# Topics in Noncommutative Geometry

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*Guillermo Cortiñas*

A co-publication of the AMS and Clay Mathematics Institute

Luis Santaló Winter Schools are organized yearly by the Mathematics
Department and the Santaló Mathematical Research Institute of the
School of Exact and Natural Sciences of the University of Buenos Aires
(FCEN). This volume contains the proceedings of the third Luis
Santaló Winter School which was devoted to noncommutative geometry and
held at FCEN July 26–August 6, 2010.

Topics in this volume concern noncommutative geometry in a broad
sense, encompassing various mathematical and physical theories that
incorporate geometric ideas to the study of noncommutative phenomena.
It explores connections with several areas including algebra,
analysis, geometry, topology and mathematical physics.

Bursztyn and Waldmann discuss the classification of star products
of Poisson structures up to Morita equivalence. Tsygan explains the
connections between Kontsevich's formality theorem, noncommutative
calculus, operads and index theory. Hoefel presents a concrete
elementary construction in operad theory. Meyer introduces the
subject of \(\mathrm{C}^*\)-algebraic crossed products.
Rosenberg introduces Kasparov's \(KK\)-theory and
noncommutative tori and includes a discussion of the Baum-Connes
conjecture for \(K\)-theory of crossed products, among other
topics. Lafont, Ortiz, and Sánchez-García carry out a
concrete computation in connection with the Baum-Connes
conjecture. Zuk presents some remarkable groups produced by finite
automata. Mesland discusses spectral triples and the Kasparov product
in \(KK\)-theory. Trinchero explores the connections between
Connes' noncommutative geometry and quantum field theory. Karoubi
demonstrates a construction of twisted \(K\)-theory by means of
twisted bundles. Tabuada surveys the theory of noncommutative
motives.

Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

#### Readership

Graduate students and research mathematicians interested in various aspects of noncommutative geometry.