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Published online: 2008-09-30
Noncommutative GeometryAlain Connes, Joachim Cuntz and Marc A. Rieffel (1) Institut des Hautes Études Scientifiques, Bures-Sur-Yvette, France
(2) Universität Münster, Germany
(3) University of California, Berkeley, USA
Noncommutative geometry applies ideas from geometry to
mathematical structures determined by noncommuting variables.
Within mathematics, it is a highly interdisciplinary subject
drawing ideas and methods from many areas of mathematics and
physics. Natural questions involving noncommuting variables arise
in abundance in many parts of mathematics and quantum
mathematical physics. On the basis of ideas and methods from
algebraic topology and Riemannian geometry, as well as from the
theory of operator algebras and from homological algebra, an
extensive machinery has been developed which permits the
formulation and investigation of the geometric properties of
noncommutative structures. This includes K-theory, cyclic
homology and the theory of spectral triples. Areas of intense
research in recent years are related to topics such as index
theory, quantum groups and Hopf algebras, the Novikov- and
Baum-Connes conjectures as well as to the study of specific
questions in other fields such as number theory, modular forms,
topological dynamical systems, renormalization theory,
theoretical high-energy physics and string theory. Many results
elucidate important properties of fascinating specific classes of
examples that arise in many applications.
The talks covered substantial new results and insights in several of the different areas in Noncommutative Geometry.
The workshop was attended by 53 participants including 6 young researchers
supported by the European Union.
The talks covered substantial new results and insights in several of the different areas in Noncommutative Geometry. The workshop was attended by 53 participants including 6 young researchers supported by the European Union.
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Connes Alain, Cuntz Joachim, Rieffel Marc: Noncommutative Geometry. Oberwolfach Rep. 4 (2007), 2543-2608. doi: 10.4171/OWR/2007/43