学部・大学院区分
Undergraduate / Graduate
理学部
時間割コード
Registration Code
0615830
科目区分
Course Category
専門科目
Specialized Courses
科目名 【日本語】
Course Title
数理科学展望Ⅲ
科目名 【英語】
Course Title
Perspectives in Mathematical Science III
コースナンバリングコード
Course Numbering Code
担当教員 【日本語】
Instructor
HESSELHOLT LARS ○ 菱田 俊明 谷本 祥
担当教員 【英語】
Instructor
HESSELHOLT LARS ○ HISHIDA Toshiaki TANIMOTO Sho
単位数
Credits
2
開講期・開講時間帯
Term / Day / Period
春 火曜日 4時限
Spring Tue 4
授業形態
Course style
講義
Lecture
学科・専攻
Department / Program
Department of Mathematics
必修・選択
Compulsory / Selected
Selected


授業の目的 【日本語】
Goals of the Course(JPN)
See English text.
授業の目的 【英語】
Goals of the Course
The first part (Hishida) of the course provides an elementary functional-analytic approach to linear evolutionary partial differential equations (PDEs).
First of all, those PDEs arising in mathematical physics are introduced. We next study the Yosida-Hille theory for strongly continuous semigroups of operators.
With the aid of knowledge about Sobolev spaces, we then apply the theory to initial-boundary 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 semi-simple 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 semi-simple. 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))
See English text.
到達目標 【英語】
Objectives of the Course
See Goals of the Course.
授業の内容や構成
Course Content / Plan
Part 1
1. Introduction to PDEs
2. Yosida-Hille 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: Semi-simple 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.
不可(F)と欠席(W)の基準
Criteria for "Fail (F)" & "Absent (W)" grades
A sufficient score on weekly problem sets is necessary to pass the course.
参考書
Reference Book
N/A
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.
課外学習等(授業時間外学習の指示)
Study Load(Self-directed 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
N/A
他学科聴講の可否
Propriety of Other department student's attendance
N/A
他学科聴講の条件
Conditions for Other department student's attendance
N/A
レベル
Level
2
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 on-demand 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 (on-demand class)
You should have access to NUCT.