課程資訊

INTRODUCTION TO GEOMETRICAL METHODS AND THOUGHT

97-2

MATH5422

221 U5380

Ceiba 課程網頁
http://ceiba.ntu.edu.tw/972geom

This is a one term course.
Chapter 1 begins with a reflection on Euclidean Geometry as a revolutionary step in human civilization, then on criticisms of logical structure of Euclid's Elements, and finally on reformulation of the axiom system given by D. Hilbert in 1899.
Chapter 2 starts with the so called Absolute Geometry-geometry without Euclid's parallel axiom, and a few equivalent forms of parallel axioms. It discuss Non-Euclidean Geometry before the fomulation of Lobachevsky and Bolyai.
Chapter 3 focus on the most remarkable invention of Lobachevsky and Bolyai, namely the idea of horocycles and horospheres, its use in Non-Euclidean geometry, including explicit formula for angle of parallelism etc.
Chapter 4 discuss Spherical Geometry in both Euclidean and Non-Euclidean case, problems on consistency and Beltrami-Klein model and Poincare model.
Chapter 5 shall study in some detail B. Riemann's famous lecture on "On the hypotheses which lie at the foundations of geometry" in 1854, generally own as Riemann' s Habilitation lecture.

Course Goal：
It was said that the creation of Non-Euclidean geometry was the most consequential and revolutionary step in mathematics since Greek times. In the book "Geometry: Euclid and beyond" R. Hartshorne in preface wrote" I hope this material will become familiar to every student of mathematics, and in particular to those who will be future teachers". The planning of the current course was partially inspired by these remarks. Hopefully it may fill in some gaps in current mathematics curriculum.

Knowledge from Advanced Calculus (real number system) and Linear algebras would be helpful, but not absolutely necessary.

Office Hours

Kulczycki, S., "Non-Euclidean Geometry";
Greenberg, M.J., "Euclidean and non-Euclidean geometry, Development and History"

(僅供參考)

 No. 項目 百分比 說明 1. 期中考 40% 2. 期末考 40% 3. 隨堂測驗 0% 4. 作業 20% 5. 報告 0%

 課程進度
 週次 日期 單元主題 第9週 4/15 期中考 第17週 6/10 最後一堂課 第18週 6/17 提交essay 期末考