2025 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra G1
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Shou Yoshikawa
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MTH.A507
- Number of credits
- 100
- Course offered
- 2025
- Offered quarter
- 3Q
- Syllabus updated
- Mar 19, 2025
- Language
- English
Syllabus
Course overview and goals
Perfectoid was introduced to solve problems in both ring theory and algebraic geometry in mixed characteristic.
In the course, I introduce the definition, basic properties, and applications to ring theory of perfectoid.
This course is followed by "Advanced topics in Algebra H1" in the fourth quarter.
Course description and aims
Students are expected to:
- understand the difference between ring theory in characteristic zero or positive and mixed characteristic,
- understand the definition and basic properties of perfectoid,
- understand applications of perfectoid.
Keywords
ring theory, mixed characteristic, perfectoid
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 homework assignments.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Ring theory in characteristic zero and positive: part 1 | Details will be provided during each class session |
| Class 2 | Ring theory in characteristic zero and positive: part 2 | Details will be provided during each class session |
| Class 3 | Ring theory in characteristic zero and positive: part 3 | Details will be provided during each class session |
| Class 4 | Definition and basic properties of prisms: part 1 | Details will be provided during each class session |
| Class 5 | Definition and basic properties of prisms: part 2 | Details will be provided during each class session |
| Class 6 | Definition and basic properties of prisms: part 3 | Details will be provided during each class session |
| Class 7 | Definition and basic properties of perfectoid: part 1 | Details will be provided during each class session |
| Class 8 | Definition and basic properties of perfectoid: part 2 | 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.
B. Bhatt, M. Morrow, and P. Scholze, Integral p-adic Hodge theory, Publ. Math. Inst.BMS18 Hautes ´Etudes Sci. 128 (2018), 219–397. 7. MR3905467,
B. Bhatt and P. Scholze, Prisms and prismatic cohomology, Ann. of Math. (2) 196BS (2022), no. 3, 1135–1275. MR4502597.
Evaluation methods and criteria
Course scores are evaluated by homework assignments (100%). Details will be announced during the course.
Related courses
- MTH.A508 ： Advanced topics in Algebra H1
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
- MTH.A211 ： Advanced Linear Algebra I
- MTH.A212 ： Advanced Linear Algebra II
Prerequisites
Advanced linear algebra and basic undergraduate algebra