2023 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on advanced topics in Mathematics I
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Asuka Takatsu / Kotaro Yamada
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive (本館2階201数学系セミナー室)
- Class
- -
- Course Code
- MTH.E533
- Number of credits
- 200
- Course offered
- 2023
- Offered quarter
- 1Q
- Syllabus updated
- Jul 8, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The main subject of this course is the relation between optimaltransport theory and lower curvature bound. If we define the optimality of transport, then an optimal transport path becomes a minimal geodesic. The behavior of minimal geodesics tells us how a space is curved. In this course, among curvartures, we mainly discuss Ricci curvature, which controls the growth of the volume of metric ball.
We first study how to formulate an optimal transport problem as a variational problem on the space of probability measures. Then we briefly review Riemannian geometry. Finally, we investigate how Ricci curvature appears in optimal transport world.
Course description and aims
・State the mathematical formulation of optimal transport problem
・Determine the optimality of transport plan
・Be familiar with Riemannian geometry
・Provide examples of Riemannian manifold with positive Ricci curvature
・Understand the relation between optimal transport theory and Ricci curvature
Keywords
optimal transport theory, variatinal problem, metric space, complete, separble, geodesic, metric measure space, Riemannian geometry, Ricci curvature, Jacobi field, entropy, convexity, Brunn--Minkowski inequality
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | The following topics will be covered in this order : -- optimal transport theory on a finite set -- optimal transport theory on Euclidean space -- manifold -- Riemannian manifold -- curvature -- Riemannian distance function -- Riemannian volume form -- Jacobi field -- entropy -- curvature-dimension condition -- Brunn--Minkowski inequality | Details will be provided during each class session. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None required
Reference books, course materials, etc.
Cédic Villani, Optimal Transport: Old and New, Springer, 2000.
Evaluation methods and criteria
Assignments (100%)
Related courses
- -
Prerequisites
No prerequisites.