授業の目的 【日本語】
Goals of the Course(JPN) | | 科目の名前：楕円型作用素と指数理論
この科目は英語で行います．
Atiyah-Singer定理は20世紀に数学の素晴らしい結果の一つです．科目はこの定理に関わる若干の要素を入門します．特にユークリッド空間とリーマン多様体において楕円型という線型微分作用素を勉強して，基礎の幾何学的特性・解析的特性を考察します．大切な例はDirac作用素で，物理学との関係が様々です． |
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授業の目的 【英語】
Goals of the Course | | Course title: Elliptic operators and index theory
The Atiyah-Singer Index Theorem is a landmark achievement of mathematics in the 20th Century. The course aims to introduce students to some of the concepts and ideas are related to this theorem and proof. Our focus will be on analytic and geometric aspects of index theory, where we will study fundamental properties of elliptic differential operators on Euclidean space and Riemannian manifolds. An important example is the Dirac operator, which has many connections to 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 | | The central objective of the course is give students a basic knowledge of index theory and an awareness of the many fields of mathematics of which the theory brings together.
By the end of the course, students should have a basic understanding of the theory of Fredholm operators and knowledge of the many examples that arise from elliptic differential operators. Students should also have a basic understanding of the Dirac operator, it’s role in the Atiyah-Singer index theorem and some of its applications in mathematics and physics. |
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授業の内容や構成
Course Content / Plan | | A tentative plan for the semester is as follows.
- Fredholm operators
- Integral and Wiener-Hopf operators
- Differential operators on Euclidean space
- Differential operators on Riemannian manifolds
- Vector bundles and connections
- Clifford algebras and Dirac operators
- Index theory and applications in geometry and physics |
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履修条件
Course Prerequisites | | | Students should be familiar with the basic definitions and properties of smooth manifolds and operators on complex Hilbert spaces. Some knowledge of Fourier theory will also be helpful. |
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関連する科目
Related Courses | | | Any courses on functional analysis or (Riemannian) geometry |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | | Grading will be based on written reports of the students. It is also expected that students actively participate in the lectures. |
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教科書・テキスト
Textbook | | D. Bleecker, B. Booss-Bavnbek: Index Theory with Applications to Mathematics and Physics, International Press, 2013. ISBN: 978-1-57146-264-0
It is not necessary to purchase this book. Additional resources and materials will be distributed to students where appropriate. |
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参考書
Reference Book | | Students are encouraged to look at several references and find a text that suits them. Some options are given below.
J. Roe: Elliptic operators, topology and asymptotic methods, Second edition, Chapmann & Hall/CRC, 1998.
H. Lawson, M. Michelsohn: Spin Geometry, Princeton University Press, 1989.
P. Gilkey: Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem, Second edition, CRC Press, 1995
N. Berline, E Getzler, M. Vergne: Heat Kernels and Dirac Operators, Springer, 1992.
T. Friedrich: Dirac Operators in Riemannanian Geometry, American Mathematical Society, 2000.
古田幹雄: 整数定理，岩波書店，2018. |
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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 will be expected to independently review the lectures and consult the references for additional information. Students will also be expected to submit reports of exercises presented in the class or related topics of their own choosing. |
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注意事項
Notice for Students | | Information about the course will regularly updated on TACT and the following website.
https://sites.google.com/site/khomologyzone/teaching/masters-course-autumn-2023 |
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他学科聴講の可否
Propriety of Other department student's attendance | | | Any student is welcome to attend the course |
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他学科聴講の条件
Conditions of Other department student's attendance | | |
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レベル
Level | | |
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キーワード
Keyword | | | Fredholm operator, Index theory, Differential operator |
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履修の際のアドバイス
Advice | | The course is a good opportunity to practice listening to and doing mathematics in English.
この科目は英語での数学を使う機会があって，学生さんが挑戦してみてほしいです． |
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授業開講形態等
Lecture format, etc. | | | The course will be in-person. If necessary, online and remote methods will also be implemented. |
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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) | | | If need arises, lectures will be recorded or held online. |
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