Special lectures on advanced topics in Mathematics C

Academic unit or major

Graduate major in Mathematics

Instructor(s)

Hisashi Kasuya / Toshiaki Hattori

Class Format

Lecture (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

Intensive (Mathematics Seminar Room 201)

Class

-

Course Code

MTH.E433

Number of credits

200

Course offered

2026

Offered quarter

1Q

Syllabus updated

Mar 5, 2026

Language

Japanese

Syllabus

Course overview and goals

A homogeneous space is a space represented as the quotient of a Lie group (a smooth group structure on a manifold) by a closed subgroup. Examples of such spaces include Euclidean space, spheres, and hyperbolic space. By replacing Lie groups and their closed subgroups, various spaces are obtained, leading to a rich study of geometric structures. The geometric structure of homogeneous spaces can be studied using Lie algebras. A Lie algebra is roughly like a matrix, and thus, the homogeneous space can be studied in depth using methods from linear algebra (such as eigenvalues). In this lecture, we will focus on cohomology, which is a fundamental invariant in the study of geometry. When a Lie group is nilpotent and the closed subgroup is discrete, a homogeneous space is called a nilmanifold. It is known that the cohomology of a nilmanifold is isomorphic to the cohomology of the Lie algebra (Nomizu's theorem). Nomizu's theorem is a classical result from the 1950s, but it still has many applications today, and various extensions are actively being studied. When a Lie group is solvable (a broader class of groups than nilpotent), a homogeneous space is called a solvmanifold. One of the goals of this lecture is to explain in detail the extension of Nomizus theorem to solvmanifolds, which was developed by Kasuya.

Course description and aims

Students can calculate the cohomology of various homogeneous spaces and study geometric structures. In particular, I hope that students master the cohomology calculation method for solvmanifolds established by Kasuya, and can research the geometry of solvmanifolds, which has not yet been studied.

Keywords

Lie group, Lie algebra, homogeneous space, nilmanifold, solvmanifold, cohomology, complex geometry

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

Study advice (preparation and review)

Textbook(s)

None required.

Reference books, course materials, etc.

We will provide possible references in the lectures.

Evaluation methods and criteria

Assignments (100%).

Related courses

• MTH.B301 ： Geometry I
• MTH.B302 ： Geometry II
• MTH.B331 ： Geometry III
• MTH.B341 ： Topology

Prerequisites

Basic knowledge about smooth manifolds and topology