Special Mathematics Lecture (Differential geometry)

Goals of the Course

【Standardized across all programs】 In a global and interconnected society, mathematics play a central role. They are at the root of all communications tools, and serve as a universal language for natural and social sciences. All current technological developments are based on them. Mathematics can also be learned, enjoyed, and shared by everybody. Studying mathematics is also learning humility, tolerance, and diversity.

The aim of these Special Mathematics Lectures is to provide a solid knowledge on various mathematical topics. Concomitantly, students will develop a rigorous argumentation and a critical thinking. Each semester, a new subject is proposed, and these lectures are accessible to all students of Nagoya University, independently of their major and of their academic age. These special lectures also offer a unique opportunity to study in an international, transdisciplinary, and friendly environment.

Objectives of the Course

Differential geometry plays a central role in many physical theory, as for example in classical mechanics, in solid states physics or in general relativity. During this one semester course, many essential notions will be introduced, among them the definitions of a manifold, of the curvature, of the parallel transport, of the holonomy, etc. By attending this course, students will get familiar with these notions, and will master the fundamental difference between extrinsic and intrinsic geometrical properties.

Course Contents or Plan

1. Differentiable manifolds
2. Tensors, tensor fields and differential forms
3. Integration on manifolds
4. Riemannian manifolds
5. Curvature
6. General relativity

Course Prerequisites and Related Courses

Basic knowledge on calculus and linear algebra, as provided in Calculus I & II and in Linear algebra I & II. Motivated 1st year students can also attend without these prerequisites but after a discussion with the instructor.

Course Evaluation Method and Criteria

The final grade will be based on the active participation during the lectures and on some written reports. Students will be encouraged to work on applications related to their major during the semester. Students need to notify the course withdrawal to the instructor when they have no intention of finishing the course during the semester.

Study Load (Self-directed Learning Outside Course Hours)

Students are expected to read their notes, and to be familiar with the content of the previous lectures before each new lecture.

Notice for students

It is expected that the students will show a certain maturity in studying independently and in choosing some exercises and problems to solve. Study sessions might be organized on a weekly basis.

Textbook

Lecture Notes

Additional information

Reference Website

This lecture is provided under Creative Commons Attribution-Non Commercial-ShareAlike 4.0 International.

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