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Torelli Theorem
In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) ''C'' is determined by its Jacobian variety ''J''(''C''), when the latter is given in the form of a principally polarized abelian variety. In other words, the complex torus ''J''(''C''), with certain 'markings', is enough to recover ''C''. The same statement holds over any algebraically closed field. From more precise information on the constructed isomorphism of the curves it follows that if the canonically principally polarized Jacobian varieties of curves of genus \geq 2 are ''k''-isomorphic for ''k'' any perfect field, so are the curves. This result has had many important extensions. It can be recast to read that a certain natural morphism, the period mapping, from the moduli space of curves of a fixed genus, to a moduli space of abelian varieties, is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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] |
Daniel Huybrechts
Daniel Huybrechts (9 November 1966) is a German mathematician, specializing in algebraic geometry. Education and career Huybrechts studied mathematics from 1985 at the Humboldt University of Berlin, where in 1989 he earned his Diplom with Diplom thesis supervisor Herbert Kurke. In 1990–1992 Huybrechts studied at the Max Planck Institute for Mathematics in Bonn, where he earned his PhD in 1992 under Herbert Kurke with thesis ''Stabile Vektorbündel auf algebraischen Flächen. Tjurins Methode zum Studium der Geometrie der Modulräume''. In the academic year 1994–1995 he was at the Institute for Advanced Study and in 1996 at IHES. In 1996 he was a research assistant at the University of Essen, where in 1998 he earned his Habilitierung. In 1997–1998 he was at the École normale supérieure. He was a professor in 1998–2002 at the University of Cologne and in 2002–2005 at the École polytechnique (Chargé de Cours) and, simultaneously, at the University of Paris VII. Since 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eyal Markman
Eyal ( he, אֱיָל; ''lit.'' power) is a kibbutz in the Central District of Israel. Located close to the Green line, it falls under the jurisdiction of the Drom HaSharon Regional Council. In it had a population of . Geography Eyal is located in central Israel within the green line in the central Sharon region, and just to the east of Highway 6. It is approximately 6 km north-east of the city of Kfar Saba. Just to its north-east is the city of Kokhav Ya'ir, and west of the city of Tzur Yigal. To its north-west is the Israeli Arab city of Tira, and to its south is the Palestinian city of Qalqilyah. History Eyal was established in 1949 by Nahal volunteers. Israel sought to establish security settlements along its borders, and Eyal was established on what was then the Jordanian border. It is just north of the West Bank town of Qalqilyah. Attractions Keren Sahar Vintage Auto Museum houses a collection of vintage cars, featuring British automobiles from the 1930s and 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Misha Verbitsky
Misha Verbitsky (russian: link=no, Ми́ша Верби́цкий, born June 20, 1969 in Moscow) is a Russian mathematician. He works at the Instituto Nacional de Matemática Pura e Aplicada in Rio de Janeiro. He is primarily known to the general public as a controversial critic, political activist and independent music publisher. Scientific activities Verbitsky graduated from a Math class at the Moscow State School 57 in 1986, and has been active in mathematics since then. His principal area of interest in mathematics is differential geometry, especially geometry of hyperkähler manifolds and locally conformally Kähler manifolds. He proved an analogue of the global Torelli theorem for hyperkähler manifolds and the mirror conjecture in hyperkähler case. He also contributed to the theory of Hodge structures. His PhD thesis, titled ''Cohomology of compact Hyperkaehler Manifolds'', was defended in 1995 at Harvard University under the supervision of David Kazhdan. He has hel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperkähler Manifold
In differential geometry, a hyperkähler manifold is a Riemannian manifold (M, g) endowed with three integrable almost complex structures I, J, K that are Kähler with respect to the Riemannian metric g and satisfy the quaternionic relations I^2=J^2=K^2=IJK=-1. In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds. Hyperkähler manifolds were defined by Eugenio Calabi in 1979. Equivalent definition in terms of holonomy Equivalently, a hyperkähler manifold is a Riemannian manifold (M, g) of dimension 4n whose holonomy group is contained in the compact symplectic group . Indeed, if (M, g, I, J, K) is a hyperkähler manifold, then the tangent space is a quaternionic vector space for each point of , i.e. it is isomorphic to \mathbb^n for some integer n, where \mathbb is the algebra of quaternions. The compact symplectic group can be considered as the group of orthogonal transformations of \mathbb^n whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fedor Bogomolov
Fedor Alekseyevich Bogomolov (born 26 September 1946) (Фёдор Алексеевич Богомолов) is a Russian and American mathematician, known for his research in algebraic geometry and number theory. Bogomolov worked at the Steklov Institute in Moscow before he became a professor at the Courant Institute in New York. He is most famous for his pioneering work on hyperkähler manifolds. Born in Moscow, Bogomolov graduated from Moscow State University, Faculty of Mechanics and Mathematics, and earned his doctorate (''"candidate degree"'') in 1973, at the Steklov Institute. His doctoral advisor was Sergei Novikov. Geometry of Kähler manifolds Bogomolov's Ph.D. thesis was entitled ''Compact Kähler varieties''. In his early papers Bogomolov studied the manifolds which were later called Calabi–Yau and hyperkähler. He proved a decomposition theorem, used for the classification of manifolds with trivial canonical class. It has been re-proven using the Calabi–Y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Igor Shafarevich
Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. Outside mathematics, he wrote books and articles that criticised socialism and other books which were (controversially) described as anti-semitic. Mathematics From his early years, Shafarevich made fundamental contributions to several parts of mathematics including algebraic number theory, algebraic geometry and arithmetic algebraic geometry. In particular, in algebraic number theory, the Shafarevich–Weil theorem extends the commutative reciprocity map to the case of Galois groups, which are central extensions of abelian groups by finite groups. Shafarevich was the first mathematician to give a completely self-contained formula for the Hilbert pairing, thus initiating an important branch of the study of explicit formulas in number theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ilya Pyatetskii-Shapiro
Ilya Piatetski-Shapiro (Hebrew: איליה פיאטצקי-שפירו; russian: Илья́ Ио́сифович Пяте́цкий-Шапи́ро; 30 March 1929 – 21 February 2009) was a Soviet-born Israeli mathematician. During a career that spanned 60 years he made major contributions to applied science as well as pure mathematics. In his last forty years his research focused on pure mathematics; in particular, analytic number theory, group representations and algebraic geometry. His main contribution and impact was in the area of automorphic forms and L-functions. For the last 30 years of his life he suffered from Parkinson's disease. However, with the help of his wife Edith, he was able to continue to work and do mathematics at the highest level, even when he was barely able to walk and speak. Moscow years: 1929–1959 Ilya was born in 1929 in Moscow, Soviet Union. Both his father, Iosif Grigor'evich, and mother, Sofia Arkadievna, were from traditional Jewish families, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viktor S
The name Victor or Viktor may refer to: * Victor (name), including a list of people with the given name, mononym, or surname Arts and entertainment Film * ''Victor'' (1951 film), a French drama film * ''Victor'' (1993 film), a French short film * ''Victor'' (2008 film), a 2008 TV film about Canadian swimmer Victor Davis * ''Victor'' (2009 film), a French comedy * ''Victor'', a 2017 film about Victor Torres by Brandon Dickerson * ''Viktor'' (film), a 2014 Franco/Russian film Music * ''Victor'' (album), a 1996 album by Alex Lifeson * "Victor", a song from the 1979 album ''Eat to the Beat'' by Blondie Businesses * Victor Talking Machine Company, early 20th century American recording company, forerunner of RCA Records * Victor Company of Japan, usually known as JVC, a Japanese electronics corporation originally a subsidiary of the Victor Talking Machine Company ** Victor Entertainment, or JVCKenwood Victor Entertainment, a Japanese record label ** Victor Interactive So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K3 Surface
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface :x^4+y^4+z^4+w^4=0 in complex projective 3-space. Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of the classification of algebraic surfaces, between the positively curved del Pezzo surfaces (which are easy to classify) and the negatively curved surfaces of general type (which are essentially unclassifiable). K3 surfaces can be considered the simplest algebraic varieti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Torelli Theorem
Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administration * Local news, coverage of events in a local context which would not normally be of interest to those of other localities * Local union, a locally based trade union organization which forms part of a larger union Arts, entertainment, and media * ''Local'' (comics), a limited series comic book by Brian Wood and Ryan Kelly * ''Local'' (novel), a 2001 novel by Jaideep Varma * Local TV LLC, an American television broadcasting company * Locast, a non-profit streaming service offering local, over-the-air television * ''The Local'' (film), a 2008 action-drama film * '' The Local'', English-language news websites in several European countries Computing * .local, a network address component * Local variable, a variable that is given loca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
