授業の目的 【日本語】 Goals of the Course(JPN)   

授業の目的 【英語】 Goals of the Course   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 (ISHII Akira) Kleinian singularities and their resolutions
Part II (NAGAO Taro) Introduction to the theory of orthogonal polynomials
Part III (NAKANISHI Tomoki) Mysterious dilogarithm 


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

到達目標 【英語】 Objectives of the Course   Part I (ISHII Akira) Students should learn some basic ideas of algebraic geometry through examples of Kleinian singularities and their resolutions.
Part II (NAGAO Taro) Students should become familiar with basic concepts of orthogonal polynomials, and ready to apply them to scientific problems.
Part III (NAKANISHI Tomoki) Students should open their eyes for dilogarithm identities and their hidden algebraic perspective. 


授業の内容や構成 Course Content / Plan   Part I (ISHII Akira) I explain finite subgroups of SL(2,C), invariant rings, quotient singularities and their resolutions. If time permits, I will discuss the McKay correspondence.
Part II (NAGAO Taro) In this part of the lecture, basic properties of orthogonal polynomial, such as recurrence relations, are given and wellknown examples are introduced.
Part III (NAKANISHI Tomoki) I explain dilogarithm identities and their relation to root systems, Ysystems, and cluster algebras. 


履修条件 Course Prerequisites   Part I (ISHII Akira) Undergraduate linear algebra and algebra
Part II (NAGAO Taro) Undergraduate linear algebra, calculus and complex analysis
Part III (NAKANISHI Tomoki) Nothing other than basic undergraduate calculus 


関連する科目 Related Courses   Part I (ISHII Akira) Undergraduate linear algebra, algebra and complex analysis
Part II (NAGAO Taro) Undergraduate linear algebra, calculus and complex analysis
Part III (NAKANISHI Tomoki) Undergraduate calculus 


成績評価の方法と基準 Course Evaluation Method and Criteria   In each part, the instructor will determine the grade in the following way.
Part I (ISHII Akira) Grading is based on report.
Part II (NAGAO Taro) Grading is based on a report (or reports).
Part III (NAKANISHI Tomoki) Grading is based on a report (or possibly an exam in the last lecture).
At the end of the semester, by using the grades of the instructors, the final grade of the course will be determined. 


教科書・テキスト Textbook   Part I (ISHII Akira) Textbooks will not be used.
Part II (NAGAO Taro) Textbooks will not be used.
Part III (NAKANISHI Tomoki) Textbooks will not be used. 


参考書 Reference Book   Part I (ISHII Akira) Recommended books will be introduced in the lecture.
Part II (NAGAO Taro) Recommended books will be introduced in the lecture.
Part III (NAKANISHI Tomoki) Recommended books will be introduced in the lecture. 


課外学習等（授業時間外学習の指示） Study Load(Selfdirected Learning Outside Course Hours)   Part I (ISHII Akira) Students are encouraged to compute examples.
Part II (NAGAO Taro) Students are recommended to study concrete examples.
Part III (NAKANISHI Tomoki) Some exercises are given. 


注意事項 Notice for Students   If necessary, NUCT will be used to make announcements. 


他学科聴講の可否 Propriety of Other department student's attendance   

他学科聴講の条件 Conditions of Other department student's attendance   

レベル Level   

キーワード Keyword   Part I (ISHII Akira) invariant rings, resolutions of quotient singularities
Part II (NAGAO Taro) orthogonal polynomials, recurrence relations
Part III (NAKANISHI Tomoki) dilogarithm, root system, Ysystem, cluster algebra 


履修の際のアドバイス Advice   

授業開講形態等 Lecture format, etc.   Part I (ISHII Akira) Facetoface lecture (+ NUCT when necessary)
Part II (NAGAO Taro) Facetoface lecture (+ NUCT when necessary)
Part III (NAKANISHI Tomoki) Facetoface lecture (+ NUCT when necessary) 


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 (ondemand class)   
