2020/2021
Research Seminar "Functional Analysis and Noncommutative Geometry 1"
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
1, 2 module
Instructors:
Alexei Yu. Pirkovskii
Language:
English
ECTS credits:
3
Contact hours:
30
Course Syllabus
Abstract
The students who participate in the seminar give talks on functional analytic aspects of noncommutative geometry. Talks devoted to noncommutative algebraic geometry and to “pure” functional analysis (preferably with an algebraic or geometric flavour) are also welcome. The topics of talks are usually taken from the literature, but sometimes the participants present their own results. Occasionally, talks are given by the seminar advisor or by an invited speaker.
Learning Objectives
- Students will be introduced to some modern topics of noncommutative geometry, mostly from a functional-analytic perspective. Some related areas will also be discussed
Expected Learning Outcomes
- Each student is supposed to learn a piece of theory (for example, a paper in a journal, or a preprint, or a series of papers/preprints, or a chapter in a book, etc.) related to the topic of the seminar, and to give a talk.
Course Contents
- Quantum bounded symmteric domains and noncommutative complex analysis in the spirit of L. L. Vaksman.
- Strict deformation quantization (M. Rieffel et al.).
- Deformations of C*-algebras (in a broad sense).
- Noncommutative complex analytic geometry (A. Polishchuk, A. Schwarz, P. Smith, M. Khalkhali, G. Landi, et al.).
- An operator-theoretic approach to noncommutative complex analysis (W. Arveson, G. Popescu, et al.).
- Noncommutative complex structures and positive Hochschild cocycles (A. Connes, M. Khalkhali, G. Landi, et al.).
- Noncommutative integration, noncommutative L^p-spaces.
- Noncommutative geometry (algebraic and analytic) of PI algberas.
- Bivariant K-theory and bivariant periodic cyclic homology (G. Kasparov, J. Cuntz, R. Meyer, et al.).
- C*-superalgebras (P. Bieliavsky et al.).
- DQ-modules (M. Kashiwara, P. Schapira).
- Holomorphic functions of several free variables (J. Taylor, D. S. Kaliuzhnyi-Verbovetskyi, V. Vinnikov).
- “Physical” aspects of noncommutative geometry (including Bost-Connes systems).
Interim Assessment
- Interim assessment (2 module)To get a positive grade, a student should give (at least) one talk at the seminar.
Bibliography
Recommended Core Bibliography
- Connes, A. (1994). Noncommutative Geometry. San Diego: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=453747
Recommended Additional Bibliography
- Alain Connes, & Matilde Marcolli. (2007). Noncommutative Geometry, Quantum Fields and Motives. [N.p.]: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1495118
- Alain Connes, & Matilde Marcolli. (2008). Noncommutative geometry, quantum fields and motives. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.F1CD3BB5
- Kaliuzhnyi-Verbovetskyi, D. S., & Vinnikov, V. (2012). Foundations of Free Noncommutative Function Theory. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1212.6345
- Kaliuzhnyi-Verbovetskyi, D. S., & Vinnikov, V. (2014). Foundations of Free Noncommutative Function Theory. Providence, Rhode Island: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=971092
- Kashiwara, M., & Schapira, P. (2012). Deformation quantization modules. Luxembourg, Europe: Société Mathématique de France. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.F2E05E73
- Rieffel, M. A. (1993). Deformation Quantization for Actions of Rd. Providence, RI: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=838566