In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, fields. Speaking, it is the property of having no infinitely large or infinitely small elements, it was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder. The notion arose from the theory of magnitudes of Ancient Greece. An algebraic structure in which any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is said to be Archimedean. A structure which has a pair of non-zero elements, one of, infinitesimal with respect to the other, is said to be non-Archimedean. For example, a linearly ordered group, Archimedean is an Archimedean group; this can be made precise in various contexts with different formulations. For example, in the context of ordered fields, one has the axiom of Archimedes which formulates this property, where the field of real numbers is Archimedean, but that of rational functions in real coefficients is not.
The concept was named by Otto Stolz after the ancient Greek geometer and physicist Archimedes of Syracuse. The Archimedean property appears in Book V of Euclid's Elements as Definition 4: Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another; because Archimedes credited it to Eudoxus of Cnidus it is known as the "Theorem of Eudoxus" or the Eudoxus axiom. Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs. Let x and y be positive elements of a linearly ordered group G. X is infinitesimal with respect to y if, for every natural number n, the multiple nx is less than y, that is, the following inequality holds: x + ⋯ + x ⏟ n terms < y. The group G is Archimedean. Additionally, if K is an algebraic structure with a unit — for example, a ring — a similar definition applies to K. If x is infinitesimal with respect to 1 x is an infinitesimal element. If y is infinite with respect to 1 y is an infinite element.
The algebraic structure K is Archimedean if it has no infinite elements and no infinitesimal elements. An ordered field has some additional properties. One may assume. If x is infinitesimal 1/x is infinite, vice versa. Therefore, to verify that a field is Archimedean it is enough to check only that there are no infinitesimal elements, or to check that there are no infinite elements. If x is infinitesimal and r is a rational number r x is infinitesimal; as a result, given a general element c, the three numbers c/2, c, 2c are either all infinitesimal or all non-infinitesimal. In this setting, an ordered field K is Archimedean when the following statement, called the axiom of Archimedes, holds: Let x be any element of K. There exists a natural number n such that n > x. Alternatively one can use the following characterization: ∀ε > 0 ∈ K: ∃n ∈ N:1/n < ε. The qualifier "Archimedean" is formulated in the theory of rank one valued fields and normed spaces over rank one valued fields as follows. Let F be a field endowed with an absolute value function, i.e. a function which associates the real number 0 with the field element 0 and associates a positive real number | x | with each non-zero x ∈ F and satisfies | x y | = | x | | y | and | x + y | ≤ | x | + | y |.
F is said to be Archimedean if for any non-zero x ∈ F there exists a natural number n such that | x + ⋯ + x ⏟ n terms | > 1. A normed space is Archimedean if a sum of n terms, each equal to a non-zero vector x, has norm greater than one for sufficiently large n. A field with an absolute value or a normed space is either Archimedean or satisfies the stronger condition, referred to as the ultrametric triangle inequality, | x + y | ≤ max,respectively. A field or normed space satisfying the ultrametric triangle inequality is called non-Archimedean; the concept of a non-Archimedean normed linear space was introduced by A. F. Monna; the field of the rational numbers can be assigned one of a number of absolute value functions, including the trivial function | x |
Alan Baker (mathematician)
Alan Baker was an English mathematician, known for his work on effective methods in number theory, in particular those arising from transcendental number theory. Alan Baker was born in London on 19 August 1939, he attended Stratford Grammar School, East London, his academic career started as a student of Harold Davenport, at University College London and at Trinity College, where he received his PhD. He was a visiting scholar at the Institute for Advanced Study in 1970 when he was awarded the Fields Medal at the age of 31. In 1974 he was appointed Professor of Pure Mathematics at Cambridge University, a position he held until 2006 when he became an Emeritus, he was a fellow of Trinity College from 1964 until his death. His interests were in number theory, logarithmic forms, effective methods, Diophantine geometry and Diophantine analysis. In 2012 he became a fellow of the American Mathematical Society, he has been made a foreign fellow of the National Academy of Sciences, India. Baker generalized the Gelfond–Schneider theorem, itself a solution to Hilbert's seventh problem.
Baker showed that if α 1... Α n are algebraic numbers, if β 1.. Β n are irrational algebraic numbers such that the set are linearly independent over the rational numbers the number α 1 β 1 α 2 β 2 ⋯ α n β n is transcendental. Baker, Alan, "Linear forms in the logarithms of algebraic numbers. I", Mathematika, 13: 204–216, doi:10.1112/S0025579300003971, ISSN 0025-5793, MR 0220680 Baker, Alan, "Linear forms in the logarithms of algebraic numbers. II", Mathematika, 14: 102–107, doi:10.1112/S0025579300008068, ISSN 0025-5793, MR 0220680 Baker, Alan, "Linear forms in the logarithms of algebraic numbers. III", Mathematika, 14: 220–228, doi:10.1112/S0025579300003843, ISSN 0025-5793, MR 0220680 Baker, Transcendental number theory, Cambridge Mathematical Library, Cambridge University Press, ISBN 978-0-521-39791-9, MR 0422171. 1975. Baker, Alan. Alan Baker at the Mathematics Genealogy Project Masser, David. "Alan Baker 1939–2018". Notices of the American Mathematical Society. 66: 32–35
In mathematics, the Manin conjecture describes the conjectural distribution of rational points on an algebraic variety relative to a suitable height function. It was proposed by Yuri I. Manin and his collaborators in 1989 when they initiated a program with the aim of describing the distribution of rational points on suitable algebraic varieties, their main conjecture is. Let V be a Fano variety defined over a number field K, let H be a height function, relative to the anticanonical divisor and assume that V is Zariski dense in V. There exists a non-empty Zariski open subset U ⊂ V such that the counting function of K -rational points of bounded height, defined by N U, H = # for B ≥ 1, satisfies N U, H ∼ c B ρ − 1, as B → ∞. Here ρ is the rank of the Picard group of V and c is a positive constant which received a conjectural interpretation by Peyre. Manin's conjecture has been decided for special families of varieties, but is still open in general
André Weil was an influential French mathematician of the 20th century, known for his foundational work in number theory and algebraic geometry. He was the de facto early leader of the mathematical Bourbaki group; the philosopher Simone Weil was his sister. André Weil was born in Paris to agnostic Alsatian Jewish parents who fled the annexation of Alsace-Lorraine by the German Empire after the Franco-Prussian War in 1870–71; the famous philosopher Simone Weil was Weil's only sibling. He studied in Paris, Rome and Göttingen and received his doctorate in 1928. While in Germany, Weil befriended Carl Ludwig Siegel. Starting in 1930, he spent two academic years at Aligarh Muslim University. Aside from mathematics, Weil held lifelong interests in classical Greek and Latin literature, in Hinduism and Sanskrit literature: he taught himself Sanskrit in 1920. After teaching for one year in Aix-Marseille University, he taught for six years in Strasbourg, he married Éveline in 1937. Weil was in Finland, his wife Éveline returned to France without him.
Weil was mistakenly arrested in Finland at the outbreak of the Winter War on suspicion of spying. Weil returned to France via Sweden and the United Kingdom, was detained at Le Havre in January 1940, he was charged with failure to report for duty, was imprisoned in Le Havre and Rouen. It was in the military prison in Bonne-Nouvelle, a district of Rouen, from February to May, that Weil completed the work that made his reputation, he was tried on 3 May 1940. Sentenced to five years, he requested to be attached to a military unit instead, was given the chance to join a regiment in Cherbourg. After the fall of France, he met up with his family in Marseille, he went to Clermont-Ferrand, where he managed to join his wife Éveline, living in German-occupied France. In January 1941, Weil and his family sailed from Marseille to New York, he spent the remainder of the war in the United States, where he was supported by the Rockefeller Foundation and the Guggenheim Foundation. For two years, he taught undergraduate mathematics at Lehigh University, where he was unappreciated and poorly paid, although he didn't have to worry about being drafted, unlike his American students.
But, he hated Lehigh much for their heavy teaching workload and he swore that he would never talk about "Lehigh" any more. He quit the job at Lehigh, he moved to Brazil and taught at the Universidade de São Paulo from 1945 to 1947, where he worked with Oscar Zariski, he returned to the United States and taught at the University of Chicago from 1947 to 1958, before moving to the Institute for Advanced Study, where he would spend the remainder of his career. He was a Plenary Speaker at the ICM in 1950 in Cambridge, Massachusetts, in 1954 in Amsterdam, in 1978 in Helsinki. In 1979, Weil shared the second Wolf Prize in Mathematics with Jean Leray. Weil made substantial contributions in a number of areas, the most important being his discovery of profound connections between algebraic geometry and number theory; this began in his doctoral work leading to the Mordell–Weil theorem. Mordell's theorem had an ad hoc proof. Both aspects of Weil's work have developed into substantial theories. Among his major accomplishments were the 1940s proof of the Riemann hypothesis for zeta-functions of curves over finite fields, his subsequent laying of proper foundations for algebraic geometry to support that result.
The so-called Weil conjectures were hugely influential from around 1950. Weil introduced the adele ring in the late 1930s, following Claude Chevalley's lead with the ideles, gave a proof of the Riemann–Roch theorem with them. His'matrix divisor' Riemann–Roch theorem from 1938 was a early anticipation of ideas such as moduli spaces of bundles; the Weil conjecture on Tamagawa numbers proved resistant for many years. The adelic approach became basic in automorphic representation theory, he picked up another credited Weil conjecture, around 1967, which under pressure from Serge Lang became known as the Taniyama–Shimura conjecture based on a formulated question of Taniyama at the 1955 Nikkō conference. His attitude towards conjectures was that one should not dignify a guess as a conjecture and in the Taniyama case, the evidence was only there after extensive computational work carried out from the late 1960s. Other significant results were on Pontryagin differential geometry, he introduced the concept of a uniform space in general topology, as a by-product of his collaboration with Nicolas Bourbaki.
His work on sheaf theory hardly appears in his published papers, but correspondence with Henri Cartan in the late 1940s, reprinted in his collected papers, proved most influential. He created the ∅, he discovered that the so-called Weil representation introduced in quantum mechanics by Irving Segal an
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G to the complex numbers, invariant under the action of a discrete subgroup Γ ⊂ G of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. Modular forms are automorphic forms defined over the groups SL or PSL with the discrete subgroup being the modular group, or one of its congruence subgroups. More one can use the adelic approach as a way of dealing with the whole family of congruence subgroups at once. From this point of view, an automorphic form over the group G, for an algebraic group G and an algebraic number field F, is a complex-valued function on G, left invariant under G and satisfies certain smoothness and growth conditions. Poincaré first discovered automorphic forms as generalizations of trigonometric and elliptic functions. Through the Langlands conjectures automorphic forms play an important role in modern number theory.
An automorphic form is a function F on G, subject to three kinds of conditions: to transform under translation by elements γ ∈ Γ according to the given factor of automorphy j. It is the first of these that makes F automorphic, that is, satisfy an interesting functional equation relating F with F for γ ∈ Γ. In the vector-valued case the specification can involve a finite-dimensional group representation ρ acting on the components to'twist' them; the Casimir operator condition says. The third condition is to handle the case where G/Γ has cusps; the formulation requires the general notion of factor of automorphy j for Γ, a type of 1-cocycle in the language of group cohomology. The values of j may be complex numbers, or in fact complex square matrices, corresponding to the possibility of vector-valued automorphic forms; the cocycle condition imposed on the factor of automorphy is something that can be checked, when j is derived from a Jacobian matrix, by means of the chain rule. Before this general setting was proposed, there had been substantial developments of automorphic forms other than modular forms.
The case of Γ a Fuchsian group had received attention before 1900. The Hilbert modular forms were proposed not long after that, though a full theory was long in coming; the Siegel modular forms, for which G is a symplectic group, arose from considering moduli spaces and theta functions. The post-war interest in several complex variables made it natural to pursue the idea of automorphic form in the cases where the forms are indeed complex-analytic. Much work was done, in particular by Ilya Piatetski-Shapiro, in the years around 1960, in creating such a theory; the theory of the Selberg trace formula, as applied by others, showed the considerable depth of the theory. Robert Langlands showed how the Riemann-Roch theorem could be applied to the calculation of dimensions of automorphic forms, he produced the general theory of Eisenstein series, which corresponds to what in spectral theory terms would be the'continuous spectrum' for this problem, leaving the cusp form or discrete part to investigate.
From the point of view of number theory, the cusp forms had been recognised, since Srinivasa Ramanujan, as the heart of the matter. The subsequent notion of an "automorphic representation" has proved of great technical value when dealing with G an algebraic group, treated as an adelic algebraic group, it does not include the automorphic form idea introduced above, in that the adelic approach is a way of dealing with the whole family of congruence subgroups at once. Inside an L2 space for a quotient of the adelic form of G, an automorphic representation is a representation, an infinite tensor product of representations of p-adic groups, with specific enveloping algebra representations for the infinite prime. One way to express the shift in emphasis is that the Hecke operators are here in effect put on the same level as the Casimir operators, it is this concept, basic to the formulation of the Langlands philosophy. One of Poincaré's first discoveries in mathematics, dating to the 1880s, was automorphic forms.
He named them Fuchsian functions, after the mathematician Lazarus Fuchs, because Fuchs was known for being a good teacher and had researched on differential equations and the theory of functions. Poincaré developed the concept of these functions as part of his doctoral thesis. Under Poincaré's definition, an automorphic function is one, analytic in its domain and is invariant under a discrete infinite group of linear fractional transformations. Automorphic functions generalize both trigonometric and elliptic functions. Poincaré explains how he discovered Fuchsian functions: For fifteen days I strove to prove
In number theory, the Mordell conjecture is the conjecture made by Mordell that a curve of 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, is now known as Faltings's theorem; the conjecture was generalized by replacing Q by any number field. Let C be a non-singular algebraic curve of genus g over Q; the set of rational points on C may be determined as follows: Case g = 0: no points or infinitely many. Case g = 1: no points, or C is an elliptic curve and its rational points form a finitely generated abelian group. 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 has only a finite number of rational points. Faltings's original proof used the known reduction to a case of the Tate conjecture, a number of tools from algebraic geometry, including the theory of Néron models. A different proof, based on diophantine approximation, was found by Vojta.
A more elementary variant of Vojta's proof was given by Bombieri. Faltings's 1983 paper had as consequences a number of statements, conjectured: The Mordell conjecture that a curve of genus greater than 1 over a number field has only finitely many rational points; the reduction of the Mordell conjecture to the Shafarevich conjecture was due to A. N. Paršin. A sample application of Faltings's theorem is to a weak form of Fermat's Last Theorem: for any fixed n > 4 there are at most finitely many primitive integer solutions to an + bn = cn, since for such n the curve xn + yn = 1 has genus greater than 1. Because of the Mordell–Weil theorem, Faltings's theorem can be reformulated as a statement about the intersection of a curve C with a finitely generated subgroup Γ of an abelian variety A. Generalizing by replacing C by an arbitrary subvariety of A and Γ by an arbitrary finite-rank subgroup of A leads to the Mordell–Lang conjecture, proved by Faltings. Another higher-dimensional generalization of Faltings's theorem is the Bombieri–Lang conjecture that if X is a pseudo-canonical variety over a number field k X is not Zariski dense in X.
More general conjectures have been put forth by Paul Vojta. The Mordell conjecture for function fields was proved by Manin and by Grauert. In 1990, Coleman found and fixed a gap in Manin's proof. Bombieri, Enrico. "The Mordell conjecture revisited". Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17: 615–640. MR 1093712. Coleman, Robert F.. "Manin's proof of the Mordell conjecture over function fields". L'Enseignement Mathématique. Revue Internationale. IIe Série. 36: 393–427. ISSN 0013-8584. MR 1096426. Archived from the original on 2011-10-02. Cornell, Gary. Arithmetic geometry. Papers from the conference held at the University of Connecticut, Connecticut, July 30 – August 10, 1984. New York: Springer-Verlag. Doi:10.1007/978-1-4613-8655-1. ISBN 0-387-96311-1. MR 0861969. → Contains an English translation of Faltings Faltings, Gerd. "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern". Inventiones Mathematicae. 73: 349–366. Doi:10.1007/BF01388432. MR 0718935. Faltings, Gerd. "Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern".
Inventiones Mathematicae. 75: 381. Doi:10.1007/BF01388572. MR 0732554. Faltings, Gerd. "Diophantine approximation on abelian varieties". Ann. of Math. 133: 549–576. Doi:10.2307/2944319. MR 1109353. Faltings, Gerd. "The general case of S. Lang's conjecture". In Cristante, Valentino. Barsotti Symposium in Algebraic Geometry. Papers from the symposium held in Abano Terme, June 24–27, 1991. Perspectives in Mathematics. San Diego, CA: Academic Press, Inc. ISBN 0-12-197270-4. MR 1307396. Grauert, Hans. "Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper". Publications Mathématiques de l'IHÉS: 131–149. ISSN 1618-1913. MR 0222087. Hindry, Marc. Diophantine geometry. Graduate Texts in Mathematics. 201. New York: Springer-Verlag. Doi:10.1007/978-1-4612-1210-2. ISBN 0-387-98981-1. MR 1745599. → Gives Vojta's proof of Faltings's Theorem. Lang, Serge. Survey of Diophantine geometry. Springer-Verlag. Pp. 101–122. ISBN 3-540-61223-8. Manin, Ju. I.. "Rational points on algebraic curves over function fields".
Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya. 27: 1395–1440. ISSN 0373-2436. MR 0157971. Mordell, Louis J.. "On the rational solutions of the indeterminate equation of the third and fourth degrees". Proc. Cambridge Philos. Soc. 21: 179–192. Paršin, A. N.. "Quelques conjectures de finitude en géométrie diophantienne". Actes du Congrès International des Mathématiciens. Tome 1. Nice: Gauthier-Villars. Pp. 467–471. MR 0427323. Archived from the original on 2016-09-24. Retrieved 2016-06-11. Parshin, A. N. "Mordell conje
Algebraic varieties are the central objects of study in algebraic geometry. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.:58Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility; the concept of an algebraic variety is similar to that of an analytic manifold. An important difference is that an algebraic variety may have singular points, while a manifold cannot; the fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial in one variable with complex number coefficients is determined by the set of its roots in the complex plane.
Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory; this correspondence is a defining feature of algebraic geometry. An affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define quasi-projective varieties in a similar way; the most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, but Nagata gave an example of such a new variety in the 1950s. For an algebraically closed field K and a natural number n, let An be affine n-space over K; the polynomials f in the ring K can be viewed as K-valued functions on An by evaluating f at the points in An, i.e. by choosing values in K for each xi.
For each set S of polynomials in K, define the zero-locus Z to be the set of points in An on which the functions in S vanish, to say Z =. A subset V of An is called an affine algebraic set if V = Z for some S.:2 A nonempty affine algebraic set V is called irreducible if it cannot be written as the union of two proper algebraic subsets.:3 An irreducible affine algebraic set is called an affine variety.:3 Affine varieties can be given a natural topology by declaring the closed sets to be the affine algebraic sets. This topology is called the Zariski topology.:2Given a subset V of An, we define I to be the ideal of all polynomial functions vanishing on V: I =. For any affine algebraic set V, the coordinate ring or structure ring of V is the quotient of the polynomial ring by this ideal.:4 Let k be an algebraically closed field and let Pn be the projective n-space over k. Let f in k be a homogeneous polynomial of degree d, it is not well-defined to evaluate f on points in Pn in homogeneous coordinates.
However, because f is homogeneous, meaning that f = λd f , it does make sense to ask whether f vanishes at a point. For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in Pn on which the functions in S vanish: Z =. A subset V of Pn is called a projective algebraic set if V = Z for some S.:9 An irreducible projective algebraic set is called a projective variety.:10Projective varieties are equipped with the Zariski topology by declaring all algebraic sets to be closed. Given a subset V of Pn, let I be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.:10A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every affine variety is quasi-projective. Notice that the complement of an algebraic set in an affine variety is a quasi-projective variety. In classical algebraic geometry, a