Compact course for PhD-students
5-9 September 2005, University of Aarhus, Denmark
Sponsor: The Danish Graduate School in Mathematics and Applications
Organizer : Henning Haahr Andersen
Programme:
This compact course is aimed at PhD students in mathematics. There will be two separate series of lectures and problem sessions (See Programme) . The topics are as follows :
Alexander Odesskii : Introduction to the theory of elliptic algebras.
The purpose of this course is to make an informal and nontechnical introduction to the theory of elliptic algebras. These algebras are associative N-graded algebras presented by n generators and n(n-1)/2 quadratic relations and satisfying the so-called Poincare-Birkhoff-Witt condition (PBW-algebras). We will consider examples of such algebras depending on two continuous parameters (namely, on an elliptic curve and a point on this curve) which are flat deformations of the polynomial ring in n variables. Diverse properties of these algebras will be described, together with their relations to integrable systems, deformation quantization, moduli spaces and other directions of modern investigations.
The following topics will be covered :
1. Poisson structures and quantization. Review of general theory and main examples.
2. Quadratic algebras and quadratic Poisson structures. General definition, review of known examples.
3. Elliptic algebras. I begin with the most simple example of such algebras: algebra with three generators. I explain some methods of studying elliptic algebras by this example. After that I will introduce and study more general elliptic algebras.
4. Elliptic algebras and integrable systems. Quantum Yang-Baxter equation. Elliptic solutions of this equation. Elliptic algebras as algebraic structures connected with these solutions.
All necessary notions are introduced during the course. However, some knowledge about Poisson structures, quantum groups, theta-functions in one variable would be useful but not necessary for the participants.
Click here for list of exercises. These problems are to a large extent independent of the course (all concepts are defined). They will be (part of) the topics in the exercise sessions in elliptic algebras.
NOTICE! : Revised version of lecture notes ( pdf , ps ) and exercises ( pdf , ps ) are now available.
Simon Goodwin : An introduction to derived categories in representation theory.
This course will give an elementary introduction to derived categories with an emphasis on derived categories of module categories. The course will require few prerequisites, for example knowledge of chapters 1 and 2 of [P.J. Hilton and U. Stammbach, A course in homological algebra] is ample. The highlights of the course (time permitting) will be Rickard's Morita theory for derived categories and Happel's Theorem, which concerns tilting in derived categories. In the first half of the course we will define the derived category of an abelian category and prove that it is a triangulated category. We will also cover projective (and injective) resolutions, derived functors and derived equivalences. The remainder of the course will concern Rickard's Morita theory, Happel's Theorem and applications of the theory of derived categories.
More specific information about the course you may find here.
Information:
Here you will find a collection of pictures from the compact course.
All lectures will take place at the Department of Mathematical Sciences, where lunch is also available.
Accomodation will be arrangered for students from outside of Aarhus.
A list of participants.
For visitor information for the University of Aarhus see here.
To register please contact Henning Haahr Andersen. When registering please include your name, institute, address, email, the subject of your PhD-project and the name of your supervisor. Deadline for registration is 15. August 2005.