### 2020　Special lectures on advanced topics in Mathematics B

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Instructor(s)
Yasuda Takehiko  Kato Fumiharu
Course component(s)
Lecture
Mode of instruction
ZOOM
Day/Period(Room No.)
Intensive (Zoom)
Group
-
Course number
MTH.E432
Credits
2
2020
Offered quarter
3Q
Syllabus updated
2020/9/18
Lecture notes updated
-
Language used
English
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### Course description and aims

The main subject of this course is the study of the McKay correspondence by using motivic integration. Firstly we explain motivic integration over algebraic varieties and formulation of the McKay correspondence in terms of motivic invariants. After that, we explain the proof of the McKay correspondence by generalization of motivic integration. Lastly we explain generalization of the McKay correspondence to positive and mixed characteristics and explain its relation with the number theory as well as application to singularities.

We find that there is a beautiful (but non-trivial) relation between representations of finite groups and geometry of singularities. Moreover we learn that by generalizing to positive and mixed characteristics, we get number-theoretic phenomena and a new approach to singularities in positive and mixed characteristics.

### Student learning outcomes

Understand the overview of motivic integration theory and become able to compute simple integrals.
Understand construction of quotient singularties and their basic properties.
Understand the formulation of the McKay correspondence in terms of motivic invariants and become able to derive properites of quotient singularities by applying it.

### Keywords

McKay correspondence, quotient singularities, motivic integration, positive and mixed characteristics

### 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 Motivic integration 1 Details will be provided during each class session.
Class 2 Motivic integration 2 Details will be provided during each class session.
Class 3 Motivic integration 3 Details will be provided during each class session.
Class 4 Quotient singularities 1 Details will be provided during each class session.
Class 5 Quotient singularities 2 Details will be provided during each class session.
Class 6 Quotient singularities 3 Details will be provided during each class session.
Class 7 McKay correspondence in terms of motivic invariants 1 Details will be provided during each class session.
Class 8 McKay correspondence in terms of motivic invariants 2 Details will be provided during each class session.
Class 9 McKay correspondence in terms of motivic invariants 3 Details will be provided during each class session.
Class 10 Generalization of motivic integration and the proof of the McKay correspondence 1 Details will be provided during each class session.
Class 11 Generalization of motivic integration and the proof of the McKay correspondence 2 Details will be provided during each class session.
Class 12 Generalization to positive and mixed characteristics 1 Details will be provided during each class session.
Class 13 Generalization to positive and mixed characteristics 2 Details will be provided during each class session.
Class 14 Application to singularities 1 Details will be provided during each class session.
Class 15 Application to singularities 2 Details will be provided during each class session.

### Textbook(s)

None in particular.

### Reference books, course materials, etc.

Reference articles will be mentioned during the course

### Assessment criteria and methods

Assignments (100%)

### Related courses

• MTH.A201 ： Introduction to Algebra I
• MTH.A202 ： Introduction to Algebra II
• MTH.A203 ： Introduction to Algebra III
• MTH.A204 ： Introduction to Algebra IV
• MTH.A301 ： Algebra I
• MTH.A302 ： Algebra II

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

None in particular