数理科学展望III-2015 講師 山上滋 教授 理学部／理学研究科 2015年度 前期 理学部数理学科4年

授業の目的

In his celebrated Erlangen Program in 1872, F. Klein opened a way to synthesize geometric objects based on group symmetry. Since then the notion of group has been playing signiﬁcant roles in the study of various geometries. Among them, fundamental is the so-called projective geometry, which is intimately related to that of vector spaces. Interrelations of geometric positions of ﬂat objects such as lines and planes in Euclidean spaces are described most aesthetically in the framework of projective geometry. The fundamental theorem of projective geometry then states that the three-point collinearity is enough to recover the linear group structure behind them. Its importance is not just restricted within purely mathematical subjects and we shall review here, in quantum theory and special relativity, two fundamentals in physics, how their symmetries can be realized as linear groups as applications of the fundamental theorem.

授業の工夫

Symmetry is such a vast subject that its overall description requires lots of words. As a perspective course, I present here an interdisciplinary topic which ranges from geometry to algebra with physical backgrounds. It is just a tiny part of the subject but still provides good impetus to the typical way of thinking of symmetry.

The Method of Evaluation

Grading in this part is based on submitted reports on homeworks which will be assigned during the course. (No submission therefore means Nonattendance.)

Course notes will be provided at the ﬁrst lecture time or you can directly refer to the source papers below.
 C.-A. Faure, An elementary proof of the fundamental theorem of projective geometry, Geom. Dedicata, 90(2002), 145–151.
 P.G. Vroegindewey, An algebraic generalization of a theorem of E.C. Zeeman, Indagationes Mathematica, 77(1974), 77–81.

The Plan of the Course

Part 1 is scheduled to be 4/14, 4/21, 4/28, 5/12.

1. Review on aﬃne spaces. 2. Touch of projective spaces. 3. The fundamental theorem of projective geometry. 4. Wigner’s theorem on describing symmetry in quantum mechanics. 5. Alexandrov-Zeeman’s theorem on describing symmetry in special relativity.

Keywords

Projective geometry, aﬃne geometry, symmetry in physics.

Required Knowledge

Basic knowledge and skills in linear algebra and set theory.

Attendance

This course is open for all students in Nagoya University as a part of open subject program. Certain amount of experience in the set-theoretic framework of mathematics is required, however, to get beneﬁts from this part of the course.