Program
 Topics on quadratic forms Student No.： 50 Time： Mon/Wed 15:10-17:00, 2014-07-09 ~ 2014-07-21 Instructor： Yong Hu  [Universite de Rennes 1, France] Place： Conference Room 1, Floor 1, Jin Chun Yuan West Building Starting Date： 2014-7-9 Ending Date： 2014-7-21

Description:

In this course we will study quadratic forms from an algebraic and arithmetic point of view. It is intended for a wide range of students. The final goal is to give an introduction to recent progress on arithmetic problems about quadratic forms (e.g. the determination of the u-invariant of p-adic function fields).

The course will be comprised of three parts. After a brief introduction to general problems in this domain, the first part will be concerned with basic notions in the algebraic theory of quadratic forms. In the second part, we will discuss interaction with division algebras, Galois cohomology and Milnor K-theory. We will also introduce some field invariants associated to quadratic forms. In the third part, we will sketch some of the existing methods of determining the u-invariant of a p-adic function field.

Prerequisite:

For the first part of the course, we only require a solid background on abstract algebra. For the second part, students are assumed to have basic knowledge of commutative algebra and algebraic number theory, in particular, modest familarity with notions like tensor products, local rings, discrete valuation rings, number fields and local fields. Knowledge ofalgebraic geometry will be helpful but not required at least before the last 2 or 3 lectures (part 3). In the third part, we will have to use schemes, but a good understanding of Sections II.1-6 of Hartshorne's Algebraic Geometry will be sufficient.

Reference:

1. T.Y. Lam, Introduction to quadratic forms over fields

2. P. Gille and T. Szamuely, Central simple algebras and Galois cohomology

3. J.P. Serre, Local fields

4. J.P. Serre, Galois cohomology