
住所: 〒464-8602 愛知県名古屋市千種区不老町

ファイル更新日：2022年12月22日

教育・就職

■少人数クラスシラバス■

●J. イェーリッシュ

学部・大学院区分 Undergraduate / Graduate	多・前期
時間割コード Registration Code
科目区分 Course Category	B類(講究) C類(実習)/Category B Category C
科目名【日本語】 Course Title	確率論講究1 確率論講究2 確率論講究3 確率論講究4 確率論実習1 確率論実習2 確率論実習3 確率論実習4
科目名【英語】 Course Title	Seminar on Probability Theory 1 Seminar on Probability Theory 2 Seminar on Probability Theory 3 Seminar on Probability Theory 4 Practical Class on Probability Theory 1 Practical Class on Probability Theory 2 Practical Class on Probability Theory 3 Practical Class on Probability Theory 4
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 Bernhard
担当教員【英語】 Instructor	JAERISCH Johannes Klaus Bernhard
単位数 Credit	B類4単位 C類1単位
開講期・開講時間帯 Term / Day / Period	前期課程1年春学期(講究1・実習1) 前期課程1年秋学期(講究2・実習2) 前期課程2年春学期(講究3・実習3) 前期課程2年秋学期(講究4・実習4)
授業形態 Course style	セミナー
学科・専攻 Department / Program	多元数理科学研究科
必修・選択 Compulsory / Selected	選択必修

授業の目的【日本語】 Goals of the Course(JPN)	Title: Ergodic theory and fractal geometry for conformal dynamical systems Dynamical systems provide mathematical models to analyse the time dependence of systems in basically all sciences and our daily life. In this seminar, we study chaotic conformal dynamical systems from the view point of ergodic theory and with applications to fractal geometry. The aim is to enable the students to perform research in the area of ergodic theory and dynamcal systems with applications to fractal geometry. We provide the necessary guidance so that the students can enter this research area and develop their own ideas. A further aim 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 students will develop deeper understanding and knowledge of mathematics especially related to ergodic theory and fractal geometry for conformal dynamical systems. Through study and research in this area, the students will develop their learning and problem-solving skills. Further, the students will acquire the capability to contribute to the research on topics of this course. The students will improve their communication skills as well as written and oral presentation skills by giving regular reports on their study and research progress. A thesis is written in the second half of the course.
到達目標【英語】 Objectives of the Course
授業の内容や構成 Course Content / Plan	A dynamical systems is given by a self-map on a (state) space. Chaos refers to the sensitive dependence of orbits on the initial value. For chaotic systems it is difficult to predict and to analyse the individual behavior of orbit points. The same applies to large particle systems. Ergodic theory aims to analyse the average long-term behavior of such systems from the viewpoint of probability measures. We focus in particular on the so-called thermodynamic formalism originating from statistical physics, which allows us to determine meaningful measures for a given dynamical system. Chaotic behavior of dynamical systems is often related to highly irregular sets (such as attractors of conformal iterated functions systems, limit sets of Kleinian groups, or Julia sets of complex polynomials) which are called fractal sets. In this seminar, we study the interplay of ergodic theory, dynamical systems and fractal geometry for conformal dynamical systems. The seminar is used to discuss research problems and ideas how to deal with them. Moreover, the seminar helps to monitor progress and timeline of the research project. For the main part of this course, the students are expected to take the responsibility to organise their research activities, and to monitor and manage their time.
履修条件 Course Prerequisites	Background in analysis, measure theory, probability theory, functional analysis is required. Knowledge of ergodic theory is desirable.
関連する科目 Related Courses	Lectures on probability and measure (in particular, Lebesgue integration and measures on topological spaces), and functional analysis (spectral theory for bounded linear operators).
成績評価の方法と基準 Course Evaluation Method and Criteria	Grading is based on seminar performance (attendance, presentation, discussion) and Master's thesis.
教科書・テキスト Textbook	Recommended books will be introduced on an individual basis.
参考書 Reference Book	R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, Vol. 470. Springer-Verlag, Berlin-New York, 1975. F. Dal’ Bo. Geodesic and horocyclic trajectories: Universitext. Springer-Verlag London, Ltd.,London; EDP Sciences, Les Ulis, 2011. K. Falconer. Mathematical foundations and applications. Third edition. John Wiley and Sons,Ltd., Chichester, 2014 A. Katok, B. Hasselblatt. A first course in dynamics. With a panorama of recent developments. Cambridge University Press, New York, 2003. D. Mauldin, M. Urbanski. Graph directed Markov systems. Geometry and dynamics of limitsets. Cambridge Tracts in Mathematics, 148. Cambridge University Press, Cambridge, 2003. Y. Pesin. Dimension theory in dynamical systems. Contemporary views and applications.Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1997. F. Przytycki, M. Urbanski. Conformal fractals: ergodic theory methods. London MathematicalSociety Lecture Note Series, 371. Cambridge University Press, Cambridge, 2010. P. Walters. An introduction to ergodic theory. Graduate Texts in Mathematics, 79. SpringerVerlag, New York-Berlin, 1982.
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	This seminar is in English.
質問への対応方法 How to Ask Questions	Please send me an email.
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 for Other department student’s attendance	Every student with strong motivation to explore mathematics is welcome.
レベル Level	2
キーワード Keyword	Fractals, Hausdorff dimension, Dynamical systems, Ergodic theory, Thermodynamic formalism
履修の際のアドバイス Advice	Enjoy maths.
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)