Introduction to noncommutative geometry 1000-1M22WGN
1. Noncommutative topology, differential geometry, and measure theory (Gelfand-Naimark theorem, Connes' spectral geometry, von Neumann algebras)
2. Index theorems and cyclic cohomology (Fredholm operator index, Atiyah-Singer theorem, Riemann-Roch theorem, Novikov hypothesis, and Connes-Moscovici theorem)
3. Deformation quantization (Poisson manifolds and the Kontsevich formality theorem)
4. Quantum groups (compact quantum groups and the Woronowicz-Peter-Weyl theorem, noncommutative generalization of Pontryagin duality, Poisson groups, and quantum Lie groups deformations)
Main fields of studies for MISMaP
physics
Type of course
Mode
Blended learning
Self-reading
Prerequisites (description)
Course coordinators
Learning outcomes
1) knows the basic concepts of noncommutative geometry,
2) is able to relate them to classical problems of geometry (topology, differential geometry, etc.),
3) understands the role of the idea of noncommutative geometry in their contemporary reformulation and solving in much greater generality,
4) can give examples of mutual inspirations between noncommutative geometry and quantum physics,
5) is prepared to read suggested literature on its own, as necessary to understand the latest results in this field.
Assessment criteria
Credit on the basis of active participation in exercises and giving a talk on one topic selected from the list of tasks.
Bibliography
1. John Madore: Noncommutative Geometry for Pedestrians, gr-qc / 9906059
2. Joseph C. Varilly: An Introduction to Noncommutative Geometry, gr-
qc / 9909059
3. Daniel Sternheimer: Deformation Quantization: Twenty Years After,
math / 9809056
and selected excerpts from the following items with suggestions for further reading:
4. Alain Connes: Noncommutative Geometry, Academic Press 1994.
5. Jose M. Gracia-Bondia, Joseph C. Varilly and Hector Figueroa: Elements of
Noncommutative Geometry Birkhauser 2001.
6. Giovanni Landi: Noncommutative Spaces and their Geometry, Lecture Notes
in physics, Springer 2002, heptth / 9701078.
7. John Madore: An Introduction to Noncommutative Differenial Geometry and
its Physical Applications. Second Edition Cambridge University Press, Cambridge 1999
8. Simone Gutt: Variations on Deformation Quantization, math / 0003107.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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