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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 differentia ... [...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] |
Formal Group
In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between Lie groups (or algebraic groups) and Lie algebras. They are used in algebraic number theory and algebraic topology. Definitions A one-dimensional formal group law over a commutative ring ''R'' is a power series ''F''(''x'',''y'') with coefficients in ''R'', such that # ''F''(''x'',''y'') = ''x'' + ''y'' + terms of higher degree # ''F''(''x'', ''F''(''y'',''z'')) = ''F''(''F''(''x'',''y''), ''z'') (associativity). The simplest example is the additive formal group law ''F''(''x'', ''y'') = ''x'' + ''y''. The idea of the definition is that ''F'' should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the id ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology Theories
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do wi ... [...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 empire. * February 20 – Tyrolean rebel leader Andreas Hofer is executed. * March 4 – Peninsular War: The French Army, under the command of André Masséna, retreats from Portugal. * March 11 – Napoleon marries Marie-Louise of Austria by proxy in Vienna. April–June * April – Kaumualii receives an assurance of the continued independence of the Kingdom of Hawaii. * April 2 - Napoleon Bonaparte marries Marie Louise of Austria, D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2002
File:2002 Events Collage.png, From left, clockwise: The 2002 Winter Olympics are held in Salt Lake City; Queen Elizabeth The Queen Mother and her daughter Princess Margaret, Countess of Snowdon die; East Timor gains East Timor independence, independence from Indonesia and is admitted to the United Nations; an Armenia, Armenian postage stamp depicts the 2002 FIFA World Cup, which was held in South Korea and Japan; the Department of Homeland Security is created in the wake of 9/11 to counter further terrorist threats against the United States; the 2002 Überlingen mid-air collision kills 71 people; FBI agents investigate a crime scene related to the D.C. sniper attacks; the Euro becomes the official currency of the European Union., 300x300px, thumb rect 0 0 200 200 2002 Winter Olympics rect 200 0 400 200 Death and funeral of Queen Elizabeth The Queen Mother rect 400 0 600 200 East Timor independence rect 0 200 300 400 Euro rect 300 200 600 400 2002 FIFA World Cup rect 200 400 400 600 ... [...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 All S ... [...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 The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a g ... [...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 theories from the "chromatic" point of view, which is based on 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 tmf. Chromatic convergence theorem In algebraic topology, the chromatic convergence theorem states the homotopy limit of the chromatic tower (defined below) of a finite ''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-local spectrum. Then there is a tower associated to the localizations :\cdots \rightarrow L ... [...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 manifold, 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 ... [...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, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 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"; 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. Formally, a scheme is a topological space together with commut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
