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.
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.
David William Masser is Professor Emeritus in the Department of Mathematics and Computer Science at the University of Basel. He is known for his work in transcendental number theory, Diophantine approximation, and Diophantine geometry. With Joseph Oesterlé in 1985, Masser formulated the abc conjecture, which has been called "the most important unsolved problem in Diophantine analysis".
Hugh Lowell Montgomery is an American mathematician, working in the fields of analytic number theory and mathematical analysis. As a Marshall scholar, Montgomery earned his Ph.D. from the University of Cambridge. For many years, Montgomery has been teaching at the University of Michigan.
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.
Gisbert Wüstholz is a German mathematician internationally known for his fundamental contributions to number theory and arithmetic geometry.
Per H. Enflo is a Swedish mathematician working primarily in functional analysis, a field in which he solved problems that had been considered fundamental. Three of these problems had been open for more than forty years:
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.
In mathematics, specifically in transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions. The existence of these polynomials was proven by Axel Thue; Thue's proof used Dirichlet's box principle. Carl Ludwig Siegel published his lemma in 1929. It is a pure existence theorem for a system of linear equations.
In mathematics, the Bombieri norm, named after Enrico Bombieri, is a norm on homogeneous polynomials with coefficient in or . This norm has many remarkable properties, the most important being listed in this article.
Jean-Michel Bismut is a French mathematician who has been a professor at the Université Paris-Sud since 1981. His mathematical career covers two apparently different branches of mathematics: probability theory and differential geometry. Ideas from probability play an important role in his works on geometry.
Lucien Serge Szpiro was a French mathematician known for his work in number theory, arithmetic geometry, and commutative algebra. He formulated Szpiro's conjecture and was a Distinguished Professor at the CUNY Graduate Center and an emeritus Director of Research at the CNRS.
Minhyong Kim is a South Korean mathematician who specialises in arithmetic geometry and anabelian geometry.
Enrico Giusti, is an Italian mathematician mainly known for his contributions to the fields of calculus of variations, regularity theory of partial differential equations, minimal surfaces and history of mathematics. He has been professor of mathematics at the Università di Firenze; he also taught and conducted research at the Australian National University at Canberra, at the Stanford University and at the University of California, Berkeley. After retirement, he devoted himself to the managing of the "Giardino di Archimede", a museum entirely dedicated to mathematics and its applications. Giusti is also the editor-in-chief of the international journal, dedicated to the history of mathematics "Bollettino di storia delle scienze matematiche".
Henri Antoine Gillet is an American mathematician, specializing in arithmetic geometry and algebraic geometry.
Umberto Zannier is an Italian mathematician, specializing in number theory and Diophantine geometry.
Pietro Corvaja is an Italian mathematician working in Diophantine geometry. He is a professor of geometry at the University of Udine.
