
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th
ANNIVERSARY
YEAR

January–June
2013
Thematic Program on Torsors,
Nonassociative Algebras and Cohomological Invariants
Graduate Courses
Location:
Stewart Library, Fields Institute

April,
2013 Course Schedule 

April
1012:
Wed & Fri, 10 a.m.12 p.m.
April 1626:
Tues.& Thurs, 13 p.m.

Jan.
10 to Apr 5, 2013
Graduate Course on Algebraic
and Geometric Theory of Quadratic Forms
Lecturer: Nikita Karpenko (Dean's Distinguished Visitor)

April
25:
Tues.& Thurs, 10a.m.12 p.m.

January
14 to April 12, 2013
Graduate course on Algebraic
Groups over arbitrary fields
Lecturers: Vladimir Chernousov and Nikita Semenov

April
15:
Everday, 1 p.m. 3 p.m.
April 811:
Mon.  Thurs,
13 p.m.

March
422, and April 826, 2013
Graduate Course on Reductive
group schemes
Lecturer: Philippe Gille

April
911:
Tues. & Thurs, 10 a.m.12 p.m.
April 1526:
Mon, Wed, Fri, 13 p.m. 
To
be informed of course schedule changes please subscribe to the
Fields mail list for information about
the Thematic
Program on Torsors, Nonassociative Algebras and Cohomological
Invariants.


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 finitedimensional Lie algebras and affine
KacMoody 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 fixedpoint theorem concerning the
action of a solvable group on a quasiprojective 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 midJanuary until the
beginning of March so that students are wellprepared 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. uinvariants.
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 422 and April 826, 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(at)fields.utoronto.ca

