授業の目的 【日本語】
Goals of the Course(JPN) | | |
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授業の目的 【英語】
Goals of the Course | | Title: Clifford algebras, spin groups and Dirac operators
The aim of the course is to develop the necessary mathematics to further study the interplay of geometry and topology via the Clifford/Dirac paradigm. These perspectives underlie much of index theory as developed by Atiyah, Singer and many others, as well as being essential for many areas of geometric analysis and mathematical physics. |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | |
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到達目標 【英語】
Objectives of the Course | | By the end of the course, students will be able to:
1) Describe the internal structure and classification of Clifford algebras.
2) Understand the representations of spin groups and their relation to rotation groups.
3) Grasp the fundamentals of the theory of connections on vector bundles.
4) Define Dirac operators of spin manifolds. |
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授業の内容や構成
Course Content / Plan | | Clifford algebras encode representations of the spin groups in an accessible way, along with a range of other geometric information. Using Clifford algebras we will describe representations of Euclidean and Lorentzian spin groups, and their Dirac operators. We will conclude with the role of Dirac operators in topology via index theory.
The topics of Clifford algebras, spin groups and their representations will be covered in the first two-thirds, and their use in constructing Dirac operators in geometry will be the last third.
1) Overview + Clifford algebras
2) Clifford algebras + representations
3) Spin groups, spin manifolds
4) Vector bundles and their connections
5) Construction of Dirac operators
6) Weitzenboeck formulae and applications in topology |
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履修条件
Course Prerequisites | | ＊Professor Adam Rennie will be the actual instructor for this lecture.
Students are expected have a solid understanding of linear and multi-linear algebra of finite dimensional vector spaces. Basic familiarity with differential geometry and homotopy theory is also expected.
The course will be conducted in English. この講義を英語で行います． |
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関連する科目
Related Courses | | | Any course on differential geometry or geometric analysis |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | | Students will submit solutions to 3 questions from a question bank available from the beginning of the subject. |
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教科書・テキスト
Textbook | | | Lecture notes will be provided |
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参考書
Reference Book | | | H. B. Lawson and M. L. Michelsohn, Spin Geometry, Princeton Univ. Press, Princeton, NJ, 1989. |
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課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours) | | | Students are expected to review the notes outside of lectures and try practice problems to gain a fuller understanding of the lecture content. Solutions to 3 exercises from a question bank will also be submitted after the lectures. |
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注意事項
Notice for Students | | This course is designed for both 4th year undergraduate and graduate students. Students are encouraged to ask questions to clarify concepts.
＊Professor Adam Rennie will be the actual instructor for this lecture. |
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他学科聴講の可否
Propriety of Other department student's attendance | | |
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他学科聴講の条件
Conditions of Other department student's attendance | | | Any student with the appropriate pre-requisites is welcome |
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レベル
Level | | |
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キーワード
Keyword | | | Clifford algebras, spin groups, Dirac operator |
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履修の際のアドバイス
Advice | | |
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授業開講形態等
Lecture format, etc. | | | Lectures will be in-person |
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遠隔授業（オンデマンド型）で行う場合の追加措置
Additional measures for remote class (on-demand class) | | |
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