The Friedlander–Iwaniec theorem is a result in number theory, specifically in the area of additive number theory concerning the distribution of prime numbers. It was established by the mathematicians J. Friedlander and H. Iwaniec in the early 1990s.
Chen's theorem is a result in number theory, specifically in the area of prime numbers. It states that every sufficiently large even integer can be expressed as the sum of a prime and the product of at most two primes. The theorem can be seen as a refinement of the Goldbach conjecture, which posits that every even integer greater than 2 can be expressed as the sum of two primes.
The Brun–Titchmarsh theorem is a result in analytic number theory that provides an asymptotic estimate for the number of primes in an arithmetic progression. More specifically, it addresses the distribution of prime numbers in the form \( a + nd \), where \( a \) and \( d \) are coprime integers, and \( n \) ranges over the natural numbers.
The Barban–Davenport–Halberstam theorem is a result in number theory, specifically in the area of additive number theory and the distribution of prime numbers. It provides a way to estimate the size of the prime numbers in certain ranges. More formally, the theorem deals with the distribution of prime numbers in arithmetic progressions and gives a bound on the number of primes in intervals of certain lengths.
Whitehead's Lemma is a result in the field of algebraic topology, particularly in the study of homotopy theory and the properties of topological spaces. It deals with the question of when a certain kind of map induces an isomorphism on homotopy groups.
The Structure Theorem for finitely generated modules over a principal ideal domain (PID) is a fundamental result in abstract algebra, specifically in the study of modules over rings. It describes the classification of finitely generated modules over a PID in terms of simpler components. Here’s a concise statement of the theorem: Let \( R \) be a principal ideal domain, and let \( M \) be a finitely generated \( R \)-module.
Strassmann's theorem is a result in complex analysis that provides conditions under which a sequence of complex functions converges uniformly on compact sets. Specifically, it addresses the uniform convergence of power series in the context of multivariable functions, but it also applies to single-variable functions.
Segal's conjecture is a significant statement in the field of algebraic topology, particularly in the study of stable homotopy theory. Formulated by Graeme Segal in the 1960s, the conjecture concerns the relationship between the stable homotopy groups of spheres and the representation theory of finite groups.
The Quillen–Suslin theorem, also known as the vanishing of the topological K-theory of the field of rational numbers, is a fundamental result in algebraic topology and the theory of vector bundles. It states that every vector bundle over a contractible space is trivial. More specifically, it can be expressed in the context of finite-dimensional vector bundles over real or complex spaces.
The Primitive Element Theorem is a fundamental result in field theory, which deals with field extensions in algebra.
The Latimer–MacDuffee theorem is a result in the field of algebra, specifically concerning finite abelian groups and their decompositions. It states that any finite abelian group can be expressed as a direct sum of cyclic groups, and the number of different ways to express a finite abelian group as such a direct sum is given by a specific combinatorial expression related to its invariant factors.
Joubert's theorem is a result in the field of geometry, particularly in the study of cyclic quadrilaterals. The theorem states that if a quadrilateral is cyclic (i.e., all its vertices lie on a single circle), then the angles opposite each other conform to a specific relationship in terms of their sine values.
Generic flatness is a concept from algebraic geometry and commutative algebra, often used in the context of schemes and modules over rings. In simple terms, it describes a condition on a family of algebraic objects that ensures they behave "nicely" with respect to flatness in a way that is uniform across a given parameter space.
The Gabriel–Popescu theorem is a result in the field of category theory, particularly in the study of module categories and ring theory. It provides a characterization of when a category of modules can be represented as the module category over a certain ring.
The Fundamental Theorem on Homomorphisms, often referred to in the context of group theory or algebra in general, states that there is a specific relationship between a group, a normal subgroup, and the quotient group formed by the subgroup. In summary, it describes how to relate the structure of a group to its quotient by a normal subgroup.
The Fundamental Lemma is a key result in the Langlands program, which is a vast and influential set of conjectures and theories in number theory and representation theory that seeks to relate Galois groups and automorphic forms. The Langlands program is named after Robert P. Langlands, who initiated these ideas in the late 1960s.
The Eckmann–Hilton argument is a concept in category theory and homotopy theory that plays a role in the context of algebraic structures such as monoids and operads. It particularly addresses the interactions between two operations defined on a space or an algebraic structure when these operations are defined in a certain way, especially in relation to commutativity and associativity.
The Dold–Kan correspondence is a fundamental theorem in algebraic topology and homological algebra that establishes a relationship between two important categories: the category of simplicial sets and the category of chain complexes of abelian groups (or modules). It is named after mathematicians Alfred Dold and D. K. Kan, who formulated it in the context of homotopy theory.
The Dimension Theorem for vector spaces is a fundamental result in linear algebra that relates the dimensions of certain components of vector spaces and their subspaces.