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.
See English text. |
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授業の目的 【英語】
Goals of the Course | | This course is an omnibus lecture given by 3 separate instructors.
The subjects are as follows.
Part 1 (Nakaoka): Mackey functors on finite groups
Part 2 (Okada): Algebraic approach to combinatorics
Part 3 (Uzawa): Fourier analysis and the Heisenberg group/algebra |
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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 1: Students should become familiar with the category of finite
G-sets and acquire an understanding of the notion of Mackey functor, for
finite group G.
Part 2: Students should understand a relationship between combinatorics
and linear algebra (representation theory of Lie algebras).
Part 3: Students should become familiar with the interplay between algebra
and analysis by looking at the relationship between Fourier analysis
and the Heisenberg group/algebra |
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授業の内容や構成
Course Content / Plan | | Part 1(Nakaoka):
I will explain some properties of the category of finite G-sets for
finite group G, and introduce the notion of a Mackey functor on G.
Part 2(Okada):
In this part, I introduce the very basics of representation
theory of the Lie algebra sl_2 and explain how to resolve purely
combinatorial problems by applying the representation theory.
Part 3 (Uzawa):
In this part, I will show how the very simple commutation relationship
[d/dx, x](f) = d/dx(xf) - x (d/dx f) = f gives rise to the Heisenberg algebra,
and how this simple relationship "governs" many phenomenon in Fourier
analysis. |
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履修条件
Course Prerequisites | | Part 1: Undergraduate linear algebra and algebra
Part 2: Undergraduate linear algebra and algebra
Part 3: Undergraduate linear algebra, algebra, and analysis |
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関連する科目
Related Courses | | Part 1: Undergraduate linear algebra and algebra
Part 2: Undergraduate linear algebra and algebra
Part 3: Undergraduate linear algebra, algebra, and analysis |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | Grading is based on reports for the 3 parts.
The grade is based on the two best reports for each student. |
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不可(F)と欠席(W)の基準
Criteria for "Fail (F)" & "Absent (W)" grades | | | If less than 2 reports are submitted the grade will be W. |
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参考書
Reference Book | | Part 1: Recommended books will be introduced in the lecture.
Part 2: Recommended books will be introduced in the lecture.
Part 3: "Fourier Series and Integrals", by Dym and McKean is highly recommended. |
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教科書・テキスト
Textbook | | Part 1: Textbooks will not be used.
Part 2: Textbooks will not be used.
Part 3: Textbooks will not be used. |
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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 1: Review of each lecture is recommended.
Part 2: Students are encouraged to review each lecture and to compute
examples.
Part 3: Students are encouraged to work out their notes after lectures. |
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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 for Other department student's attendance | | | Ask the instructors first. |
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レベル
Level | | |
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キーワード
Keyword | | Part 1: finite group, finite G-set, Mackey functor
Part 2: Lie algebra, partially ordered set, Sperner property
Part 3: Lie algebra, groups, Fourier transforms |
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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. | | | 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) | | | Access to TACT is needed. |
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