Non-commutative geometry

In mathematics, non-commutative geometry is the study of non-commutative C ^* algebras using invariants derived from topology such as K-theory and homology theories . It was essentially founded and expanded by Alain Connes with preparatory work that goes back to Israel Gelfand .

motivation

Topological spaces (more precisely: locally compact Hausdorff spaces ) correspond to commutative C ^* algebras. Namely, one can assign to each locally compact topological space the algebra of the infinitely vanishing , complex-valued , continuous functions (with the supremum norm as the norm and the complex conjugation as the involution), which is a commutative C ^* algebra. Conversely, Gelfand-Neumark's theorem says that for every commutative C ^* ‑ algebra there is a locally compact Hausdorff space with a C ^* ${\displaystyle X}$ ${\displaystyle C_{0}(X,\mathbb {C} )}$ ${\displaystyle A}$ ${\displaystyle X}$Isomorphism . ${\displaystyle A\simeq C_{0}(X,\mathbb {C} )}$

In particular, invariants of topological spaces can also be understood as invariants of commutative C ^* -algebras.

The idea of ​​non-commutative geometry is now to define invariants based on the definitions of the topology for non-commutative C ^* algebras and to make these usable for the investigation and classification of C ^* algebras. The invariants examined in this context include topological K-theory , cyclic homology, and Hochschild homology of C ^* algebras.

A more specific topic is the theory of spectral triples , these are intended to generalize the differential geometry of Riemannian spin manifolds .

The theory was also applied to number theory by Connes and Matilde Marcolli . The attempt to apply non-commutative geometry to physics also led to the non-commutative standard model and to non-commutative incarnations of M-theory .

literature

• Alain Connes: Noncommutative geometry. Academic Press, Inc., San Diego, CA, 1994. ISBN 0-12-185860-X online (pdf)
• José M. Gracia-Bondía, Joseph C. Várilly, Héctor Figueroa: Elements of noncommutative geometry. Birkhäuser Advanced Texts: Basel textbooks. Birkhauser Boston, Inc., Boston, MA, 2001. ISBN 0-8176-4124-6
• Alain Connes, Matilde Marcolli: Noncommutative geometry, quantum fields and motives. American Mathematical Society Colloquium Publications, 55. American Mathematical Society, Providence, RI; Hindustan Book Agency, New Delhi, 2008. ISBN 978-0-8218-4210-2 online (pdf)
• Masoud Khalkhali: Basic noncommutative geometry. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zurich, 2009. ISBN 978-3-03719-061-6
• Joseph C. Várilly: An introduction to noncommutative geometry. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zurich, 2006. ISBN 978-3-03719-024-1 ; 3-03719-024-8

Web links

• Andrew Lesniewski, Noncommutative geometry, Notices AMS, 1997, No. 7
• Matilde Marcolli: Non-commutative geometry and number theory. Max Planck Society
• Alexander Schenkel: Non-Commutative Geometry and Gravitation. Report colloquium Graduiertenkolleg 1147. Institute for Theoretical Physics and Astrophysics, University of Würzburg, July 13, 2009
• Journal of Noncommutative Geometry

Individual evidence

1. ^ Review by Henri Moscovici, Vaughan Jones, Notices AMS, August 1997