授業の目的【日本語】
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作用素環論と非可換幾何学は歴史的に量子力学に基づいて発展された数学です．このセミナーでは作用素環，非可換幾何学の基礎を勉強して，物理への応用も学ぶ機会です．
セミナーにおけるテーマは以下のことは可能です．
作用素環論と$K$理論
指数理論，非可換幾何学
数理物理への応用
テーマの間では固い境界ではなく，勉強する分野を自由に選べます． |
授業の目的【英語】
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Operator algebras and noncommutative geometry are mathematical fields that arose out of the study of quantum mechanics. The aim of the seminar is to introduce to students to the mathematical foundations of these subjects as well as their application in quantum physics.
Some possible subjects/themes for the seminar are as follows:
Operator algebras and $K$-Theory
Index theory and noncommutative geometry
Applications in mathematical physics
There is no strict boundary between these themes and students are free to choose whichever topic they find the most interesting. |
到達目標【日本語】
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作用素環論，$K$理論，指数理論，非可換幾何学の基礎を勉強することです．また独立の勉強の技を磨いて，英語でのコミュニケーションと発表の能力も開発することです． |
到達目標【英語】
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To study and learn the fundamental aspects of operator algebras, index theory, $K$-theory and their applications in physics.
A further aim is to develop a student’ s independent learning skills as well as their English presentation skills. |
授業の内容や構成
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このセミナーはできる限り英語で行います．
The seminar consists of regular meetings and oral presentations made by the students. Participants may work on different themes and subjects (with different reference material), though presentations will be held as a group so we can all benefit from the work of each other. |
履修条件
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Standard knowledge of undergraduate mathematics (such as calculus, linear algebra and complex analysis) will be assumed. More specialised knowledge in functional analysis, topology and geometry may be helpful, but is not necessary. |
関連する科目
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There are master courses given by the instructor that will have connections to the content of the seminar. The master courses given by Prof. Richard are also closely related. |
成績評価の方法と基準
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Grades will be determined by student presentations, a report and participation of the student. |
教科書・テキスト
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The different themes of the textbook have different (but related) textbooks. Some possible options include:
K. Davidson - $C^{*}$-Algebras by Example
G. J. Murphy - $C^{*}$-Algebras and Operator Theory
N. E. Wegge-Olsen - $K$-Theory and $C^{*}$-Algebras: A Friendly Approach
N. Higson, J. Roe - Analytic $K$-Homology
J. M. Gracia-Bondia, J. C. Varilly, H. Figueroa - Elements of Noncommutative Geometry
D. D. Bleecker, B. Booß-Bavnbek - Index Theory
H. B. Lawson, M.-L. Michelsohn - Spin Geometry
M. Reed, B. Simon - Methods of Modern Mathematical Physics (Vol 1-2)
O. Bratteli, D. Robinson - Operator Algebras and Quantum Statistical Mechanics
H. Tasaki - Physics and Mathematics of Quantum Many-Body Systems |
参考書
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Discussed on an individual basis |
課外学習等 (授業時間外学習の指示)
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Students are expected to study independently to prepare for the seminar. Regular meetings with the instructor may also be held for the student to ask questions and help clarify their understanding. |
注意事項
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質問への対応方法
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Email or in-person |
他学科聴講の可否
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可 |
他学科聴講の条件
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The seminar is intended for masters students in the Mathematics Department. Though any interested student is welcome to participate. |
レベル
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キーワード
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operator algebras, $K$-theory, index theory, noncommutative geometry, mathematical physics |
履修の際のアドバイス
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A small classroom environment is an ideal place to practice and develop one's English communication skills.
少人数クラスでは英語の発表を挑戦する機会として良いと思います． |
授業開講形態等
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Seminars will generally be held in a classroom. Depending on circumstances and the needs of the student, online meetings are also possible. |
遠隔授業(オンデマンド型)で行う場合の追加措置
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2024年度 
