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Bombieri–Lang Conjecture
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. Statement The weak Bombieri–Lang conjecture for surfaces states that if X is a smooth surface of general type defined over a number field k, then the points of X do not form a dense set in the Zariski topology on X. The general form of the Bombieri–Lang conjecture states that if X is a positive-dimensional algebraic variety of general type defined over a number field k, then the points of X do not form a dense set in the Zariski topology. The refined form of the Bombieri–Lang conjecture states that if X is an algebraic variety of general type defined over a number field k, then there is a dense open subset U of X such that for all number field extensions k' over k, the set of points in U is finite. History The Bombieri–Lang conjecture was independent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Geometry
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 variety, algebraic varieties. In more abstract terms, arithmetic geometry can be defined as the study of scheme (mathematics), schemes of Finite morphism#Morphisms of finite type, finite type over the spectrum of a ring, spectrum of the ring of integers. Overview The classical objects of interest in arithmetic geometry are rational points: solution set, sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or Algebraic function field, function fields, i.e. field (mathematics), fields that are not algebraically closed excluding the real numbers. Rational points can be directly characterized by height functions which measure their arithmetic complexity. The structure of algebraic varieties defined over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Faltings's Theorem
In arithmetic geometry, the Mordell conjecture is the conjecture made by Louis Mordell that a curve of Genus (mathematics), genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by Gerd Faltings, and is now known as Faltings's theorem. The conjecture was later generalized by replacing Q by any number field. Background Let ''C'' be a non-singular algebraic curve of genus (mathematics), genus ''g'' over Q. Then the set of rational points on ''C'' may be determined as follows: * Case ''g'' = 0: no points or infinitely many; ''C'' is handled as a conic section. * Case ''g'' = 1: no points, or ''C'' is an elliptic curve and its rational points form a finitely generated abelian group (''Mordell's Theorem'', later generalized to the Mordell–Weil theorem). Moreover, Mazur's torsion theorem restricts the structure of the torsion subgroup. * Case ''g'' > 1: according to the Mordell conjecture, now Faltings's theorem, ''C'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Geometry
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. Four theorems in Diophantine geometry which are of fundamental importance include: * Mordell–Weil Theorem * Roth's Theorem * Siegel's Theorem * Faltings's Theorem Background Serge Lang published a book ''Diophantine Geometry'' in the area in 1962, and by this book he coined the term "Diophantine Geometry". The traditional arrangement of material on Diophantine equations was by degree and number of variables, as in Mordell's ''Diophantine Equations'' (1969). Mordell's book starts with a remark on homogeneous equations ''f'' = 0 over the rational field, attributed to C. F. Gauss, that non-zero solutions in integers (even primitive lattice points) exist if non-zero rational solutions do, and notes a caveat of L. E. D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Field Extension
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently. Definition and notation Suppose that ''E''/''F'' is a field extension. Then ''E'' may be considered as a vector space over ''F'' (the field of scalars). The dimension of this vector space is called the degree of the field extension, and it is denoted by :F The degree may be finite or infinite, the field being called a finite extension or infinite extension accordingly. An extension ''E''/''F'' is also sometimes said to be simply finite if it is a finite extension; this should not be confused with the fields themselves being finite fields (fields with finitely many elements). The degree should not be confused with the transcendence degree of a field; for example, the field Q(''X'') o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genus (mathematics)
In mathematics, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1. Topology Orientable surfaces The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic ''χ'', via the relationship ''χ'' = 2 − 2''g'' for closed surfaces, where ''g'' is the genus. For surfaces with ''b'' boundary components, the equation reads ''χ'' = 2 − 2''g'' − ''b''. In layman's terms, it's the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A torus has 1 such h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Boundedness Conjecture For Rational Points
In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field K and a positive integer g \geq 2 that there exists a number N(K,g) depending only on K and g such that for any algebraic curve C defined over K having genus equal to g has at most N(K,g) K-rational points. This is a refinement of Faltings's theorem, which asserts that the set of K-rational points C(K) is necessarily finite. Progress The first significant progress towards the conjecture was due to Caporaso, Harris, and Mazur. They proved that the conjecture holds if one assumes the Bombieri–Lang conjecture. Mazur's Conjecture B A variant of the conjecture, due to Mazur, asserts that there should be a number N(K,g,r) such that for any algebraic curve C defined over K having genus g and whose Jacobian variety J_C has Mordell–Weil rank over K equal to r, the number of K-rational points of C is at most N(K,g,r). This variant of the conjecture is known as Mazur's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joe Harris (mathematician)
Joseph Daniel Harris (born August 17, 1951) is a mathematician at Harvard University working in the field of algebraic geometry. After earning an AB from Harvard College, he continued at Harvard to study for a PhD under Phillip Griffiths. Work During the 1980s, he was on the faculty of Brown University, moving to Harvard around 1988. He served as chair of the department at Harvard from 2002 to 2005. His work is characterized by its classical geometric flavor: he has claimed that nothing he thinks about could not have been imagined by the Italian geometers of the late 19th and early 20th centuries, and that if he has had greater success than them, it is because he has access to better tools. Harris is well known for several of his books on algebraic geometry, notable for their informal presentations: * ''Principles of Algebraic Geometry'' , with Phillip Griffiths * ''Geometry of Algebraic Curves, Vol. 1'' , with Enrico Arbarello, Maurizio Cornalba, and Phillip Griffiths * , wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barry Mazur
Barry Charles Mazur (; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in number theory, Mazur's torsion theorem in arithmetic geometry, the Mazur swindle in geometric topology, and the Mazur manifold in differential topology. Life Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. He was nonetheless accepted for graduate studies at Princeton University, from where he received his PhD in mathematics in 1959 after completing a doctoral dissertation titled "On embeddings of spheres." He then became a Junior Fellow at Harvard University from 1961 to 1964. He is the Gerhard Gade University Professor and a Senior Fellow at Harvard. He is the brother of Joseph Mazur and the father of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucia Caporaso
Lucia Caporaso is an Italian mathematician, holding a professorship in mathematics at Roma Tre University. She was born in Rome, Italy, on May 22,1965. Her research includes work in algebraic geometry, arithmetic geometry, tropical geometry and enumerative geometry. Education and career Caporaso earned a laurea from Sapienza University of Rome in 1989. She completed her Ph.D. at Harvard University in 1993. Her dissertation, ''On a Compactification of the Universal Picard Variety over the Moduli Space of Stable Curves'', was supervised by Joe Harris. She became a Benjamin Pierce Assistant Professor of Mathematics at Harvard, a researcher at the University of Rome Tor Vergata, an assistant professor at the Massachusetts Institute of Technology, and an associate professor at the University of Sannio, before moving to Roma Tre as a professor in 2001. From 2013 to 2018, she has headed the Department of Mathematics and Physics at Roma Tre. Recognition Caporaso was the 1997 winner of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete & Computational Geometry
'' Discrete & Computational Geometry'' is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational geometry. Abstracting and indexing The journal is indexed in: * ''Mathematical Reviews'' * ''Zentralblatt MATH'' * ''Science Citation Index'' * ''Current Contents''/Engineering, Computing and Technology Notable articles The articles by Gil Kalai with a proof of a subexponential upper bound on the diameter of a polyhedron and by Samuel Ferguson on the Kepler conjecture, both published in Discrete & Computational geometry, earned their author the Fulkerson Prize The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at e .... References External link ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Ulam Problem
In mathematics, the Erdős–Ulam problem asks whether the plane contains a dense set of points whose Euclidean distances are all rational numbers. It is named after Paul Erdős and Stanislaw Ulam. Large point sets with rational distances The Erdős–Anning theorem states that a set of points with integer distances must either be finite or lie on a single line. However, there are other infinite sets of points with rational distances. For instance, on the unit circle, let ''S'' be the set of points :(\cos\theta,\sin\theta) where \theta is restricted to values that cause \tan\tfrac to be a rational number. For each such point, both \sin\tfrac and \cos\tfrac\theta 2 are themselves both rational, and if \theta and \varphi define two points in ''S'', then their distance is the rational number : \left, 2\sin\frac \theta 2 \cos\frac \varphi 2 -2\sin\frac \varphi 2 \cos\frac \theta 2 \. More generally, a circle with radius \rho contains a dense set of points at rational distances to e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
