JournalsowrVol. 1 / No. 4DOI 10.4171/owr/2004/45

Nichtkommutative Geometrie

  • Joachim Cuntz

    Universität Münster, Germany
  • Alain Connes

    Institut des Hautes Études Scientifiques, Bures-Sur-Yvette, France
  • Marc A. Rieffel

    University of California, Berkeley, USA
Nichtkommutative Geometrie cover

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Abstract

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. Areas of intense research in recent years are related to topics such as index theory, K-theory, cyclic homology, 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. The meeting was attended by 45 participants, including a fair number of young mathematicians but also many of the leading experts in the field. The exchange of ideas was very lively and quite a few significant new results were presented in the talks, which covered many of the aspects of noncommutatve geometry mentioned above.