SCIENTIFIC PROGRAMS AND ACTIVITIES

Starting Thursday, January 10 to April 26, 2013
Graduate course on Affine and Extended Affine Lie Algebras
Lecturer: E. Neher

The aim of this course is to provide the participants of Concentration period I with the necessary background from the structure theory of affine and extended affine Lie algebras.
Contents. Review of split simple finite-dimensional Lie algebras and affine Kac-Moody Lie algebras. Extended affine Lie algebras: Definition, examples, first properties. Reflection systems, in particular affine reflection systems and extended affine root systems. Lie tori: Definition, properties, examples. Relation between Lie tori and extended affine Lie algebras. Classification of Lie tori.

Starting Monday, January 14 to April 12, 2013
Graduate Course on Algebraic Groups over Arbitrary Fields
Lecturers: V. Chernousov and N. Semenov

The primary goal of the course is to provide an introduction to the theory of reductive algebraic groups over arbitrary fields and local regular rings. The main objectives are to give some basic material on their structure and classification.

Background:
Linear algebraic groups have been investigated for over 100 years. They first appeared in a paper of Picard related to differential equations. The subject was later developed by Cartan, Killing, Weyl and others who studied and classified semisimple Lie groups and Lie algebras over the complex and real numbers. With the development of algebraic geometry, it became important to study algebraic groups in a more general setting. The fundamental work of Weil and Chevalley in the 1940s and 1950s initiated the development of the theory of algebraic groups over arbitrary fields. Over the next thirty years, the foundations of this theory (and of the even more general theory of group schemes) over arbitrary fields and rings led to many important results by Borel, Chevalley, Grothendieck, Demazure, Serre, Springer, Steinberg, Tits and others. The motivation for this generalization was to establish a synthesis between different parts of mathematics such as number theory, the theory of finite groups, representation theory, invariant theory, the theory of Brauer groups, the algebraic theory of quadratic forms, and the study of Jordan algebras. Indeed, using the language of the theory of algebraic groups, many outstanding problems and conjectures can be reformulated in a uniform way. Nowadays this branch of mathematics is a very interesting mixture of group theory and algebraic geometry. Over finite fields it classifies almost all simple finite groups, over number fields it studies important arithmetic properties of different algebraic objects such as quadratic and hermitian forms, central simple algebras, arithmetic groups, discrete subgroups, modular forms, over real numbers it clarifies the theory of Lie groups, and so on.
The primary goal of the course is to provide an introduction to the theory of reductive algebraic groups over arbitrary fields and local regular rings. The main objectives are to give some basic material on their structure and classification.

The course will begin with an overview of some notions and objects in algebraic groups over algebraically closed fields and their properties (part I) such as: subgroups, homomorphisms, Lie algebras, semisimple and unipotent elements, tori, solvable groups, semisimple and unipotent elements, Jordan decomposition. After that it will pass to the Borel fixed-point theorem concerning the action of a solvable group on a quasi-projective variety. They lead to the important conjugacy theorems and from them to the long road of the classification of reductive groups over algebraically closed fields in terms of root systems.

Then the main direction of the course will shift to the theory of algebraic groups over arbitrary fields (part II). This will be based on the celebrated paper by Tits on the classification of semisimple linear algebraic groups and the Book of Involution by Knus, Merkurjev, Rost and Tignol. As was shown by Tits, any semisimple group G over a field is determined by its anisotropic kernel and a combinatorial datum, called the Tits index. In the course these two concepts will be systematically studied. In particular, the notions of an inner/outer, strongly inner forms of linear algebraic groups will be introduced together with explicit links to the theory of central simple algebras, Jordan algebras and quadratic forms.

Prerequisites: The main prerequisite is some familiarity with Lie algebras and algebraic geometry, like for example the first part of the book Linear Algebraic Groups by James E. Humphreys.

Course structure: The course will run from mid-January until the beginning of March so that students are well-prepared to follow the remainder of the thematic program. Both parts will have approximately 20 hours. Arrangements will be made so that the course can be taken for credit by participating students. The final grade will be based on homework assignments. The solutions of the homework problems will be discussed in tutorials.

Starting Thursday, January 10 to April 5, 2013
Graduate Course on Algebraic and Geometric Theory of Quadratic Forms
Lecturer: N. Karpenko, Dean's Distinguished Visitor

Following [1, Part 1], we develop the basics of the theory of quadratic forms over arbitrary fields. In the second half of the course we briefly introduce the Chow groups and then apply them to get some of more advanced results of [1, Part 3].

Here is the program in more details:
1. Bilinear forms.
2. Quadratic forms.
3. Forms over rational function fields.
4. Function fields of quadrics.
5. Forms and algebraic extensions.
6. u-invariants.
7. Applications of the Milnor conjecture.
8. Chow groups.
9. Cycles on powers of quadrics.
10. Izhboldin dimension.

References:
1. R. Elman, N. Karpenko, A. Merkurjev.
The Algebraic and Geometric Theory of Quadratic Forms.
American Mathematical Society Colloquium Publications, 56. American Mathematical Society, Providence, RI, 2008. 435 pp.

Starting March 4-22 and April 8-26, 2013
Graduate Course on Reductive group schemes
Lecturer: P. Gille
Course Notes

Definition of affine group schemes, group actions, representations. Link with Hopf algebras and comodules. Descent, quotients, examples of representable functors (e.g. centralizers, normalizers). Diagonalisable groups and groups of multiplicative type. Grothendieck's theorem of existence of tori locally for Zariski topology, applications. Split subtori, root data, parabolic subgroups, Levi subgroups. Classification of reductive group schemes by cohomology, examples of forms.

Taking the Institute's Courses for Credit
As graduate students at any of the Institute's University Partners, you may discuss the possibility of obtaining a credit for one or more courses in this lecture series with your home university graduate officer and the course instructor. Assigned reading and related projects may be arranged for the benefit of students requiring these courses for credit.

For additional information contact thematic(PUT_AT_SIGN_HERE)fields.utoronto.ca

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