Научно-исследовательский семинар "Функциональный анализ и некоммутативная геометрия 1"

2020/2021

Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»

Лучший по критерию «Новизна полученных знаний»

Статус: Дисциплина общефакультетского пула

Кто читает: Факультет математики

Где читается: Факультет математики

Когда читается: 1, 2 модуль

Преподаватели: Пирковский Алексей Юльевич

Язык: английский

Кредиты: 3

Контактные часы: 30

Full Syllabus Ask Question

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).

Assessment Elements

A talk on a seminar
A talk on a seminar

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

Course Syllabus