Goals of the Course(JPN)
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.
Objectives of the Course(JPN))
Objectives of the Course
Course Content / Plan
1. Introduction to PDEs
2. Yosida-Hille theory
3. Analytic semigroups
4. Sobolev space
5. Applications to PDEs
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
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
|Familiarity of linear algebra and calculus is desirable, but not strictly necessary.
|Part 1 : calculus, differential equations, functional analysis
Part 2 : Linear algebra, algebra, representation theory.
Part 3 : Algebra
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.
Criteria for "Fail (F)" & "Absent (W)" grades
|A sufficient score on weekly problem sets is necessary to pass the course.
|Lecture notes will be uploaded weekly to the course homepage on NUCT.
Lecture notes will be provided at the beginning of the third part.
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.
Notice for Students
Propriety of Other department student's attendance
Conditions for Other department student's attendance
|Part 1 : partial differential equations, semigroups of operators,Sobolev space
Part 2 : Linear algebra, semisimple rings, discrete Fourier transform, ideal class group.
|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.
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.
Lectures of the third part will be given via Zoom.
Additional measures for remote class (on-demand class)
|You should have access to NUCT.