学部・大学院区分

多・博前

時間割コード

3211002

科目区分

Ａ類Ⅰ（基礎科目）
Category A-1

科目名　【日本語】

数理科学展望Ⅱ

科目名　【英語】

Perspectives in Mathematical Science II

コースナンバリングコード

担当教員　【日本語】

大平　徹 ○ 藤江　双葉 JAERISCH Johannes Klaus B

担当教員　【英語】

OHIRA Toru ○ FUJIE Futaba JAERISCH Johannes Klaus Bernhard

単位数

開講期・開講時間帯

秋火曜日３時限
Fall Tue 3

授業形態

学科・専攻

多元数理科学研究科

必修・選択

授業の目的　【日本語】

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 I: Bayes’ Theorem (Toru OHIRA), Part II: Dynamics of continued fractions (Johannes JAERISCH), Part III: Graphs and matrices (Futaba FUJIE).

授業の目的　【英語】

到達目標　【日本語】

Basic understanding of concepts in each part of the lecture

到達目標　【英語】

授業の内容や構成

Part I: Bayes’ Theorem (Toru OHIRA)
Dates
Oct. 6, 13, 20, 27

In this part of the lecture, we aim to understand the Bayes' theorem, which gives a way to infer a cause from outcomes statistically. The basic concepts such as expectation, conditional probability of the probability theory are reviewed as a preparation. We will discuss concrete examples and applications of the theorem as well.

Part II: Dynamics of continued fractions (Johannes JAERISCH)
Dates
Nov. 10, 17, 24, Dec. 1

From ancient times it is known that the number pi can be well approximated by the rational number 355/113 up to 6 decimal places. In fact, this approximation is best in the sense that there exists no rational number closer to pi and with denominator at most 113. Best approximations will lead us to continued fraction expansions of real numbers.
We will investigate properties of continued fractions from several viewpoints:
circle rotations and closest returns to the initial value,
Gauss map defining an ergodic measure-preserving transformation on [0,1], and
cutting sequences of hyperbolic rays with respect to the Farey tessellation of the upper half plane.

Keywords
Continued fractions, circle rotations, Gauss map, cutting sequence, hyperbolic geometry.

References
B. Hasselblatt, A. Katok. A First Course in Dynamics: with a Panorama of Recent Developments. 2003.
A. Ya. Khinchin. Continued Fractions. Dover Publications; Revised edition, 1997.
C. Series. The modular surface and continued fractions. Journal of the London Mathematical Society, 2(1):69?80, 1985.

Part III: Graphs and matrices (Futaba FUJIE)
Dates
Dec. 8, 15, 22, and Jan. 12

One can find many results in the area of graph theory where linear algebra plays an important role. In this course, we will study some well-known matrices associated with a graph and their properties as well as what they can tell us about the original graph. We will first introduce ourselves to some well-known concepts and terminology in graph theory, then will focus on three graph-related matrices. This is a 4-lecture course; our tentative plan is to cover basics of graphs, incidence matrices, adjacency matrices, Laplacian matrices (and other related topics if time permits).

Key words
incidence matrix, adjacent matrix, Laplacian matrix.

References
(There will be no specific books that we will stick with, but the topics will be mainly selected from the following. When necessary, handouts will be provided.)
[1] R.B. Bapat, Graphs and Matrices, Springer (2011)
[2] D.B. West, Introduction to Graph Theory, Prentice Hall (2000)

Jan 19, 26, Feb 2
Reserved days for extra lectures if necessary

履修条件

Prerequisite Working knowledge of basic undergraduate mathematics including calculus and linear algebra is required.