HB340000 Topics in Algebra

Hiroshima University Syllabus

Japanese

Academic Year 2025Year School/Graduate School School of Science

Lecture Code HB340000 Subject Classification Specialized Education

Subject Name 代数学特殊講義

Subject Name
（Katakana）ダイスウガクトクシュコウギ

Subject Name in
English Topics in Algebra

Instructor KIMURA SHIYUNICHI

Instructor
(Katakana) キムラ　シュンイチ

Campus Higashi-Hiroshima Semester/Term 4th-Year,  First Semester,  2Term

Days, Periods, and Classrooms (2T) Tues7-8,Fri5-6：SCI E102

Lesson Style Lecture Lesson Style
(More Details) Face-to-face, Online (simultaneous interactive), Online (on-demand)

Credits 2.0 Class Hours/Week 4 Language of Instruction J : Japanese

Course Level 4 : Undergraduate Advanced

Course Area（Area） 25 : Science and Technology

Course Area（Discipline） 01 : Mathematics/Statistics

Eligible Students

Keywords Combinatorial Games, Infinite, Infinitesimals, Surreal Number, NIM

Special Subject for Teacher Education 　 Special Subject 　

Class Status
within Educational
Program
(Applicable only to targeted subjects for undergraduate students)
Criterion referenced
Evaluation
(Applicable only to targeted subjects for undergraduate students) Mathematics
（Knowledge and Understanding）
・Acquiring knowledge and vision on advanced theories as an extension of core theory of modern mathematics.

Class Objectives
/Class Outline Conway's book "On Numbers and Games" and Knuth's book "Surreal Numbers" treats very interesting mathematical Object, namely Surreal numbers, but are mathematically not rigorous enough.  In this lecture, we try to fix their problems and give a logical introduction to the theory of Surreal Numbers.  If time permits, we also discuss NIM values.

Class Schedule lesson1 Introduction: notion of numbers, notion of infinity
lesson2 Finite Hackenbush
lesson3 Definition of Combinatorial Games, Sum of games, Inverse games, and Outcomes
lesson4 Outcome of sums of games
lesson5 Definition of Numbers in short games, totally ordered Abelian group structure
lesson6 Game values of finite Hackenbush, and their products
lesson7 Simplicity Theorem
lesson8 Transfinite Ordinal Numbers
lesson9 Transfinite Hackenbush and Surreal Numbers
lesson10 Products of Surreal Numbers
lesson11 Quotients of Surreal Nubers
lesson12 Cantor Normal Form
lesson13 Sturm's Theorem
lesson14 Algebraic Equations and real closedness
lesson15 Characteristic 2 case, Transfinite NIM values

Text/Reference
Books,etc. Lecture is based on Conway's "On Numbers and Games", but will try to fix the problems there,
so students do not have to prepare the book.  Recommended Text Books are Abuku et at "組合せゲーム理論の世界", and in English, Siegel "Combinatorial Game Theory".

PC or AV used in
Class,etc.

(More Details)

Learning techniques to be incorporated

Suggestions on
Preparation and
Review The tools are games, which may sound like an easy topic, but this lecture is supposed to be very logical and mathematical.  No prior knowledge is required except for rudimentary abstract mathematical knowledge like Cantor's Set Theory, but advanced mathematical thinking will be required to follow the lectures.

Requirements

Grading Method Reports

Practical Experience

Summary of Practical Experience and Class Contents based on it

Message

Other

Please fill in the class improvement questionnaire which is carried out on all classes.
Instructors will reflect on your feedback and utilize the information for improving their teaching.

Academic Year	2025Year	School/Graduate School	School of Science
Lecture Code	HB340000	Subject Classification	Specialized Education
Subject Name	代数学特殊講義
Subject Name （Katakana）	ダイスウガクトクシュコウギ
Subject Name in English	Topics in Algebra
Instructor	KIMURA SHIYUNICHI
Instructor (Katakana)	キムラ　シュンイチ
Campus	Higashi-Hiroshima	Semester/Term	4th-Year, First Semester, 2Term
Days, Periods, and Classrooms	(2T) Tues7-8,Fri5-6：SCI E102
Lesson Style	Lecture	Lesson Style (More Details)	Face-to-face, Online (simultaneous interactive), Online (on-demand)

Credits	2.0	Class Hours/Week	4	Language of Instruction	J : Japanese
Course Level	4 : Undergraduate Advanced
Course Area（Area）	25 : Science and Technology
Course Area（Discipline）	01 : Mathematics/Statistics
Eligible Students
Keywords	Combinatorial Games, Infinite, Infinitesimals, Surreal Number, NIM
Special Subject for Teacher Education		Special Subject
Class Status within Educational Program (Applicable only to targeted subjects for undergraduate students)
Criterion referenced Evaluation (Applicable only to targeted subjects for undergraduate students)	Mathematics （Knowledge and Understanding）・Acquiring knowledge and vision on advanced theories as an extension of core theory of modern mathematics.
Class Objectives /Class Outline	Conway's book "On Numbers and Games" and Knuth's book "Surreal Numbers" treats very interesting mathematical Object, namely Surreal numbers, but are mathematically not rigorous enough. In this lecture, we try to fix their problems and give a logical introduction to the theory of Surreal Numbers. If time permits, we also discuss NIM values.
Class Schedule	lesson1 Introduction: notion of numbers, notion of infinity lesson2 Finite Hackenbush lesson3 Definition of Combinatorial Games, Sum of games, Inverse games, and Outcomes lesson4 Outcome of sums of games lesson5 Definition of Numbers in short games, totally ordered Abelian group structure lesson6 Game values of finite Hackenbush, and their products lesson7 Simplicity Theorem lesson8 Transfinite Ordinal Numbers lesson9 Transfinite Hackenbush and Surreal Numbers lesson10 Products of Surreal Numbers lesson11 Quotients of Surreal Nubers lesson12 Cantor Normal Form lesson13 Sturm's Theorem lesson14 Algebraic Equations and real closedness lesson15 Characteristic 2 case, Transfinite NIM values
Text/Reference Books,etc.	Lecture is based on Conway's "On Numbers and Games", but will try to fix the problems there, so students do not have to prepare the book. Recommended Text Books are Abuku et at "組合せゲーム理論の世界", and in English, Siegel "Combinatorial Game Theory".
PC or AV used in Class,etc.
(More Details)
Learning techniques to be incorporated
Suggestions on Preparation and Review	The tools are games, which may sound like an easy topic, but this lecture is supposed to be very logical and mathematical. No prior knowledge is required except for rudimentary abstract mathematical knowledge like Cantor's Set Theory, but advanced mathematical thinking will be required to follow the lectures.
Requirements
Grading Method	Reports
Practical Experience
Summary of Practical Experience and Class Contents based on it
Message
Other
Please fill in the class improvement questionnaire which is carried out on all classes. Instructors will reflect on your feedback and utilize the information for improving their teaching.

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