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. Hermann Minkowski initiated this line of research at the age of 26 in his work The Geometry of Numbers.
Jean Louis, baron Bourgain was a Belgian mathematician. He was awarded the Fields Medal in 1994 in recognition of his work on several core topics of mathematical analysis such as the geometry of Banach spaces, harmonic analysis, ergodic theory and nonlinear partial differential equations from mathematical physics.
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.
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:
John Friedlander is a Canadian mathematician specializing in analytic number theory. He received his B.Sc. from the University of Toronto in 1965, an M.A. from the University of Waterloo in 1966, and a Ph.D. from Pennsylvania State University in 1972. He was a lecturer at M.I.T. in 1974–76, and has been on the faculty of the University of Toronto since 1977, where he served as Chair during 1987–91. He has also spent several years at the Institute for Advanced Study. In addition to his individual work, he has been notable for his collaborations with other well-known number theorists, including Enrico Bombieri, William Duke, Andrew Granville, and especially Henryk Iwaniec.
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.
Zlil Sela is an Israeli mathematician working in the area of geometric group theory. He is a Professor of Mathematics at the Hebrew University of Jerusalem. Sela is known for the solution of the isomorphism problem for torsion-free word-hyperbolic groups and for the solution of the Tarski conjecture about equivalence of first-order theories of finitely generated non-abelian free groups.
Lucien 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 French National Centre for Scientific Research.
Enrico Giusti was 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 was 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 was also the editor-in-chief of the international journal dedicated to the history of mathematics Bollettino di storia delle scienze matematiche.
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.
The Joseph L. Doob Prize of the American Mathematical Society (AMS) awards $5,000 (U.S.) every three years for "a single, relatively recent, outstanding research book that makes a seminal contribution to the research literature, reflects the highest standards of research exposition, and promises to have a deep and long-term impact in its area." The prize, endowed in 2005 by Paul and Virginia Halmos, is named in honor of AMS President Joseph L. Doob, who was Paul Halmos's doctoral advisor in the department of mathematics at the University of Illinois. According to Paul Halmos, "Doob was the first well-informed modern mathematician in the department". In order for a mathematical research book to be eligible for the prize, it must have been published within the past 6 calendar years of the year of its nomination. The prize was originally named the AMS Book Prize, but after the first award was renamed the Doob Prize.
