2024 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on current topics in Mathematics M
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Shunsuke Takagi / Kazuma Shimomoto
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive
- Class
- -
- Course Code
- MTH.E643
- Number of credits
- 200
- Course offered
- 2024
- Offered quarter
- 2Q
- Syllabus updated
- Mar 17, 2025
- Language
- Japanese
Syllabus
Course overview and goals
"F-singularities" refer to singularities defined by the Frobenius map. There are four basic classes of F-singularities: F-regular, F-pure, F-rational, and F-injective. These are expected to correspond to major classes of singularities in complex birational geometry. In this course, I will outline the recent development of this correspondence.
Course description and aims
The aim is to learn and become accustomed to various concepts that appear in the singularity theory of algebraic varieties, with a focus on positive characteristic methods.
Keywords
F-singularities, BCM test ideals, multiplier ideals, reduction modulo p
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 : -- Basic notions in the theory of F-singularities (F-regular, F-pure, F-rational, and F-injective singulariteis) -- Singularities in birational geometry (log terminal, log canonical, rational, and Du Bois singularities) -- Absolute integral closures and BCM test ideals -- Reduction modulo p technique due to Deligne-Illusie-Raynaud -- Weak ordinarity conjecture | 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.
S. Takagi and K.-i. Watanabe, F-singularities: applications of characteristic p methods to singularity theory, Sugaku Expositions 31 (2018), no.1, 1–42.
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.A301 ： Algebra I
- LAS.M106 ： Linear Algebra II
- MTH.A331 ： Algebra III
Prerequisites
None required