学部・大学院区分
Undergraduate / Graduate

多・博前

時間割コード
Registration Code

3211002

科目区分
Course Category

Ａ類Ⅰ（基礎科目）
Category A-1

科目名　【日本語】
Course Title

数理科学展望Ⅱ

科目名　【英語】
Course Title

Perspectives in Mathematical Science II

コースナンバリングコード
Course Numbering Code

担当教員　【日本語】
Instructor

中岡宏行 ○ 宇澤達岡田聡一

担当教員　【英語】
Instructor

NAKAOKA Hiroyuki ○ UZAWA Tohru OKADA Soichi

単位数
Credits

開講期・開講時間帯
Term / Day / Period

秋火曜日３時限
Fall Tue 3

授業形態
Course style

学科・専攻
Department / Program

多元数理科学研究科

必修・選択
Required / Selected

選択

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.

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

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.

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

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.

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