2018 Special lectures on advanced topics in Mathematics A

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Kato Fumiharu  Kondo Shigeyuki 
Course component(s)
Lecture
Mode of instruction
 
Day/Period(Room No.)
Intensive ()  
Group
-
Course number
MTH.E431
Credits
2
Academic year
2018
Offered quarter
2Q
Syllabus updated
2018/3/20
Lecture notes updated
-
Language used
Japanese
Access Index

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.

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