Information om | Engelska ordet K-THEORY

Exempel på hur man kan använda K-THEORY i en mening

• John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems.
• Examples of results gleaned from the K-theory approach include the Grothendieck–Riemann–Roch theorem, Bott periodicity, the Atiyah–Singer index theorem, and the Adams operations.
• The Atiyah-Jänich theorem identifies the K-theory K(X) of a compact topological space X with the set of homotopy classes of continuous maps from X to the space of Fredholm operators H→H, where H is the separable Hilbert space and the set of these operators carries the operator norm.
• Before his work in defining higher algebraic K-theory, Quillen worked on the Adams conjecture, formulated by Frank Adams, in homotopy theory.
• Intersection theory is still a motivating force in the development of (higher) algebraic K-theory through its links with motivic cohomology and specifically Chow groups.
• Among other advances he showed how the Adams spectral sequence, a powerful tool for proceeding from homology theory to the calculation of homotopy groups, could be adapted to the new (at that time) cohomology theory typified by cobordism and K-theory.
• In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres.
• By Bloch, Lichtenbaum, Friedlander, Suslin, and Levine, there is a spectral sequence from motivic cohomology to algebraic K-theory for every smooth scheme X over a field, analogous to the Atiyah-Hirzebruch spectral sequence in topology:.
• In algebraic geometry one can also define analogous Stiefel–Whitney classes for vector bundles with a non-degenerate quadratic form, taking values in etale cohomology groups or in Milnor K-theory.
• In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s.
• 1990 Raoul Bott for having been instrumental in changing the face of geometry and topology, with his incisive contributions to characteristic classes, K-theory, index theory, and many other tools of modern mathematics.
• proved a far-reaching generalization of Goodwillie's result, stating that for a commutative ring A so that the Henselian lemma holds with respect to the ideal I, the relative K-theory is isomorphic to relative topological cyclic homology (without tensoring both with Q).
• This construction was extended to equivariant K-theory and to holomorphic K-theory by Mathai and Stevenson.
• Using the superconnection formalism of Quillen, they obtained a refinement of the Riemann–Roch formula, which links together the Thom classes in K-theory and cohomology, as an equality on the level of differential forms.
• Milnor took the hypothesis that these were the only relations, hence he gave the following "ad-hoc" definition of Milnor K-theory as.
• Such stacks of branes are inconsistent in a non-torsion Neveu–Schwarz (NS) 3-form background, which, as was highlighted by , complicates the extension of the K-theory classification to such cases.
• In 1982, Beilinson published his own conjectures about the existence of motivic cohomology groups for schemes, provided as hypercohomology groups of a complex of abelian groups and related to algebraic K-theory by a motivic spectral sequence, analogous to the Atiyah–Hirzebruch spectral sequence in algebraic topology.
• Under Atiyah's tutelage Morava concentrated on the relation between K-theory and cobordism, and when Daniel Quillen's work on that subject appeared he saw that ideas of Sergei Novikov implied close connections between the stable homotopy category and the derived category of quasicoherent sheaves on the moduli stack of one-dimensional formal groups; in particular, that the category of spectra is naturally stratified by height.
• It's named after Friedhelm Waldhausen, who introduced this notion (under the term category with cofibrations and weak equivalences) to extend the methods of algebraic K-theory to categories not necessarily of algebraic origin, for example the category of topological spaces.
• This can be encoded as a (p + 1)-functional on A satisfying some algebraic conditions and give Hochschild / cyclic cohomology cocycles, which describe the above maps from K-theory to the integers.

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