K-theory

English K-theory Etymology From circa 1960. The mul K stands for en de Klasse class. The theory developed out of algebraic geometry after the 1957 publication of work by German-born French mathematician Alexander Grothendieck. Noun - K-theory en uncountable algebra algebraic geometry algebraic topology The study of rings R generated by the set of vector bundles over some topological space or scheme;en dated obsolete that part of algebraic topology comprising what is now called topological K-theory. en 1994 Jonathan Rosenberg (mathematician) Jonathan Rosenberg Algebraic K-Theory and Its Applications https://books.google.com.au/books?id=W_HjBwAAQBAJ&printsec=frontcover&dq=%22K-theory%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjO6-TS3tnsAhVUjuYKHaDKCVQQ6AEwAXoECAEQAg#v=onepage&q=%22K-theory%22&f=false 1 Springer K-theory as an independent discipline is a fairly new subject, only about 35 years old. en 2000 M. Rørdam; F. Larsen; Flemming Larsen; N. Laustsen An Introduction to K-Theory for C*-Algebras https://books.google.com.au/books?id=SMiB8VIB5UIC&printsec=frontcover&dq=%22K-theory%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjO6-TS3tnsAhVUjuYKHaDKCVQQ6AEwAnoECEwQAg#v=onepage&q=%22K-theory%22&f=false ix w:Cambridge University Press K-theory was developed by Atiyah and Hirzebruch in the 1960s based on work of Grothendieck in algebraic geometry. It was introduced as a tool in C*-algebra theory in the early 1970s through some specific applications described below. Very briefly, K-theory (for C*-algebras) is a pair of functors, called K0 and K1, that to each C*-algebra A associate two Abelian groups K0(A) and K1(A). en countable The cohomology generated by the set of vector bundles over some topological space or scheme. en 1978 Michiel Hazewinkel Formal Groups and Applications Harcourt Brace Jovanovich (Academic Press) https://books.google.com.au/books?id=ZuAaMhSMEEkC&pg=PR11&dq=%22K-theories%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjXrPXkys7sAhXCgeYKHX3vBbsQ6AEwIXoECFEQAg#v=onepage&q=%22K-theories%22&f=false xi The theory of formal groups has found a number of rather spectacular applications in recent years in number theory, arithmetical algebraic geometry, algebraic geometry, and algebraic topology, ranging from congruences for the coefficients of modular forms and local class field theory to extraordinary K-theories and (indirectly) results on the homotopy groups of spheres. en 2007 Mathematical Reviews w:American Mathematical Society https://books.google.com.au/books?id=1pfjAAAAMAAJ&q=%22K-theories%22&dq=%22K-theories%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjXrPXkys7sAhXCgeYKHX3vBbsQ6AEwJnoECFYQAg 4338 This has changed in recent years: on the one hand, bivariant K-theories were defined by the author for other categories of algebras [Doc. Math. 2 (1997), 139–182 (electronic); MR1456322 (98h: 19006)]; on the other hand, the local cyclic homology theory by M. Puschnigg works for small algebras and C*-algebras alike. en 2014 Stephan Stolz Topology and Field Theories w:American Mathematical Society https://books.google.com.au/books?id=2-tgAwAAQBAJ&pg=PA102&dq=%22K-theories%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjXrPXkys7sAhXCgeYKHX3vBbsQ6AEwA3oECAUQAg#v=onepage&q=%22K-theories%22&f=false 102 In particular, the Morava K-theory appears as a direct summand of 'complex K-theory modulo '. However, the higher Morava K-theories are much more mysterious from a geometric point of view.REMARK 5.8. For a fixed prime number , the Morava K-theories are defined for . Usage notes In mathematics: Following Alexander Grothendieck's 1957 publication of a proof of the Grothendieck–Riemann–Roch theorem, K-theory was developed as a subfield of algebraic topology (notably in relation to C*-algebra theory). This subfield is now known as en topological K-theory. Subsequently, K-theory was generalised as a subfield of algebraic geometry, now often called en algebraic K-theory. This development allows a revisionist definition of topological K-theory as "the application of tools from K-theory to problems in algebraic topology." In its current state, K-theory (in the area of study sense) comprises algebraic K-theory and topological K-theory. The term K-theory also refers (in its countable sense) to the cohomology (a ring whose multiplication operation is tensor product) that is the central subject of K-theory. Applications have also been found in: The study of operator algebras (within functional analysis), where K-theory is a fundamental tool. The study of certain kinds of invariant of large matrices. In physics: In condensed matter physics, K-theory is a tool used to classify certain materials of importance. In string theory, a variation called en twisted K-theory (aka K-theory with local coefficients) is the basis of an application called K-theory classification, by which certain (hypothetical) objects are classified. Hyponyms study of rings generated by vector bundles en algebraic K-theory algebraic K-theory, en topological K-theory topological K-theory Derived terms en algebraic K-theory complex K-theory K-theory classification Morava K-theory real K-theory topological K-theory twisted K-theory Translations study of rings generated by vector bundles cohomology generated by vector bundles Further reading Algebraic K-theory Algebraic K-theory Topological K-theory Topological K-theory Twisted K-theory Twisted K-theory Cohomology Grothendieck–Riemann–Roch theorem K-theory (physics) K-theory (physics) Algebraic K-theory on Encyclopedia of Mathematics K-theory on Encyclopedia of Mathematics en terms containing italics