a polynomial with real coefficients, the complex conjugate of that number will also be a root.

(mathematics) In Ramsey theory, a theorem stating that, for any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression whose elements are of the same color.

(mathematics) A result on the combinatorics of block designs, stating that, if a (v, b, r, k, λ)-design exists with v = b (a symmetric block design), then: (i) if v is even, then k − λ is a square; (ii) if v is odd, then the following Diophantine equation has a nontrivial solution: x² − (k − λ)y² − (−1)^((v−1)/2) λ z² = 0.