学部・大学院区分

多・博前

時間割コード

3211001

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Ⅰ

科目名　【英語】

Perspectives in Mathematical Science I

コースナンバリングコード

担当教員　【日本語】

担当教員　【英語】

単位数

開講期・開講時間帯

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 provides to graduate and undergraduate students. The course is directed not only at international students but also at domestic students, who wish to study abroad and/or improve communication skills in English. All course activities including lectures, homework assignments, questions, and consultations are conducted 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: Topics in Galois theory (K. Fujiwara)
Part 2: Linear algebra over general rings (L. Hesselholt )
Part 3: Algebraic complexity theory and matrix multiplication (F. Le Gall).

授業の目的　【英語】

到達目標　【日本語】

The goal of the course is to introduce participants to topics of current interest in mathematics that are typically not covered in the standard curriculum. Lectures and instruction take place in English with the goal of facilitate participants to discuss mathematics organically in the English language.

到達目標　【英語】

授業の内容や構成

Part 1 (Fujiwara): Topics in Galois theory

In the 19th century, Galois discovered that there is a symmetry which controls algebraic equations of one variable, which is now known as Galois theory. The topic of this part is to give explicit examples of the theory, in particular examples related to regular polyhedra (after Klein). Students attending this course will be able to learn algebraic and geometric aspects of Galois theory by examples.

Part 2 (Hesselholt): Linear algebra over general rings

In 1847, Lam? and Cauchy announced proofs of Fermat?s last theorem in a meeting of the French Academy of Sciences. However, shortly thereafter, Kummer pointed out a fatal error in the proofs. In a way, this was a most fortunate turn of events, for some very important parts of modern mathematics grew out of Kummer?s work. This portion of the course will present some parts of this mathematics. In the end, I will present a conjecture of Kummer?or as he wrote, “a theorem still to be proved”?that to this day remains an important open problem.

Part 3 (Le Gall): Algebraic complexity theory and matrix multiplication

Algebraic complexity theory is the study of computation and algorithms using algebraic models. This part of the course will give an overview of this field and in particular describe several powerful techniques to analyze the complexity of important computational problems from linear algebra. The presentation of these techniques will follow the history of progress on constructing fast algorithms for matrix multiplication, and also include some of its recent developments.

履修条件

A working knowledge of basic undergraduate mathematics including calculus and linear algebra is required.