授業の目的 【日本語】
Goals of the Course(JPN) | | この講義は講師三人によるオムニバス形式の英語講義である.
内容は英文を参照すること.
日程等詳しいことは8月頃に追って通知する．
See English text. |
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授業の目的 【英語】
Goals of the Course | | The purpose of this course is to introduce and explain various concepts and methods in mathematical sciences.
This year, the course is provided by three instructors.
Further details such as the schedule will be announced around August.
Part I (OHTA Hiroshi) Glimpse of Lagrangian Floer theory and related topics
Part II (NAYATANI Shin) Spectral graph theory
Part III (HIRAI Hiroshi) Matrix scaling and generalizations |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | 各講師ごとに与えられたテーマに則した問題について考察し, レポートを作成, 提出することによってテーマの理解を深める. 各テーマの内容は英文を参照すること.
See English text. |
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到達目標 【英語】
Objectives of the Course | | Part I (OHTA Hiroshi) Students should become familiar with basic materials of Floer theory and related topics.
Part II (NAYATANI Shin) Students should become familiar with basic materials of spectral graph theory.
Part III (HIRAI Hiroshi) Students should become familiar with mathematical concepts behind matrix scaling and generalizations. |
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授業の内容や構成
Course Content / Plan | | Part I (OHTA Hiroshi) Lagrangian Floer theory is a kind of an infinite dimensional Morse theory associated to Lagrangian submanifolds in a symplectic manifold. After a brief introduction to Floer theory, I will give an overview of some recent developments in the field and related topics.
Part II (NAYATANI Shin) In this part of the lecture, after preliminaries, some topics of spectral graph theory are discussed.
Part III (HIRAI Hiroshi) Matrix scaling is the problem of multiplying positive diagonal matrices X,Y to a given nonnegative matrix A so that XAY is a doubly stochastic matrix. This part of the lecture will present the mathematical theory behind. With the generalization to operator scaling and recent development, the interplay between optimization, combinatorics, invariant theory, differential geometry, theoretical computer science will be explained. |
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履修条件
Course Prerequisites | | Part I (OHTA Hiroshi) Manifold theory, complex analysis, (co)homology theory
Part II (NAYATANI Shin) Undergraduate linear algebra and calculus.
Part III (HIRAI Hiroshi) Undergraduate linear algebra and calculus. |
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関連する科目
Related Courses | | Part I (OHTA Hiroshi) Manifold theory, complex analysis, (co)homology theory
Part II (NAYATANI Shin) Undergraduate linear algebra and calculus.
Part III (HIRAI Hiroshi) Undergraduate linear algebra and calculus. |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | In each part, the instructor will assign exercises, reports, etc. during the lectures and determines grades (A+, A, B, C, F, W) independently. At the end of the semester, for students who have earned TWO or MORE grades other than F and W, the BEST will be used as the final grade of this course.
((e.g.1) A, W, C => A. (e.g. 2) F, B, A=> A. (e.g. 3) B, C, B=> B.) |
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教科書・テキスト
Textbook | | Part I (OHTA Hiroshi) Textbooks will not be used.
Part II (NAYATANI Shin) Textbooks will not be used.
Part III (HIRAI Hiroshi) Textbooks will not be used. |
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参考書
Reference Book | | Part I (OHTA Hiroshi) Recommended books and references will be introduced in the lecture.
Part II (NAYATANI Shin) Recommended books and references will be introduced in the lecture.
Part III (HIRAI Hiroshi) Recommended books and references will be introduced in the lecture. |
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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) | | Part I (OHTA Hiroshi) Students are recommended to study by themselves the references given during the lecture.
Part II (NAYATANI Shin) Review after each lecture is recommended.
Part III (HIRAI Hiroshi) Students are recommended to study by themselves the references given during the lecture. |
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注意事項
Notice for Students | | | If necessary, TACT will be used to make announcements. |
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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 | | |
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レベル
Level | | |
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キーワード
Keyword | | Part I (OHTA Hiroshi) symplectic geometry, Lagrangian submanifolds, Morse theory
Part II (NAYATANI Shin) graph, Laplacian, eigenvalue, eigenfunction
Part III (HIRAI Hiroshi) matrix scaling, operator scaling, computational complexity |
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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. | | Part I (OHTA Hiroshi) Face-to-face lecture (+ TACT when necessary)
Part II (NAYATANI Shin) Face-to-face lecture (+ TACT when necessary)
Part III (HIRAI Hiroshi) Face-to-face lecture (+ TACT when necessary) |
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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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