The program will start in January with an intensive introductory
course on algebraic groups, see below for more details. The introductory
course will be followed by two concentration periods:
I. Applications of torsors to infinite dimensional Lie theory;
II. Torsors, motives and cohomological invariants
aligned with the main themes of the program. Each period will feature
several mini-courses and lecture series aimed at young researchers
and students. At the end of each period we plan to have a mini-workshop
with more specialized talks.
We plan to organize two seminar series, one more traditional with
invited speakers (established researchers and postdocs) and a second
one in the form of learning seminars aimed at graduate students
and young researchers.
At the end of the program (June 9-15, 2013) we plan to have a
conference summarizing the activity of the semester.
Intensive
introductory course on algebraic groups
January-March 2013
Graduate course on Algebraic Groups over
arbitrary fields
Lecturers: Vladimir Chernousov and Victor Petrov
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.
Concentration
Periods
I.
February
- March, 2013
Applications of torsors to infinite dimensional
Lie theory
( organized by V. Chernousov, E.
Neher and A. Pianzola)
Graduate
course on Affine and Extended Affine Lie Algebras (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. The course will
run from early February to mid-March.
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.
Mini course 1: Galois cohomology and
descent theory (A. Pianzola)
Review of absolute Galois group of a field. Non-abelian Galois
cohomology. Twisted forms of algebras (case of fields). Faithfully
flat descent. Galois extension of rings. Twisted forms of algebras
(case of rings). Multiloop algebras. Algebraic fundamental group
of Laurent polynomials. Loop and toral torsors. Internal characterization
of multiloop algebras. Acyclicity theorems and applications
to the classification of multiloop algebras.
Mini course 2: Reductive group schemes
(P. Gille)
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.
Learning Seminar
The mini courses will be complemented by a learning seminars,
one on "Conjugacy Theorems", organized by V. Chernousov, P.
Gille and A. Pianzola.
The concentration period will end with:
March 18 - 29, 2013
Workshop on Geometric methods in Lie theory
II.
April-June,
2013
Torsors, motives
and cohomological invariants
(organized by V. Chernousov, A. Merkurjev and K. Zainoulline)
Course: Introduction to quadratic forms
and algebras with involutions (Anne Queguiner-Matthieu).
Basics on quadratic and bilinear forms. Witt ring. Quadratic
forms under field extensions. Generic splitting properties.
Witt index. Classification of quadratic forms in small dimensions.
Basics on algebras with involutions. Clifford algebras.
Mini-Course 3: Introduction to Chow
groups and Chow motives (Stefan Gille).
Chow groups (pull-back, push-forward, homotopy invariance,
localization). Characteristic classes. Basics on the Intersection
theory.
Chow motives. Motives of flag varieties (cellular decomposition,
Bruhat-Tits decomposition). Rost nilpotence and the Krull-Schmidt
Theorem.
Mini-course 4: Local-global principles
in the theory of linear algebraic groups (Julia Hartmann)
In this course, we consider local-global principles for torsors
when the base field is an algebraic function field over a complete
discretely valued field. We compute the obstructions to these
principles with respect to certain other families of overfields.
The results then give insight about the original local-global
map with respect to discrete valuations. The proofs use patching
methods.
Mini course 5: Motives and algebraic
cycles on twisted flag varieties (Kirill Zainoulline)
Motives of twisted flag varieties is an important tool in the
study of splitting properties of torsors, in the geometric theory
of quadratic forms and central simple algebras with involutions.
For instance, motives of Pfister quadrics played a key role
in the proofs of the Milnor conjecture on quadratic forms and
the Bloch-Kato conjecture. Another applications include cohomological
invariants, the theory of canonical and essential dimensions
of linear algebraic groups.
The course will survey some of this research. It will be started
with an introduction to the theory motives of twisted flag varieties
and conclude with a discussion of open problems. We will introduce
and study the discrete motivic invariant of a torsor (the J-invariant),
explain relations to canonical dimensions, K-theory, Chow groups,
algebraic cobordism of twisted flag variaties and linear algebraic
groups.
Mini-course 6: An introduction to the
theory of essential dimension. (Zinovy Reichstein)
The essential dimension of an algebraic object is the minimal
number of independent parameters one needs to define it. This
notion was initially introduced in the context where the objects
in question are finite field extensions. Essential dimension
has since been investigated in several broader contexts, by
a range of techniques, and has been found to have interesting
and surprising connections to many problems in algebra and algebraic
geometry.
The course will survey some of this research. It will be started
with the definition of essential dimension and conclude with
a discussion of open problems.
Learning Seminars
The mini courses will be complemented by two learning seminars,
one on Algebraic cycles and motives of projective homogeneous
varieties organized by N. Semenov and K. Zainoulline and the
second one on Cohomological invariants organized by S. Garibaldi
and K. Zainoulline
Mini lecture series:
During the mini-courses, we plan to have several lecture series
(2 lectures each) on various topics in the direction of the
program aimed at post-docs and researchers:
- Cohomological invariants for exceptional groups by S. Garibaldi
- Patching and a local-global principle by D. Krashen
- Quadratic spaces over local regular rings by I. Panin
- Algebraic cobordism and its applications to quadratic forms
by A. Vishik
May 6-17, 2013
Spring school and Workshop on Torsors,
Motives and Cohomological Invariants
June
10-14, 2013
Conference on Torsors, Nonassociative Algebras and Cohomological Invariants
Summarizing the activity of the semester.
Coxeter Lecture and Distinguished Lectures
TBA
Seminars
Seminars are important occasions for participants and short-term
visitors to meet and exchange ideas. There will be two types of
seminars, a more traditional seminar for established researchers
and a second one for young researchers and participating students.
The first seminar series will feature short-term visitors and
participating postdoctoral fellows and will present finished research.
The second seminar will be rather informal. Its aim is to provide
young researchers a forum to discuss their ongoing research, or
to learn together some more advanced topics.
Postdoctoral Fellows and Program Visitors
Postdoctoral fellowship
applications
We will support a number of Fields postdocs for the duration of
the program, as well as offer support towards a visitors' program,
including visiting Ph.D. students For Postdoctoral Fellowships
click
here
Program Participants requesting support
or office space
All scientific events are open to the mathematical sciences community.
Visitors who are interested in office space or funding
are requested to apply by filling out the application form
(open shortly). Additional support may be available to support
junior US visitors to this program.
Fields scientific programs are devoted to research in the mathematical
sciences, and enhanced graduate and post-doctoral training opportunities.
Part of the mandate of the Institute is to broaden and enlarge
the community, and to encourage the participation of women and
members of visible minority groups in our scientific programs.
For additional information contact thematic(at)fields.utoronto.ca
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