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School of Environment and Society
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- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Yamada Kotaro
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- Day/Period(Room No.)
- Intensive (本館2階201数学系セミナー室)
- Group
- -
- Course number
- MTH.E639
- Credits
- 2
- Academic year
- 2023
- Offered quarter
- 1Q
- Syllabus updated
- 2023/3/20
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
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.
Student learning outcomes
・Learn the definition and basic properties of equivariant homology groups
・Understand the definition and properties of convolution algebras
・Learn the mathematical definition of Coulomb branches of supersymmetric gauge theories
・Understand basics on topological quantum field theories and vaccum.
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 that will be developed
| ✔ 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/Required learning
| Course schedule | Required learning | |
|---|---|---|
| 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. |
Textbook(s)
None required
Reference books, course materials, etc.
Cédic Villani, Optimal Transport: Old and New, Springer, 2000.
Assessment criteria and methods
Assignments (100%).
Related courses
- -
Prerequisites (i.e., required knowledge, skills, courses, etc.)
No prerequisites
