学部・大学院区分
Undergraduate / Graduate
理学部
時間割コード
Registration Code
0615840
科目区分
Course Category
専門科目
Specialized Courses
科目名 【日本語】
Course Title
数理科学展望Ⅳ
科目名 【英語】
Course Title
Perspectives in Mathematical Science IV
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
2
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
講義
Lecture
学科・専攻
Department / Program
数理学科
必修・選択
Compulsory / Selected
選択


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.
関連する科目
Related Courses
Part I (Ryoichi Kobayashi) geometry. complex analysis. PDE.

Part II (Tetsuya Shiromizu) geometry, partial differential equation, mathematical physics

Part III (Johannes Jaerisch) probability theory
成績評価の方法と基準
Course Evaluation Method and Criteria
In each part, the instructor will determine grades (S, A, B, C, F) independently. At the end of the semester, the best grade will be used as the final grade of the course.

Part I (Ryoichi Kobayashi) Grading is based on reports solving problems given in the course.

Part II (Tetsuya Shiromizu) Grading is based on learning attitude.

Part III (Johannes Jaerisch) Grading is based on homework assignments.
不可(F)と欠席(W)の基準
Criteria for "Fail (F)" & "Absent (W)" grades
参考書
Reference Book
Part I (Ryoichi Kobayashi) I will submit handouts to NUCT.

Part II (Tetsuya Shiromizu) one can download from my home page http://www.math.nagoya-u.ac.jp/~shiromizu/index-e.html.

Part III(Johannes Jaerisch)
Walters, P. Introduction to Ergodic theory. Graduate texts in mathematics, 1982.
Cover, T.M., Thomas, J.A. Elements of Information Theory, 2006.
教科書・テキスト
Textbook
Recommended books will be introduced on an individual basis.
課外学習等(授業時間外学習の指示)
Study Load(Self-directed Learning Outside Course Hours)
注意事項
Notice for Students
他学科聴講の可否
Propriety of Other department student's attendance
他学科聴講の条件
Conditions for Other department student's attendance
We will welcome any motivated student.
レベル
Level
2
キーワード
Keyword
Part I (Ryoichi Kobayashi) Plateau problem. Harmonic functions. Dirichlet boundary value problem. Dirichlet principle. Area and Energy. Isothermal coordinates. Minimal surfaces.

Part II (Tetsuya Shiromizu)
spacetime, causal structure, black hole, universe

Part III (Johannes Jaerisch)
entropy, information, probability space, dynamical system
履修の際のアドバイス
Advice
Part I (Ryoichi Kobayashi) The proof needs long argument. So, each student is encouraged to reconstruct long argument by him/herself so that
he/she can explain the idea to his/her friend.

Part II (Tetsuya Shiromizu) Do not hesitate to ask me if you have any questions. This is the privilege for young peoples.

Part III (Johannes Jaerisch)
To enjoy maths.
授業開講形態等
Lecture format, etc.
NUCT等にて連絡する。
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)