Gerd Faltings is a German mathematician known for his work in arithmetic geometry. From 1972 to 1978, Faltings studied physics at the University of Münster. In 1978 he received his PhD in mathematics. In 1981 he obtained the venia legendi both from the University of Münster. During this time he was an assistant professor at the University of Münster. From 1982 to 1984, he was professor at the University of Wuppertal. From 1985 to 1994, he was professor at Princeton University. In the fall of 1988 and in the academic year 1992–1993 he was a visiting scholar at the Institute for Advanced Study. In 1986 he was awarded the Fields Medal at the ICM at Berkeley for proving the Tate conjecture for abelian varieties over number fields, the Shafarevich conjecture for abelian varieties over number fields and the Mordell conjecture, which states that any non-singular projective curve of genus g > 1 defined over a number field K contains only finitely many K-rational points. As a Fields Medalist he gave an ICM plenary talk Recent progress in arithmetic algebraic geometry.
In 1994 as an ICM invited speaker in Zurich he gave a talk Mumford-Stabilität in der algebraischen Geometrie. Extending methods of Paul Vojta, he proved the Mordell–Lang conjecture, a generalization of the Mordell conjecture. Together with Gisbert Wüstholz, he reproved Roth's theorem, for which Roth had been awarded the Fields medal in 1958. In 1994, he returned to Germany and from 1994 to 2018, he was a director of the Max Planck Institute for Mathematics in Bonn. In 1996, he received the Gottfried Wilhelm Leibniz Prize of the Deutsche Forschungsgemeinschaft, the highest honour awarded in German research. Faltings was the formal supervisor of Wieslawa Niziol, Nikolai Dourov. Fields Medal Guggenheim Fellowship Gottfried Wilhelm Leibniz Prize King Faisal International Prize Shaw Prize Foreign Member of the Royal Society Cantor Medal National Academy of Sciences foreign associate
Root of unity
In mathematics, a root of unity called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, are important in number theory, the theory of group characters, the discrete Fourier transform. Roots of unity can be defined in any field. If the characteristic of the field is zero, they are complex numbers that are algebraic integers. In positive characteristic, they belong to a finite field, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains n nth roots of unity, except if n is a multiple of the characteristic of the field. An nth root of unity, where n is a positive integer, is a number z satisfying the equation z n = 1. Unless otherwise specified, the roots of unity may be taken to be complex numbers, in this case, the nth roots of unity are exp = cos 2 k π n + i sin 2 k π n, k = 0, 1, …, n − 1; however the defining equation of roots of unity is meaningful over any field F, this allows considering roots of unity in F.
Whichever is the field F, the roots of unity in F are either complex numbers, if the characteristic of F is 0, or, belong to a finite field. Conversely, every nonzero element in a finite field is a root of unity in that field. See Root of unity modulo n and Finite field for further details. An nth root of unity is said to be primitive if it is not a kth root of unity for some smaller k, if z n = 1 and z k ≠ 1 for k = 1, 2, 3, …, n − 1. If n is a prime number, all nth roots of unity, except 1, are primitive. In the above formula in terms of exponential and trigonometric functions, the primitive nth roots of unity are those for which k and n are coprime integers. Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see Finite field § Roots of unity. For the case of roots of unity in rings of modular integers, see Root of unity modulo n; every nth root of unity z is a primitive ath root of unity for some a ≤ n, the smallest positive integer such that za = 1.
Any integer power of an nth root of unity is an nth root of unity, as n = z k n = k = 1 k = 1. This is true for negative exponents. In particular, the reciprocal of an nth root of unity is its complex conjugate, is an nth root of unity: 1 z = z − 1 = z n − 1 = z ¯. If z is an nth root of unity and a ≡ b za = zb. In fact, by the definition of congruence, a = b + kn for some integer k, z a = z b + k n = z b z k n = z b k = z b 1 k = z b. Therefore, given a power za of z, one has za = zr, where 0 ≤ r < n is the remainder of the Euclidean division of a by n. Let z be a primitive nth root of unity; the powers z, z2, ... zn−1, zn = z0 = 1 are nth root of unity and are all distinct. This implies that z, z2, ... zn−1, zn = z0 = 1 are all of the nth roots of unity, since an nth-degree polynomial equation has at most n distinct solutions. From the preceding, it follows that, if z is a primitive nth root of unity z a = z b if and only if a ≡ b. If z is not primitive a ≡ b implies z a = z b, but the converse may be false, as shown by the following example.
If n = 4, a non-primitive nth root of unity is z = –1, one has z 2 =
John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry. He is professor emeritus at Harvard University, he was awarded the Abel Prize in 2010. Tate was born in Minneapolis, his father, John Tate Sr. was a professor of physics at the University of Minnesota, a longtime editor of Physical Review. His mother, Lois Beatrice Fossler, was a high school English teacher. Tate Jr. received his bachelor's degree in mathematics from Harvard University, entered the doctoral program in physics at Princeton University. He transferred to the mathematics department and received his PhD in 1950 as a student of Emil Artin. Tate taught at Harvard for 36 years before joining the University of Texas in 1990 as a Sid W. Richardson Foundation Regents Chair, he retired from the Texas mathematics department in 2009, returned to Harvard as a professor emeritus. He resides in Cambridge, Massachusetts with his wife Carol.
He has three daughters with his first wife Karin Tate. Tate's thesis on Fourier analysis in number fields has become one of the ingredients for the modern theory of automorphic forms and their L-functions, notably by its use of the adele ring, its self-duality and harmonic analysis on it. Together with his teacher Emil Artin, Tate gave a cohomological treatment of global class field theory, using techniques of group cohomology applied to the idele class group and Galois cohomology; this treatment made more transparent some of the algebraic structures in the previous approaches to class field theory which used central division algebras to compute the Brauer group of a global field. Subsequently, Tate introduced. In the decades following that discovery he extended the reach of Galois cohomology with the Poitou–Tate duality, the Tate–Shafarevich group, relations with algebraic K-theory. With Jonathan Lubin, he recast local class field theory by the use of formal groups, creating the Lubin–Tate local theory of complex multiplication.
He has made a number of individual and important contributions to p-adic theory. He found a p-adic analogue of Hodge theory, now called Hodge–Tate theory, which has blossomed into another central technique of modern algebraic number theory. Other innovations of his include the'Tate curve' parametrization for certain p-adic elliptic curves and the p-divisible groups. Many of his results were not published and some of them were written up by Serge Lang, Jean-Pierre Serre, Joseph H. Silverman and others. Tate and Serre collaborated on a paper on good reduction of abelian varieties; the classification of abelian varieties over finite fields was carried out by Taira Tate. The Tate conjectures are the equivalent for étale cohomology of the Hodge conjecture, they relate to the Galois action on the l-adic cohomology of an algebraic variety, identifying a space of'Tate cycles' that conjecturally picks out the algebraic cycles. A special case of the conjectures, which are open in the general case, was involved in the proof of the Mordell conjecture by Gerd Faltings.
Tate has had a major influence on the development of number theory through his role as a Ph. D. advisor. His students include George Bergman, Bernard Dwork, Benedict Gross, Robert Kottwitz, Jonathan Lubin, Stephen Lichtenbaum, James Milne, V. Kumar Murty, Carl Pomerance, Ken Ribet, Joseph H. Silverman, Dinesh Thakur. In 1956 Tate was awarded the American Mathematical Society's Cole Prize for outstanding contributions to number theory. In 1995 he received the Leroy P. Steele Prize for Lifetime Achievement from the American Mathematical Society, he was awarded a Wolf Prize in Mathematics in 2002/03 for his creation of fundamental concepts in algebraic number theory. In 2012 he became a fellow of the American Mathematical Society. In 2010, the Norwegian Academy of Science and Letters, of which he is a member, awarded him the Abel Prize, citing "his vast and lasting impact on the theory of numbers". According to a release by the Abel Prize committee "Many of the major lines of research in algebraic number theory and arithmetic geometry are only possible because of the incisive contributions and illuminating insights of John Tate.
He has left a conspicuous imprint on modern mathematics."Tate has been described as "one of the seminal mathematicians for the past half-century" by William Beckner, Chairman of the Department of Mathematics at the University of Texas. Tate, Fourier analysis in number fields and Hecke's zeta functions, Princeton University Ph. D. thesis under Emil Artin. Reprinted in Cassels, J. W. S.. Algebraic number theory, London: Academic Press, pp. 305–347, MR 0215665 Tate, John, "The higher dimensional cohomology groups of class field theory", Ann. of Math. 2, 56: 294–297, doi:10.2307/1969801, MR 0049950 Lang, Serge.
In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.
A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.
Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.
Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a
Michiel Hazewinkel is a Dutch mathematician, Emeritus Professor of Mathematics at the Centre for Mathematics and Computer and the University of Amsterdam known for his 1978 book Formal groups and applications and as editor of the Encyclopedia of Mathematics. Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam, he received his BA in Mathematics and Physics in 1963, his MA in Mathematics with a minor in Philosophy in 1965 and his PhD in 1969 under supervision of Frans Oort and Albert Menalda for the thesis "Maximal Abelian Extensions of Local Fields". After graduation Hazewinkel started his academic career as Assistant Professor at the University of Amsterdam in 1969. In 1970 he became Associate Professor at the Erasmus University Rotterdam, where in 1972 he was appointed Professor of Mathematics at the Econometric Institute. Here he was thesis advisor of Roelof Stroeker, M. van de Vel, Jo Ritzen, Gerard van der Hoek. From 1973 to 1975 he was Professor at the Universitaire Instelling Antwerpen, where Marcel van de Vel was his PhD student.
From 1982 to 1985 he was appointed part-time Professor Extraordinarius in Mathematics at the Erasmus Universiteit Rotterdam, part-time Head of the Department of Pure Mathematics at the Centre for Mathematics and Computer in Amsterdam. In 1985 he was appointed Professor Extraordinarius in Mathematics at the University of Utrecht, where he supervised the promotion of Frank Kouwenhoven, Huib-Jan Imbens, J. Scholma and F. Wainschtein. At the Centre for Mathematics and Computer CWI in Amsterdam in 1988 he became Professor of Mathematics and head of the Department of Algebra and Geometry until his retirement in 2008. Hazewinkel has been managing editor for journals as Nieuw Archief voor Wiskunde since 1977, he was managing editor for the book series Mathematics and Its Applications for Kluwer Academic Publishers in 1977. Hazewinkel was member of 15 professional societies in the field of Mathematics, participated in numerous administrative tasks in institutes, Program Committee, Steering Committee, Consortiums and Boards.
In 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. Hazewinkel has authored and edited several books, numerous articles. Books, selection: 1970. Géométrie algébrique-généralités-groupes commutatifs. With Michel Demazure and Pierre Gabriel. Masson & Cie. 1976. On invariants, canonical forms and moduli for linear, finite dimensional, dynamical systems. With Rudolf E. Kalman. Springer Berlin Heidelberg. 1978. Formal groups and applications. Vol. 78. Elsevier. 1993. Encyclopaedia of Mathematics. Ed. Vol. 9. Springer. Articles, a selection: Hazewinkel, Michiel. "Moduli and canonical forms for linear dynamical systems II: The topological case". Mathematical Systems Theory. 10: 363–385. Doi:10.1007/BF01683285. Archived from the original on 12 December 2013. Hazewinkel, Michiel. "On Lie algebras and finite dimensional filtering". Stochastics. 7: 29–62. Doi:10.1080/17442508208833212. Archived from the original on 12 December 2013. Hazewinkel, M.. J.. "Nonexistence of finite-dimensional filters for conditional statistics of the cubic sensor problem".
Systems & Control Letters. 3: 331–340. Doi:10.1016/0167-691190074-9. Hazewinkel, Michiel. "The algebra of quasi-symmetric functions is free over the integers". Advances in Mathematics. 164: 283–300. Doi:10.1006/aima.2001.2017. Homepage
Encyclopedia of Mathematics
The Encyclopedia of Mathematics is a large reference work in mathematics. It is available in book form and on CD-ROM; the 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate level, the presentation is technical in nature. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer; the CD-ROM contains three-dimensional objects. The encyclopedia has been translated from the Soviet Matematicheskaya entsiklopediya edited by Ivan Matveevich Vinogradov and extended with comments and three supplements adding several thousand articles; until November 29, 2011, a static version of the encyclopedia could be browsed online free of charge online. This URL now redirects to the new wiki incarnation of the EOM. A new dynamic version of the encyclopedia is now available as a public wiki online; this new wiki is a collaboration between the European Mathematical Society. This new version of the encyclopedia includes the entire contents of the previous online version, but all entries can now be publicly updated to include the newest advancements in mathematics.
All entries will be monitored for content accuracy by members of an editorial board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Sov. Entsiklopediya, 1977. Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 1, Kluwer, 1987. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 2, Kluwer, 1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 3, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 4, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 5, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 6, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 7, Kluwer, 1991. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 8, Kluwer, 1992. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 9, Kluwer, 1993. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 10, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement I, Kluwer, 1997. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement II, Kluwer, 2000.
Hazewinkel, M. Encyclopaedia of Mathematics, Supplement III, Kluwer, 2002. Hazewinkel, M. Encyclopaedia of Mathematics on CD-ROM, Kluwer, 1998. Encyclopedia of Mathematics, public wiki monitored by an editorial board under the management of the European Mathematical Society. List of online encyclopedias Official website Publications by M. Hazewinkel, at ResearchGate