Search results
Results from the WOW.Com Content Network
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott (1957, 1959), 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.
Raoul Bott (September 24, 1923 – December 20, 2005) [1] was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem , the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem .
The symmetry classes are ordered according to the Bott clock (see below) so that the same values repeat in the diagonals. [5] An X in the table of "Symmetries" indicates that the Hamiltonian of the symmetry is broken with respect to the given operator. A value of ±1 indicates the value of the operator squared for that system.
Raoul Bott used Morse–Bott theory in his original proof of the Bott periodicity theorem. Round functions are examples of Morse–Bott functions, where the critical sets are (disjoint unions of) circles. Morse homology can also be formulated for Morse–Bott functions; the differential in Morse–Bott homology is computed by a spectral ...
In 2000 Allyn Jackson interviewed Bott, who then revealed Shapiro's part in the Periodicity Theorem. He explained that there was a controversy in dimension 10 about the homotopy of the unitary group. I hit upon a very complicated method involving the exceptional group G2 to check the conundrum independently. My good friend Arnold Shapiro and I ...
The abstract theory of Clifford modules was founded by a paper of M. F. Atiyah, R. Bott and Arnold S. Shapiro. A fundamental result on Clifford modules is that the Morita equivalence class of a Clifford algebra (the equivalence class of the category of Clifford modules over it) depends only on the signature p − q (mod 8).
Toggle the table of contents. Topological K-theory. 7 languages. Deutsch; ... The phenomenon of periodicity named after Raoul Bott (see Bott periodicity theorem) ...
The famous periodicity theorem of Raoul Bott asserts that the K-theory of any space X is isomorphic to that of the S 2 X, the double suspension of X. In algebraic geometry, one considers the K-theory groups consisting of coherent sheaves on a scheme X, as well as the K-theory groups of vector bundles on the scheme with the above equivalence ...