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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 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).
Furthermore, the table assumes the limit of an infinite number of bands, i.e. involves Hamiltonians for . The table also is periodic in the sense that the group of invariants in dimensions is the same as the group of invariants in + dimensions. In the case of no anti-unitary symmetries, the invariant groups are periodic in dimension by 2.
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 abstract algebra, the Virasoro group or Bott–Virasoro group (often denoted by Vir) [1] is an infinite-dimensional Lie group defined as the universal central extension of the group of diffeomorphisms of the circle. The corresponding Lie algebra is the Virasoro algebra, which has a key role in conformal field theory (CFT) and string theory.
Raoul Bott Robert W. Brooks (1985) Robert Wolfe Brooks (Washington, D.C., September 16, 1952 – Montreal, September 5, 2002) was a mathematician known for his work in spectral geometry , Riemann surfaces , circle packings , and differential geometry .
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 ...