授業の目的 【日本語】
Goals of the Course(JPN) | | This course is designed to be one of the English courses which the Graduate School of Mathematics is providing for the graduate and undergraduate students not only from foreign countries but also domestic students who wish to study abroad or to communicate with foreign scientists in English. All course activities including lectures, homework assignments, questions and consultations are in English. 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 1: Hamanaka
Part 2: Hesselholt
Part 3: Kimura |
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授業の目的 【英語】
Goals of the Course | | |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | | The goal of the course is to introduce to introduce some central topics from contemporary mathematics and to do so using the English language. |
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到達目標 【英語】
Objectives of the Course | | |
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授業の内容や構成
Course Content / Plan | | Part 1: Five Lectures on Determinants (Hamanaka)
Determinants play essential roles in broad areas of mathematics and physics. I would like to discuss various aspects of determinants with relation to algebraic and differential equations. First, I will lecture on the basics of determinant theory and introduce various identities, such as the Cauchy-Binet formula, the Plucker relation, Sylvester's determinant identity, and so on. Next, I will apply them to problems to solve algebraic equations and differential equations. Finally, I will discuss noncommutative versions of determinants, especially quasideterminants, with application to noncommutative integrable equations.
Part 2: Category theory (Hesselholt)
If V and W are two vector spaces, then it is not a reasonable question to ask if V and W are equal. It is better to ask if V and W are isomorphic. But it is much better to know that a specific linear map f : V --> W is an isomorphism, since this lets us translate calculations in V to calculations in W and vice versa. So we should study vector spaces and the linear maps between them together. This is a general mathematical phenomenon: given a class of mathematical objects, there is usually a preferred class of maps between them. This principle is encoded in the mathematical notion of a category, which was introduced by Eilenberg and MacLane in the mid-twentieth century. It leads to a long list of very general but extremely useful concepts and results. To wit, the notion of adjoint functors, introduced by Kan, is ubiquitous in mathematics and the results about them are among the main tools in much of modern mathematics.
Part 3: Introduction to fluid mechanics (Kimura)
Historically, fluid mechanics has provided many interesting mathematical problems. In this short course, some examples will be discussed as an introduction of fluid mechanics. The subjects will cover:
(1) two-dimensional fluid mechanics and complex function theory
(2) boundary layer theory and asymptotic analysis
(3) vortex motion and orthogonal polynomials |
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履修条件
Course Prerequisites | | | Working knowledge of basic undergraduate mathematics including calculus and linear algebra is required. |
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関連する科目
Related Courses | | | The course highlights concepts and results that are typically not included in the standard curriculum, or at least not at the same level of detail. |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | | In each part, the instructor will assign exercises, problems, etc. during the lectures and determines grades (S, A, B, C, F) independently. Therefore, each student has three letter grades after the course is done. Based on these three estimates, the final grade of the course will be determined. |
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不可(F)と欠席(W)の基準
Criteria for "Fail (F)" & "Absent (W)" grades | | |
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参考書
Reference Book | | For Part 2, the following textbook is useful:
Saunders MacLane, Categories for the Working Mathematician. Second Edition, Graduate Texts in Mathematics 5, Springer-Verlag, New York, 1998. |
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教科書・テキスト
Textbook | | |
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課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours) | | |
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注意事項
Notice for Students | | | If you are unsure whether or not your mathematical background knowledge is sufficient, then please contact the instructors for advice. |
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他学科聴講の可否
Propriety of Other department student's attendance | | |
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他学科聴講の条件
Conditions for Other department student's attendance | | | While this course is open for any students at Nagoya University as one of the "open subjects" of general education, students not majoring in mathematics should contact any of the instructors before the first lecture to see if the course is appropriate to take. |
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レベル
Level | | |
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キーワード
Keyword | | Part 1: Five lectures on determinants
Part 2: Category theory
Part 3: Introduction to fluid mechanics |
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履修の際のアドバイス
Advice | | | The English version of Wikipedia is a great source. In addition to reliable and precise definitions, it often includes very good examples that help the non-expert understand the relevant concepts. |
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授業開講形態等
Lecture format, etc. | | |
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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) | | |
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