Combinatory logicCombinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. It is based on combinators, which were introduced by Schönfinkel in 1920 with the idea of providing an analogous way to build up functions—and to remove any mention of variables—particularly in predicate logic.
Geometric invariant theoryIn mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory. Geometric invariant theory studies an action of a group G on an algebraic variety (or scheme) X and provides techniques for forming the 'quotient' of X by G as a scheme with reasonable properties.
Lambda calculusLambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. Lambda calculus consists of constructing lambda terms and performing reduction operations on them.
Fixed-point subringIn algebra, the fixed-point subring of an automorphism f of a ring R is the subring of the fixed points of f, that is, More generally, if G is a group acting on R, then the subring of R is called the fixed subring or, more traditionally, the ring of invariants under G. If S is a set of automorphisms of R, the elements of R that are fixed by the elements of S form the ring of invariants under the group generated by S. In particular, the fixed-point subring of an automorphism f is the ring of invariants of the cyclic group generated by f.
Mathematical analysisAnalysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).
System FSystem F (also polymorphic lambda calculus or second-order lambda calculus) is a typed lambda calculus that introduces, to simply typed lambda calculus, a mechanism of universal quantification over types. System F formalizes parametric polymorphism in programming languages, thus forming a theoretical basis for languages such as Haskell and ML. It was discovered independently by logician Jean-Yves Girard (1972) and computer scientist John C. Reynolds.
Computer-assisted proofA computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program.
Aix-en-ProvenceAix-en-Provence (UKˌɛks_ɒ̃_prɒˈvɒ̃s, USˌeɪks_ɒ̃_proʊˈvɒ̃s,ˌɛks-), or simply Aix (medieval Occitan: Aics), is a city and commune in southern France, about north of Marseille. A former capital of Provence, it is the subprefecture of the arrondissement of Aix-en-Provence, in the department of Bouches-du-Rhône, in the region of Provence-Alpes-Côte d'Azur. The population of Aix-en-Provence is approximately 145,000. Its inhabitants are called Aixois or, less commonly, Aquisextains.
Abstract object theoryAbstract object theory (AOT) is a branch of metaphysics regarding abstract objects. Originally devised by metaphysician Edward Zalta in 1981, the theory was an expansion of mathematical Platonism. Abstract Objects: An Introduction to Axiomatic Metaphysics (1983) is the title of a publication by Edward Zalta that outlines abstract object theory. AOT is a dual predication approach (also known as "dual copula strategy") to abstract objects influenced by the contributions of Alexius Meinong and his student Ernst Mally.
Provence wineProvence (Provençal) wine comes from the French wine-producing region of Provence in southeast France. The Romans called the area provincia nostra ("our province"), giving the region its name. Just south of the Alps, it was the first Roman province outside Italy. Wine has been made in this region for at least 2,600 years, ever since the ancient Greeks founded the city of Marseille in 600 BC. Throughout the region's history, viticulture and winemaking have been influenced by the cultures that have been present in Provence, which include the Ancient Greeks, Romans, Gauls, Catalans and Savoyards.
Hilbert's fourteenth problemIn mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated. The setting is as follows: Assume that k is a field and let K be a subfield of the field of rational functions in n variables, k(x1, ..., xn ) over k. Consider now the k-algebra R defined as the intersection Hilbert conjectured that all such algebras are finitely generated over k.
