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Elliptic Cohomology
In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms. History and motivation Historically, elliptic cohomology arose from the study of elliptic genera. It was known by Atiyah and Hirzebruch that if S^1 acts smoothly and non-trivially on a spin manifold, then the index of the Dirac operator vanishes. In 1983, Witten conjectured that in this situation the equivariant index of a certain twisted Dirac operator is at least constant. This led to certain other problems concerning S^1-actions on manifolds, which could be solved by Ochanine by the introduction of elliptic genera. In turn, Witten related these to (conjectural) index theory on free loop spaces. Elliptic cohomology, invented in its original form by Landweber, Stong and Ravenel in the late 1980s, was introduced to clarify certain issues with elliptic genera and provide a context for (conjectural) index theory of families of differe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Stack Of Elliptic Curves
In mathematics, the moduli stack of elliptic curves, denoted as \mathcal_ or \mathcal_, is an algebraic stack over \text(\mathbb) classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves \mathcal_. In particular its points with values in some field correspond to elliptic curves over the field, and more generally morphisms from a scheme S to it correspond to elliptic curves over S. The construction of this space spans over a century because of the various generalizations of elliptic curves as the field has developed. All of these generalizations are contained in \mathcal_. Properties Smooth Deligne-Mumford stack The moduli stack of elliptic curves is a smooth separated Deligne–Mumford stack of finite type over \text(\mathbb), but is not a scheme as elliptic curves have non-trivial automorphisms. j-invariant There is a proper morphism of \mathcal_ to the affine line, the coarse moduli space of elliptic curves, given by the ''j''-i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1810
Events January–March * January 1 – Major-General Lachlan Macquarie officially becomes Governor of New South Wales. * January 4 – Australian seal hunter Frederick Hasselborough discovers Campbell Island, in the Subantarctic. * January 12 – The marriage of Napoleon and Joséphine is annulled. * February 13 – After seizing Jaén, Córdoba, Seville and Granada, Napoleonic troops enter Málaga under the command of General Horace Sebastiani. * February 17 – Napoleon Bonaparte decrees that Rome would become the second capital of the French Empire. * February 20 – Tyrolean rebel leader Andreas Hofer is executed. * March 11 – Napoleon marries Marie-Louise of Austria by proxy in Vienna. April–June * April 2 – Napoleon Bonaparte marries Marie Louise of Austria, Duchess of Parma, in person, in Paris. * April 19 – Venezuela achieves home rule: Vicente Emparán, Governor of the Captaincy General of Venezuela, is removed by the people of Caracas, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2002
The effects of the September 11 attacks of the previous year had a significant impact on the affairs of 2002. The war on terror was a major political focus. Without settled international law, several nations engaged in anti-terror operations, and human rights concerns arose surrounding the treatment of suspected terrorists. Elsewhere, the Colombian conflict and the Nepalese Civil War represented some of the most severe militant conflicts, while the conflict between India and Pakistan was the only one between two sovereign nations. Religious tensions permeated the year, including violence between Hindus and Muslims in India during violent riots and other attacks and attacks on Jews in response to the Second Intifada. The Catholic Church grappled with scrutiny amid sexual abuse cases. Timor-Leste was established as a new sovereign nation, and the African Union began operating as a new intergovernmental organization. The International Criminal Court was founded in July. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graeme Segal
Graeme Bryce Segal FRS (born 21 December 1941) is an Australian mathematician, and professor at the University of Oxford. Biography Segal was educated at the University of Sydney, where he received his BSc degree in 1961. He went on to receive his D.Phil. in 1967 from St Catherine's College, Oxford; his thesis, written under the supervision of Michael Atiyah, was titled ''Equivariant K-theory''. His thesis was in the area of equivariant K-theory. The Atiyah–Segal completion theorem in that subject was a major motivation for the Segal conjecture, which he formulated. He has made many other contributions to homotopy theory in the past four decades, including an approach to infinite loop spaces. He was also a pioneer of elliptic cohomology, which is related to his interest in topological quantum field theory. Segal was an Invited Speaker at the ICM in 1970 in Nice and in 1990 in Kyoto. He was elected a Fellow of the Royal Society in 1982 and an Emeritus Fellow of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Nachrichten
''Mathematische Nachrichten'' (abbreviated ''Math. Nachr.''; English: ''Mathematical News'') is a mathematical journal published in 12 issues per year by Wiley-VCH GmbH. It should not be confused with the ''Internationale Mathematische Nachrichten'', an unrelated publication of the Austrian Mathematical Society. It was established in 1948 by East German mathematician Erhard Schmidt, who became its first editor-in-chief. At that time it was associated with the German Academy of Sciences at Berlin, and published by Akademie Verlag. After the fall of the Berlin Wall, Akademie Verlag was sold to VCH Verlagsgruppe Weinheim, which in turn was sold to John Wiley & Sons. According to the 2020 edition of Journal Citation Reports, the journal had an impact factor of 1.228, ranking it 111th among 333 journals in the category "Mathematics". As of 2021, Ben Andrews, Robert Denk, Klaus Hulek and Frédéric Klopp are the editors-in-chief of the journal. References External links * * P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Homotopy Theory
In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theory, complex-oriented cohomology theories from the "chromatic" point of view, which is based on Daniel Quillen, Quillen's work relating cohomology theories to formal groups. In this picture, theories are classified in terms of their "chromatic levels"; i.e., the heights of the formal groups that define the theories via the Landweber exact functor theorem. Typical theories it studies include: complex K-theory, elliptic cohomology, Morava K-theory and Topological modular forms, tmf. Chromatic convergence theorem In algebraic topology, the chromatic convergence theorem states the homotopy limit of the chromatic tower (defined below) of a finite local spectrum, ''p''-local spectrum X is X itself. The theorem was proved by Hopkins and Ravenel. Statement Let L_ denotes the Bousfield localization with respect to the Morava E-theory and let X be a finite, p-loca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähler manifold or Hodge structure is a complex torus that is a common generalization of the Jacobian variety of a curve and the Picard variety and the Albanese variety. It is obtained by putting a complex structure on the torus H^n(M,\R)/H^n(M,\Z) for ''n'' odd. There are several different natural ways to put a complex structure on this torus, giving several different sorts of intermediate Jacobians, including one due to and one due to . The ones constructed by Weil have natural polarizations if ''M'' is projective, and so are abelian varieties, while the ones constructed by Griffiths behave well under holomorphic deformations. A complex structure on a real vector space is given by an automorphism ''I'' with square -1. The complex structures on H^n(M,\R) are defined using the Hodge decomposition : H^(M,) \otimes = H^(M)\oplus\cdots\oplus H^(M). On H^ the Weil complex structure I_W is multiplication by i^, while the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Modular Forms
In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer ''n'' there is a topological space \operatorname^, and these spaces are equipped with certain maps between them, so that for any topological space ''X'', one obtains an abelian group structure on the set \operatorname^(X) of homotopy classes of continuous maps from ''X'' to \operatorname^. One feature that distinguishes tmf is the fact that its coefficient ring, \operatorname^(point), is almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting the primes 2 and 3, but this inversion erases a lot of torsion information in the coefficient ring. The spectrum of topological modular forms is constructed as the global sections of a sheaf of E-infinity ring spectra on the moduli stack of (generalized) elliptic curves. This theory has rela ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme (algebraic Geometry)
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations and define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise '' Éléments de géométrie algébrique'' (EGA); one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Schemes elaborate the fundamental idea that an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Algebraic Geometry
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over \mathbb), simplicial commutative rings or E_-ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements. Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for intersection theory (or motivic homotopy theory) of singular algebraic varieties and cotangent complexes in deformation theory (cf. J. Francis), among the other applications. Introduction Basic objects of study in the field are derived schemes and derived stacks. The oft-cited motivation is Serre's intersection formula. In the usual formulation, the formula involves the Tor functor and thus, unless ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
