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.
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.
Complex manifolds, K3 surfaces, Enriques surfaces, Torelli-type theorems, automorphisms, symplectic automorphisms, reflection groups, lattices, Niemeier lattice, Leech lattice, Mathieu groups
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
This is a standard lecture course. There will be some assignments.
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. |
None.
Course materials are provided during class.
Assignments (100 %).
No prerequisites are necessary, but enrollment in the related courses is desirable.