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 first part (Hishida) of the course provides an elementary functionalanalytic approach to linear evolutionary partial differential equations (PDEs). First of all, those PDEs arising in mathematical physics are introduced. We next study the YosidaHille theory for strongly continuous semigroups of operators. With the aid of knowledge about Sobolev spaces, we then apply the theory to initialboundary value problems for various PDEs.
The subject for the second part (Hesselholt) is modules over rings. We show that every (left or right) module over a division ring is free and introduce and classify semisimple rings. We prove Maschke's theorem that the group ring of a finite group over a field, the characteristic of which does not divide the order of the group, is semisimple. We study the case of cyclic groups in detail, where the description of the group ring afforded by the theorem is known as the discrete Fourier transform. Finally, we study rings of integers in number fields, where we introduce the ideal class group first considered by Kummer.
The subject of the third part of this course is an introduction to algebraic geometry. We will discuss algebraic varieties defined over complex numbers and their sheaf theoretic treatments. 


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

到達目標 【英語】 Objectives of the Course   

授業の内容や構成 Course Content / Plan   Part 1 1. Introduction to PDEs 2. YosidaHille theory 3. Analytic semigroups 4. Sobolev space 5. Applications to PDEs
Part 2 The following is a preliminary outline of the five lectures in the part of the course:
Lecture 1: Rings and modules Lecture 2: Simple modules Lecture 3: Semisimple rings Lecture 4: The discrete Fourier transform Lecture 5: The ideal class group
Part 3 The following is a preliminary outline of the third part of the lectures:
Lecture 1: Affine varieties and Hilbert Nullstellensatz Lecture 2: Regular functions and sheaves Lecture 3: Morphisms of varieties and products Lecture 4: Prevarieties and varieties Lecture 5: Projective spaces and projective varieties 


履修条件 Course Prerequisites   Familiarity of linear algebra and calculus is desirable, but not strictly necessary. 


関連する科目 Related Courses   Part 1 : calculus, differential equations, functional analysis Part 2 : Linear algebra, algebra, representation theory. Part 3 : Algebra 


成績評価の方法と基準 Course Evaluation Method and Criteria   Grades are assigned based on solutions to weekly problem sets. The course grade is based on a final numerical grade calculated as Max(I+II,I+III,II+III)/2, where I, II, and III are the number of points (between 0 and 100) in each of the three parts of the course.
Grades of the third part are based on a final report. 


教科書・テキスト Textbook   Lecture notes will be uploaded weekly to the course homepage on NUCT.
Lecture notes will be provided at the beginning of the third part. 


参考書 Reference Book   

課外学習等（授業時間外学習の指示） Study Load(Selfdirected Learning Outside Course Hours)   Expect to spend at least two hours per week to read the course notes and complete the problem set. 


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 of Other department student's attendance   

レベル Level   

キーワード Keyword   Part 1 : partial differential equations, semigroups of operators,Sobolev space
Part 2 : Linear algebra, semisimple rings, discrete Fourier transform, ideal class group. 


履修の際のアドバイス Advice   The most important thing that you learn as a mathematician is what a *definition* is and that a *proof* is. It is not enough to almost understand a definition. You really need to understand definitions completely and be able to state a definition without any mistakes at all. 


授業開講形態等 Lecture format, etc.   Lectures will be uploaded to Google Drive for ondemand viewing(Hesselhot). (You do not need a Google account to be able to view the lectures.) Links for the lectures and lecture notes are posted on NUCT along with weekly problem sets. Solutions to problem sets should be uploaded to NUCT. LaTeX'd solutions are preferable, but scanned PDF is acceptable.
Part 3 Lectures of the third part will be given via Zoom. 


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)   You should have access to NUCT. 

