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

小林　亮一 ○ 白水　徹也 JAERISCH Johannes Klaus B

担当教員　【英語】
Instructor

KOBAYASHI Ryohichi ○ SHIROMIZU Tetsuya JAERISCH Johannes Klaus Bernhard

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

授業の目的　【日本語】
Goals of the Course(JPN)

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.

Part I (Ryoichi Kobayashi) Douglas's solution to the Plateau Problem asserts that there exists a surface spanning a given counter with least area. This lecture is an account of this result.

Part II (Tetsuya Shiromizu) General relativity is based on the Riemann geometry. One can learn that it describes our universe and black hole.

Part III (Johannes Jaerisch). We give an introduction to the mathematical concept of entropy, which plays an important role in statistical physics, information theory, and dynamical systems.

授業の目的　【英語】
Goals of the Course

到達目標　【日本語】
Objectives of the Course(JPN))

Part I (Ryoichi Kobayashi) My goal of this lecture is to give a self-contained account of Douglas's solution to the classical Plateau problem. I hope this lecture will be a reasonable introduction to modern geometric analysis.

Part II (Tetsuya Shiromizu) enjoying the Riemann geometry and partial differential equation through the theory for spacetime.

Part III (Johannes Jaerisch) To give an introduction to the mathematical concept of entropy, and some of its applications.

到達目標　【英語】
Objectives of the Course

授業の内容や構成
Course Content / Plan

Part I (Ryoichi Kobayashi) Self-contained account on Douglas's solution to the Plateau Problem. ① What is minimal surface ? ② Harmonic function and Dirichlet problem.
③ Dirichlet's principle. Douglas's solution to the Plateau problem.

Part II: General relativity (Tetsuya Shiromizu)

In this part of the lecture, we give the brief introduction of the Riemann geometry and then formulate general relativity which is the theory for spacetime. As a consequence, we can discuss black hole and expanding universe in terms of the Riemann geometry.

Part III: Entropy (Johannes Jaerisch)

We give an introduction to the mathematical concept of entropy, which plays an important role in statistical physics, information theory, and dynamical systems. We discuss the theorem of Shannon-McMillan-Breiman theorem and the asymptotic equipartition property, as well the role of entropy in the classification of measure-preserving dynamical systems.

履修条件
Course Prerequisites

Part I (Ryoichi Kobayashi) undergraduate linear algebra, calculus and complex analysis.

Part II (Tetsuya Shiromizu) undergraduate linear algebra and calculus.

Part III (Johannes Jaerisch) calculus and probability theory.