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- Common courses
- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Kato Fumiharu Kondo Shigeyuki
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Intensive ()
- Group
- -
- Course number
- MTH.E643
- Credits
- 2
- Academic year
- 2018
- Offered quarter
- 2Q
- Syllabus updated
- 2018/3/20
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
K3 surfaces are interesting mathematical objects not only in algebraic geometry, but also in mathematical physic. In this lecture, we start with basics in lattice theory, an introduction to the so-called Torelli-type theorems, and then will discuss how these materials are used in the theory of K3 surfaces.
Student learning outcomes
We focus to the mathematical phenomena around automorphisms of K3 surfaces. In general, automorphism groups of K3 surfaces are discrete groups, in which both finite and infinite groups are possible. We will discuss automorphisms of K3 surfaces, and finite groups which acts on K3 surfaces as automorphisms.
Keywords
Complex manifolds, K3 surfaces, Enriques surfaces, Torelli-type theorems, automorphisms, symplectic automorphisms, reflection groups, lattices, Niemeier lattice, Leech lattice, Mathieu groups
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 | Lattice theory (unimodular lattices, Niemeier lattice, Leech lattice, Mathieu groups) 1 | Details will be provided during each class session. |
| Class 2 | Lattice theory (unimodular lattices, Niemeier lattice, Leech lattice, Mathieu groups) 2 | Details will be provided during each class session. |
| Class 3 | Lattice theory (unimodular lattices, Niemeier lattice, Leech lattice, Mathieu groups) 3 | Details will be provided during each class session. |
| Class 4 | Torelli-type theorems of K3 surfaces and automorphisms 1 | Details will be provided during each class session. |
| Class 5 | Torelli-type theorems of K3 surfaces and automorphisms 2 | Details will be provided during each class session. |
| Class 6 | Torelli-type theorems of K3 surfaces and automorphisms 3 | Details will be provided during each class session. |
| Class 7 | Finite-group action on K3 surfaces (non-symplectic case) 1 | Details will be provided during each class session. |
| Class 8 | Finite-group action on K3 surfaces (non-symplectic case) 2 | Details will be provided during each class session. |
| Class 9 | Finite-group action on K3 surfaces (non-symplectic case) 3 | Details will be provided during each class session. |
| Class 10 | Finite-group action on K3 surfaces (symplectic case, and the relation with Mathieu groups) 1 | Details will be provided during each class session. |
| Class 11 | Finite-group action on K3 surfaces (symplectic case, and the relation with Mathieu groups) 2 | Details will be provided during each class session. |
| Class 12 | Finite-group action on K3 surfaces (symplectic case, and the relation with Mathieu groups) 3 | Details will be provided during each class session. |
| Class 13 | Leech lattice and automorphisms groups of Summer surfaces 1 | Details will be provided during each class session. |
| Class 14 | Leech lattice and automorphisms groups of Summer surfaces 2 | Details will be provided during each class session. |
| Class 15 | Leech lattice and automorphisms groups of Summer surfaces 3 | Details will be provided during each class session. |
Textbook(s)
None.
Reference books, course materials, etc.
Course materials are provided during class.
Assessment criteria and methods
Assignments (100 %).
Related courses
- MTH.A302 ： Algebra II
- MTH.A403 ： Advanced topics in Algebra C
- MTH.A404 ： Advanced topics in Algebra D
- MTH.A301 ： Algebra I
Prerequisites (i.e., required knowledge, skills, courses, etc.)
No prerequisites are necessary, but enrollment in the related courses is desirable.
