学部・大学院区分
Undergraduate / Graduate		多・博前
時間割コード
Registration Code		3211018
科目区分
Course Category		Ａ類Ⅰ（基礎科目）
Category A-1
科目名　【日本語】
Course Title		解析学概論Ⅳ
科目名　【英語】
Course Title		Introduction to Analysis IV
コースナンバリングコード
Course Numbering Code
担当教員　【日本語】
Instructor		JAERISCH Johannes Klaus B ○
担当教員　【英語】
Instructor		JAERISCH Johannes Klaus Bernhard ○
単位数
Credits		2
開講期・開講時間帯
Term / Day / Period		春 火曜日 ３時限
Spring Tue 3
授業形態
Course style
学科・専攻
Department / Program
数理学科
必修・選択
Required / Selected
選択


授業の目的　【日本語】
Goals of the Course(JPN)
The aim is to introduce mathematical concepts and ideas to deal with highly irregular fractal sets. We study relations between topological, metric and measure-theoretic properties of subsets of Euclidean space R^n. A further goal is to enable the students to improve their ability to communicate in English.
授業の目的　【英語】
Goals of the Course
到達目標　【日本語】
Objectives of the Course(JPN))
The student will learn about the concept of self-similarity of fractal sets, which often implies high irregularity of the sets. The student will understand that such sets can not be investigated using the classical concepts of length and area. New ideas will be introduced. The student will learn about the framework to define fractals sets, and to develop the mathematical tools to study properties of fractals sets, such as dimension and measure.
到達目標　【英語】
Objectives of the Course
授業の内容や構成
Course Content / Plan
Preliminaries from Measure theory and Topology.
Box-counting dimension (Definition, Examples, Properties)
Hausdorff dimension (Definition, Examples, Properties)
Calculation of Hausdorff dimension using Measure theory
Iterated function systems (Banach fixed point theorem, Attractors, Bowen's formula for the Hausdorff dimension)
Interplay between Fractal Geometry and Dynamical Systems (Chaos, Repellers, Julia sets)
履修条件
Course Prerequisites
Analysis (epsilon-delta method), point set topology
関連する科目
Related Courses
Lectures on analysis, (point set) topology and measure theory.
成績評価の方法と基準
Course Evaluation Method and Criteria
Grading based on reports and quizzes.
教科書・テキスト
Textbook
Recommended books will be introduced on an individual basis.
参考書
Reference Book
M. Barnsley. Fractals everywhere. Academic Press, 1988.

K. Falconer. Mathematical foundations and applications. Third edition. John Wiley and Sons,Ltd., Chichester, 2014

Y. Pesin, V. Climenhaga. Lectures on Fractal Geometry and Dynamical Systems, AMS 2009.
課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours)
To prepare carefully for seminar presentation and discussion.
注意事項
Notice for Students
The lectures are in English.
他学科聴講の可否
Propriety of Other department student's attendance
可
他学科聴講の条件
Conditions of Other department student's attendance
Please contact me for details.
レベル
Level
2
キーワード
Keyword
Fractals, irregular sets, self-similar set, fractal dimension
履修の際のアドバイス
Advice
To review the lectures carefully.
授業開講形態等
Lecture format, etc.
We meet in classroom if the pandemic situation is acceptable. Otherwise, we meet online using zoom/skype etc., and communicate by email.
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)
-