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Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory
ESI Lectures in Mathematics and Physics

Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory

Alan Carey (Australian National University, Canberra)

ISBN print 978-3-03719-008-1, ISBN online 978-3-03719-508-6
DOI 10.4171/008
July 2011, 280 pages, softcover, 17 x 24 cm.
58.00 Euro

This collection of expository articles grew out of the workshop “Number Theory and Physics” held in March 2009 at the The Erwin Schrödinger International Institute for Mathematical Physics, Vienna. The common theme of the articles is the influence of ideas from noncommutative geometry (NCG) on subjects ranging from number theory to Lie algebras, index theory, and mathematical physics.

Matilde Marcolli’s article gives a survey of relevant aspects of NCG in number theory, building on an introduction to motives for beginners by Jorge Plazas and Sujatha Ramdorai. A mildly unconventional view of index theory from the viewpoint of NCG is described in the article by Alan Carey, John Phillips and Adam Rennie. As developed by Alain Connes and Dirk Kreimer, NCG also provides insight into novel algebraic structures underlying many analytic aspects of quantum field theory. Dominique Manchon's article on pre-Lie algebras fits into this developing research area. This interplay of algebraic and analytic techniques also appears in the articles by Christoph Bergbauer, who introduces renormalisation theory and Feynman diagram methods, and Sylvie Paycha, who focuses on relations between renormalisation and zeta function techniques.

Keywords: Arithmetic geometry, motives, noncommutative geometry, index theory, K-theory, pre-Lie algebras, Feynman integrals, zeta functions, Birkhoff–Hopf factorisation, renormalisation

Further Information

Review in MR 2841262 (2012e:58001)