Enrico Bombieri is an Italian mathematician, known for his work in analytic number theory, Diophantine geometry, complex analysis, and group theory. Bombieri is currently Professor Emeritus in the School of Mathematics at the Institute for Advanced Study in Princeton, New Jersey. Bombieri won the Fields Medal in 1974 for his contributions to large sieve mathematics, conceptualized by Linnick 1941, and its application to the distribution of prime numbers.
Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell, and known as the Mordell conjecture until its 1983 proof by Gerd Faltings. The conjecture was later generalized by replacing by any number field.
Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in and the study of these lattices provides fundamental information on algebraic numbers. The geometry of numbers was initiated by Hermann Minkowski (1910).
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations. Diophantine geometry is part of the broader field of arithmetic geometry.
In mathematics, Roth's theorem or Thue–Siegel–Roth theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number approximations that are 'very good'. Over half a century, the meaning of very good here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of Axel Thue (1909), Carl Ludwig Siegel (1921), Freeman Dyson (1947), and Klaus Roth (1955).
Ennio De Giorgi was an Italian mathematician who worked on partial differential equations and the foundations of mathematics.
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic varieties to the real numbers.
In arithmetic geometry, the Bombieri–Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type.
In mathematics, Siegel's theorem on integral points states that for a smooth algebraic curve C of genus g defined over a number field K, presented in affine space in a given coordinate system, there are only finitely many points on C with coordinates in the ring of integers O of K, provided g > 0.
In mathematics, the subspace theorem says that points of small height in projective space lie in a finite number of hyperplanes. It is a result obtained by Wolfgang M. Schmidt (1972).
In mathematics, Arakelov theory is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.
Jonathan Solomon Pila FRS is an Australian mathematician at the University of Oxford.
In mathematics, the André–Oort conjecture is a problem in Diophantine geometry, a branch of number theory, that can be seen as a non-abelian analogue of the Manin–Mumford conjecture, which is now a theorem. The conjecture concerns itself with a characterization of the Zariski closure of sets of special points in Shimura varieties. A special case of the conjecture was stated by Yves André in 1989 and a more general statement was conjectured by Frans Oort in 1995. The modern version is a natural generalization of these two conjectures.
Aldo Andreotti was an Italian mathematician who worked on algebraic geometry, on the theory of functions of several complex variables and on partial differential operators. Notably he proved the Andreotti–Frankel theorem, the Andreotti–Grauert theorem, the Andreotti–Vesentini theorem and introduced, jointly with François Norguet, the Andreotti–Norguet integral representation for functions of several complex variables.
Alessandro Faedo was an Italian mathematician and politician, born in Chiampo. He is known for his work in numerical analysis, leading to the Faedo–Galerkin method: he was one of the pupils of Leonida Tonelli and, after his death, he succeeded him on the chair of mathematical analysis at the University of Pisa, becoming dean of the faculty of sciences and then rector and exerting a strong positive influence on the development of the university.
Pietro Corvaja is an Italian mathematician working in Diophantine geometry. He is a professor of geometry at the University of Udine.
Giovanni Ricci was an Italian mathematician.
In mathematics, the Zilber–Pink conjecture is a far-reaching generalisation of many famous Diophantine conjectures and statements, such as André–Oort, Manin–Mumford, and Mordell–Lang. For algebraic tori and semiabelian varieties it was proposed by Boris Zilber and independently by Enrico Bombieri, David Masser, Umberto Zannier in the early 2000's. For semiabelian varieties the conjecture implies the Mordell–Lang and Manin–Mumford conjectures. Richard Pink proposed (again independently) a more general conjecture for Shimura varieties which also implies the André–Oort conjecture. In the case of algebraic tori, Zilber called it the Conjecture on Intersection with Tori (CIT). The general version is now known as the Zilber–Pink conjecture. It states roughly that atypical or unlikely intersections of an algebraic variety with certain special varieties are accounted for by finitely many special varieties.
