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Fourier–Deligne Transform
In algebraic geometry, the Fourier–Deligne transform, or ℓ-adic Fourier transform, or geometric Fourier transform, is an operation on objects of the derived category of ''ℓ''-adic sheaves over the affine line. It was introduced by Pierre Deligne on November 29, 1976 in a letter to David Kazhdan as an analogue of the usual Fourier transform. It was used by Gérard Laumon to simplify Deligne's proof of the Weil conjectures In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. Th .... References * * * Algebraic geometry {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proceeds on the basis that the objects of ''D''(''A'') should be chain complexes in ''A'', with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences. The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made remarkab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-adic Sheaf
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that a discret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal. Early life and education Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles (ULB), writing a dissertation titled ''Théorème de Lefschetz et critères de dégénérescence de suites spectrales'' (Theorem of Lefschetz and criteria of degeneration of spectral sequences). He completed his doctorate at the University of Paris-Sud in Orsay 1972 under the supervision of Alexander Grothendieck, with a thesis titled ''Théorie de Hodge''. Career Starting in 1972, Deligne worked with Grothendieck at the Institut des Hautes Études Scientifiques (IHÉS) near Paris, initially on the generalization within scheme theory of Zariski's main theorem. In 196 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Kazhdan
David Kazhdan ( he, דוד קשדן), born Dmitry Aleksandrovich Kazhdan (russian: Дми́трий Александро́вич Кажда́н), is a Soviet and Israeli mathematician known for work in representation theory. Kazhdan is a 1990 MacArthur Fellow. Biography Kazhdan was born on 20 June 1946 in Moscow, USSR. His father is Alexander Kazhdan. He earned a doctorate under Alexandre Kirillov in 1969 and was a member of Israel Gelfand's school of mathematics. He is Jewish, and emigrated from the Soviet Union to take a position at Harvard University in 1975. He changed his name from Dmitri Aleksandrovich to David and became an Orthodox Jew around that time. In 2002, he immigrated to Israel and is now a professor at the Hebrew University of Jerusalem as well as a professor emeritus at Harvard. On October 6, 2013, Kazhdan was critically injured in a car accident while riding a bicycle in Jerusalem. Kazhdan has four children. His son, Eli Kazhdan, was general director of N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. The term ''Fourier transform'' refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gérard Laumon
Gérard Laumon (; born 1952) is a French mathematician, best known for his results in number theory, for which he was awarded the Clay Research Award. Life and work Laumon studied at the École Normale Supérieure and Paris-Sud 11 University, Orsay. He was awarded the Silver Medal of the CNRS in 1987, and the E. Dechelle prize of the French Academy of the Sciences in 1992.2004 Clay Institute Annual report, available at http://www2.maths.ox.ac.uk/cmi/library/annual_report/ar2004/04report_clayaward.pdf In 2004, Laumon and Ngô Bảo Châu received the Clay Research Award for the proof of the fundamental lemma for unitary groups, a component in the Langlands program in number theory. In 2012, he became a fellow of the American Mathematical Society. retrieved 2013-01- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil Conjectures
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. The conjectures concern the generating functions (known as local zeta functions) derived from counting points on algebraic varieties over finite fields. A variety over a finite field with elements has a finite number of rational points (with coordinates in the original field), as well as points with coordinates in any finite extension of the original field. The generating function has coefficients derived from the numbers of points over the extension field with elements. Weil conjectured that such ''zeta functions'' for smooth varieties are rational functions, satisfy a certain functional equation, and have their zeros in restricted places. The last two parts were consciously modelled on the Riemann zeta function, a kind of generating f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Publications Mathématiques De L'IHÉS
''Publications Mathématiques de l'IHÉS'' is a peer-reviewed mathematical journal. It is published by Springer Science+Business Media on behalf of the Institut des Hautes Études Scientifiques, with the help of the Centre National de la Recherche Scientifique. The journal was established in 1959 and was published at irregular intervals, from one to five volumes a year. It is now biannual. The editor-in-chief is Claire Voisin (Collège de France). See also *''Annals of Mathematics'' *'' Journal of the American Mathematical Society'' *''Inventiones Mathematicae ''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors ...'' External links * Back issues from 1959 to 2010 Mathematics journals Publications established in 1959 Springer Science+Business Media academic journals Biannual journal ... [...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] |
