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Inter-universal Teichmüller Theory
Inter-universal Teichmüller theory (abbreviated as IUT or IUTT) is the name given by mathematician Shinichi Mochizuki to a theory he developed in the 2000s, following his earlier work in arithmetic geometry. According to Mochizuki, it is "an arithmetic version of Teichmüller theory for number fields equipped with an elliptic curve". The theory was made public in a series of four preprints posted in 2012 to his website. The most striking claimed application of the theory is to provide a proof for various outstanding conjectures in number theory, in particular the ''abc'' conjecture. Mochizuki and a few other mathematicians claim that the theory indeed yields such a proof but this has so far not been accepted by the mathematical community. History The theory was developed entirely by Mochizuki up to 2012, and the last parts were written up in a series of four preprints. Mochizuki made his work public in August 2012 with none of the fanfare that typically accompanies major advan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shinichi Mochizuki
is a Japanese mathematician working in number theory and arithmetic geometry. He is one of the main contributors to anabelian geometry. His contributions include his solution of the Grothendieck conjecture in anabelian geometry about hyperbolic curves over number fields. Mochizuki has also worked in Hodge–Arakelov theory and p-adic Teichmüller theory. Mochizuki developed inter-universal Teichmüller theory, which has attracted attention from non-mathematicians due to claims it provides a resolution of the ''abc'' conjecture. Biography Early life Shinichi Mochizuki was born to parents Kiichi and Anne Mochizuki. When he was five years old, Shinichi Mochizuki and his family left Japan to live in the United States. His father was Fellow of the Center for International Affairs and Center for Middle Eastern Studies at Harvard University (1974–76). Mochizuki attended Phillips Exeter Academy and graduated in 1985. Mochizuki entered Princeton University as an undergraduate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tokyo Institute Of Technology
is a national research university located in Greater Tokyo Area, Japan. Tokyo Tech is the largest institution for higher education in Japan dedicated to science and technology, one of first five Designated National University and selected as a Top Type university of Top Global University Project by the Japanese government. It is generally considered to be one of the most prestigious universities in Japan. Tokyo Tech's main campus is located at Ōokayama on the boundary of Meguro and Ota, with its main entrance facing the Ōokayama Station. Other campuses are located in Suzukakedai and Tamachi. Tokyo Tech is organised into 6 schools, within which there are over 40 departments and research centres. Tokyo Tech enrolled 4,734 undergraduates and 1,464 graduate students for 2015–2016. It employs around 1,100 faculty members. Tokyo Institute of Technology produced a Nobel Prize laureate in Chemistry Hideki Shirakawa Ph.D. History Foundation and early years (1881–1922) Tokyo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vojta's Conjecture
In mathematics, Vojta's conjecture is a conjecture introduced by about heights of points on algebraic varieties over number fields. The conjecture was motivated by an analogy between diophantine approximation and Nevanlinna theory (value distribution theory) in complex analysis. It implies many other conjectures in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic. Statement of the conjecture Let F be a number field, let X/F be a non-singular algebraic variety, let D be an effective divisor on X with at worst normal crossings, let H be an ample divisor on X, and let K_X be a canonical divisor on X. Choose Weil height functions h_H and h_ and, for each absolute value v on F, a local height function \lambda_. Fix a finite set of absolute values S of F, and let \epsilon>0. Then there is a constant C and a non-empty Zariski open set U\subseteq X, depending on all of the above choices, such that :: \sum_ \lambda_(P) + h_(P) \le \epsilon h_H(P) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adele Ring
Adele Laurie Blue Adkins (, ; born 5 May 1988), professionally known by the mononym Adele, is an English singer and songwriter. After graduating in arts from the BRIT School in 2006, Adele signed a record deal with XL Recordings. Her debut album, '' 19'', was released in 2008 and spawned the UK top-five singles "Chasing Pavements" and "Make You Feel My Love". The album was certified 8× platinum in the UK and triple platinum in the US. Adele was honoured with the Brit Award for Rising Star as well as the Grammy Award for Best New Artist. Adele released her second studio album, '' 21'', in 2011. It became the world's best-selling album of the 21st century, with sales of over 31 million copies. It was certified 18× platinum in the UK (the highest by a solo artist of all time) and Diamond in the US. According to ''Billboard'', ''21'' is the top-performing album in the US chart history, topping the ''Billboard'' 200 for 24 weeks (the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Szpiro's Conjecture
In number theory, Szpiro's conjecture relates to the conductor and the discriminant of an elliptic curve. In a slightly modified form, it is equivalent to the well-known ''abc'' conjecture. It is named for Lucien Szpiro, who formulated it in the 1980s. Szpiro's conjecture and its equivalent forms have been described as "the most important unsolved problem in Diophantine analysis" by Dorian Goldfeld, in part to its large number of consequences in number theory including Roth's theorem, the Mordell conjecture, the Fermat–Catalan conjecture, and Brocard's problem. Original statement The conjecture states that: given ε > 0, there exists a constant ''C''(ε) such that for any elliptic curve ''E'' defined over Q with minimal discriminant Δ and conductor ''f'', we have : \vert\Delta\vert \leq C(\varepsilon ) \cdot f^. Modified Szpiro conjecture The modified Szpiro conjecture states that: given ε > 0, there exists a constant ''C''(ε) s ... [...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] |
Frobenioid
In arithmetic geometry, a Frobenioid is a category with some extra structure that generalizes the theory of line bundles on models of finite extensions of global fields. Frobenioids were introduced by . The word "Frobenioid" is a portmanteau of Frobenius and monoid, as certain Frobenius morphisms between Frobenioids are analogues of the usual Frobenius morphism, and some of the simplest examples of Frobenioids are essentially monoids. The Frobenioid of a monoid If ''M'' is a commutative monoid, it is acted on naturally by the monoid ''N'' of positive integers under multiplication, with an element ''n'' of ''N'' multiplying an element of ''M'' by ''n''. The Frobenioid of ''M'' is the semidirect product of ''M'' and ''N''. The underlying category of this Frobenioid is category of the monoid, with one object and a morphism for each element of the monoid. The standard Frobenioid is the special case of this construction when ''M'' is the additive monoid of non-negative integers. Elemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge–Arakelov Theory
In mathematics, Hodge–Arakelov theory of elliptic curves is an analogue of classical and p-adic Hodge theory for elliptic curves carried out in the framework of Arakelov theory. It was introduced by . It bears the name of two mathematicians, Suren Arakelov and W. V. D. Hodge. The main comparison in his theory remains unpublished as of 2019. Mochizuki's main comparison theorem in Hodge–Arakelov theory states (roughly) that the space of polynomial functions of degree less than ''d'' on the universal extension of a smooth elliptic curve in characteristic 0 is naturally isomorphic (via restriction) to the ''d''2-dimensional space of functions on the ''d''- torsion points. It is called a 'comparison theorem' as it is an analogue for Arakelov theory of comparison theorems in cohomology relating de Rham cohomology to singular cohomology of complex varieties or étale cohomology of ''p''-adic varieties. In and he pointed out that arithmetic Kodaira–Spencer map and Gauss–Manin c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Teichmüller Theory
In mathematics, ''p''-adic Teichmüller theory describes the "uniformization" of ''p''-adic curves and their moduli, generalizing the usual Teichmüller theory that describes the uniformization of Riemann surfaces and their moduli. It was introduced and developed by . The first problem is to reformulate the Fuchsian uniformization of a complex Riemann surface (an isomorphism from the upper half plane to a universal covering space of the surface) in a way that makes sense for ''p''-adic curves. The existence of a Fuchsian uniformization is equivalent to the existence of a canonical indigenous bundle over the Riemann surface: the unique indigenous bundle that is invariant under complex conjugation and whose monodromy representation is quasi-Fuchsian. For ''p''-adic curves the analogue of complex conjugation is the Frobenius endomorphism, and the analogue of the quasi-Fuchsian condition is an integrality condition on the indigenous line bundle. So ''p''-adic Teichmüller theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anabelian Geometry
Anabelian geometry is a theory in number theory which describes the way in which the algebraic fundamental group ''G'' of a certain arithmetic variety ''X'', or some related geometric object, can help to restore ''X''. The first results for number fields and their absolute Galois groups were obtained by Jürgen Neukirch, Masatoshi Gündüz Ikeda, Kenkichi Iwasawa, and Kôji Uchida ( Neukirch–Uchida theorem, 1969) prior to conjectures made about hyperbolic curves over number fields by Alexander Grothendieck. As introduced in ''Esquisse d'un Programme'' the latter were about how topological homomorphisms between two arithmetic fundamental groups of two hyperbolic curves over number fields correspond to maps between the curves. These Grothendieck conjectures were partially solved by Hiroaki Nakamura and Akio Tamagawa, while complete proofs were given by Shinichi Mochizuki. Anabelian geometry can be viewed as one of the three generalizations of class field theory. Unlike two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Synthese
''Synthese'' () is a scholarly periodical specializing in papers in epistemology, methodology, and philosophy of science, and related issues. Its subject area is divided into four specialties, with a focus on the first three: (1) "epistemology, methodology, and philosophy of science, all broadly understood"; (2) "foundations of logic and mathematics, where 'logic', 'mathematics', and 'foundations' are all broadly understood"; (3) "formal methods in philosophy, including methods connecting philosophy to other academic fields"; and (4) "issues in ethics and the history and sociology of logic, mathematics, and science that contribute to the contemporary studies". As of 2022, according to Google Scholar's metrics ( h-5 index and h-5 index median), it is the top philosophy journal, but other metrics do not rank the journal as highly. Overview Published articles include specific treatment of methodological issues in science such as induction, probability, causation, statistics, symboli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematical Intelligencer
''The Mathematical Intelligencer'' is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. Volumes are released quarterly with a subset of open access articles. Springer also cross-publishes some of the articles in ''Scientific American''. Karen Parshall and Sergei Tabachnikov are currently the co-editors-in-chief. History The journal was started informally in 1971 by Walter Kaufman-Buehler, Alice Peters and Klaus Peters. "Intelligencer" was chosen by Kaufman-Buehler as a word that would appear slightly old-fashioned. An exploration of mathematically themed stamps, written by Robin Wilson, became one of its earliest columns. In 1978, the founders appointed Bruce Chandler and Harold "Ed" Edwards Jr. to serve jointly in the role of editor-in-chief. Prior to 1978, articles of the ''Intelligencer'' were not contained in regular volumes and were sent out ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
