Special lectures on current topics in Mathematics G

Academic unit or major

Graduate major in Mathematics

Instructor(s)

Tasuki Kinjo / Hironori Oya

Class Format

Lecture (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

Intensive (Mathematics Seminar Room 201)

Class

-

Course Code

MTH.E637

Number of credits

200

Course offered

2026

Offered quarter

1Q

Syllabus updated

Mar 5, 2026

Language

Japanese

Syllabus

Course overview and goals

The main subject of this lecture is the cohomological Hall algebra associated with an abelian category. First, we introduce in an elementary manner the cohomological Hall algebra associated with a quiver, and show that it admits a description as a shuffle algebra. Then, we explain the theory of stacks and derived algebraic geometry, that is, theories of generalized notions of spaces extending schemes, as well as the six-functor formalism, and describe the construction of cohomological Hall algebras for more general categories. Finally, we introduce several theorems concerning the structures of cohomological Hall algebras of 2- and 3-Calabi–Yau categories, and explain their applications to enumerative geometry.

The theory of cohomological Hall algebras has developed rapidly in recent years, and plays an important role in connecting enumerative geometry and geometric representation theory. Through cohomological Hall algebras, this lecture aims to explain the basic concepts and ideas appearing in these fields, and to provide the foundational knowledge necessary for reading recent research papers.

Course description and aims

・To become able to deduce the shuffle presentation of the cohomological Hall algebra associated with a quiver using localization formulas.
・To become familiar with the computations in the six-functor formalism and vanishing cycles.
・To learn the ideas of derived algebraic geometry and to understand the construction of two-dimensional cohomological Hall algebras.
・To understand PBW-type theorems for cohomological Hall algebras.

Keywords

Cohomological Hall algebras, derived stacks, perverse sheaves, derived categories, virtual fundamental classes, quiver varieties, Calabi-Yau categories

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.

Bu, Davison, Ibáñez Núñez, Kinjo, Padurariu "Cohomology of symmetric stacks" (available at https://arxiv.org/abs/2502.04253)
Toda "Recent Progress on the Donaldson–Thomas Theory" Springer (2021)

Evaluation methods and criteria

Assignments (100%).

Related courses

• MTH.A301 ： Algebra I
• MTH.A302 ： Algebra II

Prerequisites

None required.