授業の目的 【日本語】 Goals of the Course(JPN)   この講義は講師三人によるオムニバス形式の英語講義である. 内容は英文を参照すること. See English text. 


授業の目的 【英語】 Goals of the Course   The first part (Fujiwara) of the course provides a brief look at the classical Galois theory (theory of symmetries of algebraic equations) from a categorical viewpoint, following the idea of Grothendieck in 1960's.
In the second part (Hesselholt), more detailed basics of category theory will be discussed. The theory provides a useful language in algebraic and geometric theories in pure mathematics, and also in applied mathematics such as computer science.
The subject for the third part (Hayashi) is on information. This course briefly explain how to mathematically formulate informationtheoretical security. 


到達目標 【日本語】 Objectives of the Course(JPN))   各講師ごとに与えられたテーマに則した問題について考察し, レポートを作成, 提出することによってテーマの理解を深める. 各テーマの内容は英文を参照すること. See English text. 


到達目標 【英語】 Objectives of the Course   In part I (Fujiwara), students should learn some basic ideas on Galois theory from a categorical viewpoint.
In part II (Hesselholt), participants should learn some of the key new phenomena, which separate category theory from set theory, including limits and colimits, adjoint functors, and Grothendieck's theorem that, in the category of sets, finite limits and filtered colimits commute.
In part III (Hayashi): Understanding the basics of informationtheoretical security. 


授業の内容や構成 Course Content / Plan   The following is a preliminary outline of the lectures during the course.
Galois theory from a categorical viewpoint Lecture 1. Set theory from an axiomatic viewpoint Lecture 2. Language of category theory, including limits and colimits Lecture 3. Field extensions Lecture 4: Galois category Lecture 5: Examples
Introduction to category theory (Hesselholt) Lecture 1. Category theory Lecture 2. Limits and colimits Lecture 3. Adjoint functors Lecture 4: The Yoneda embedding Lecture 5: Grothendieck's theorem
Title: Information theoretical security (Hayashi)
Lecture 1. Notations on probability theory Lecture 2. Introduction of Information measure Lecture 3. Security measure Lecture 4. Hash function Lecture 5. Leftover hashing lemma 


履修条件 Course Prerequisites   Familiarity of linear algebra and calculus is desirable.4年生対象科目。 


関連する科目 Related Courses   Part 1: Undergraduate linear algebra, algebra, set theory Part II: Undergraduate linear algebra, algebra, set theory Part III: Undergraduate linear algebra, calculus 


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. 


不可(F)と欠席(W)の基準 Criteria for "Fail (F)" & "Absent (W)" grades   Students should get gradings from at least two lecturers, otherwise you will have W (absence). The course grade is based on the final numerical grade calculated as above. You may have F (fail) according to the result. 


参考書 Reference Book   Recommended books will be introduced in the lecture. 


教科書・テキスト Textbook   

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   Each lecturer might use NUCT for various purposes, such as distributing documents and report submissions. 


他学科聴講の可否 Propriety of Other department student's attendance   

他学科聴講の条件 Conditions for Other department student's attendance   Contact by email (to Fujiwara) is necessary before signing up to discuss prerequisites. 


レベル Level   

キーワード Keyword   Part I: algebraic equations, Galois theory, Galois category Part II: category, diagram, (co)limit, adjoint, topos. Part III: Information measure, information leakage, hash function 


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.   In person class unless there is a need for the online course. 


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. 

