授業の目的 【日本語】
Goals of the Course(JPN) | | 講義のテーマ： 位相的 K理論
位相的K理論は1960年代Atiyah-Hirzebruchによって定義され，ベクトル束から構築されたコホモロジー理論です．この理論は幾何学，位相幾何学，指数理論といった分野において重要なツールとして発展してきました．本講義の目的は位相的K理論を入門し，基礎の性質を勉強することです．またバナッハ環のベキ等元と可逆元を用いるK理論への代数的なアプローチを比較します．テーマには，ベクトル束，K群の構築，Bott周期性，Thom同型が含まれます． |
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授業の目的 【英語】
Goals of the Course | | Course title: Topological K-theory
Topological K-theory was first considered in the 1960s by Atiyah and Hirzebruch and is a cohomology theory built from vector bundles over topological spaces. The theory has become an important tool in topology, geometry and index theory. The goal of the course is to give an introduction to topological K-theory and study its fundamental properties. We will also compare the topological approach to K-theory with an algebraic approach described in terms of idempotents and invertible matrices over Banach algebras. Topics will include vector bundle theory and the construction of the K-theory group, Bott periodicity and the Thom isomorphism. |
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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, student will acquire a working knowledge of vector bundles, their elementary properties and the construction of the K-theory groups for a topological space. Students will also obtain techniques for the practical computation of K-theory groups. Finally students will gain further experience in listening to and writing mathematics in English. |
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授業の内容や構成
Course Content / Plan | | The course will be conducted in English. この講義を英語で行います．
Tentative course plan:
- Vector bundles and their operations
- Idempotents, invertible elements and other preliminaries
- The K-theory groups and basic properties
- Bott periodicity and applications
- Further structure in K-theory
- Characteristic classes and the Chern character (if time) |
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履修条件
Course Prerequisites | | The course will be conducted in English. この講義を英語で行います．
Students should be familiar with linear algebra, point set topology and elementary abstract algebra (groups and rings). Some knowledge of differential geometry and homology/cohomology will be helpful, but is not required. |
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関連する科目
Related Courses | | | Any course on algebraic topology |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | | Grading will be based on 3-4 homework assignments that will be distributed throughout the semester. It is also expected that students actively participate in the lectures. |
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教科書・テキスト
Textbook | | Lecture summary notes will be prepared and distributed. Students are also encouraged to consult the resources below.
M. F. Atiyah, K-Theory. W. A. Benjamin, 1967.
E. Park, Complex Topological K-Theory. Cambridge University Press, 2008.
M. F. アティヤ, 松尾信一郎 (監訳), K理論. 岩波書店, 2022. |
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参考書
Reference Book | | G. Luke and A. Mishchenko, Vector bundles and their applications. Springer, 1998.
A. Hatcher, Vector bundles and K-theory. Available at https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf |
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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 submit solutions to selected exercises. |
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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-2025 |
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他学科聴講の可否
Propriety of Other department student's attendance | | | Any interested 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 | | | Vector bundles, K-theory, Bott periodicity, characteristic classes |
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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り，学生さんが挑戦してほしいです． |
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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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