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


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Upcoming Seminars 
April 30, 2013
Tuesday
3:30 p.m. 
Anastasia Stavrova
Steinberg groups associated with isotropic reductive groups 
Past Seminars 
March 7, 2013
Thursday

Uladzimir Yahorau
A cohomological proof of PetersonKac's theorem on conjugacy of Cartan
subalgebras for affine KacMoody Lie algebras
We say that a subalgebra of a Lie algebra is Cartan if it
is addiagonalizable and not properly contained in a larger addiagonalizable
subalgebra. The theorem of Peterson and Kac says that Cartan subalgebras
of symmetrizable KacMoody Lie algebras are conjugate. We will discuss
the proof of this theorem for affine KacMoody Lie algebras. Unlike
the methods of Peterson and Kac, our approach is entirely cohomological
and geometric.
This is a joint project with Vladimir Chernousov, Philippe Gille and
Arturo Pianzola.

March 1, 2013 
Zhihua Chang
Differential Conformal Superalgebras and Their Twisted Forms
In this talk, I will describe the axiomatic definition of a conformal
superalgebra introduced by V. G. Kac and its generalization, a differential
conformal superalgebra, due to V. G. Kac, M. Lau, and A. Pianzola.
This generalization leads to the definition of twisted forms of conformal
superalgebras, which can be classified using certain Hˆ1. Finally,
I will state our recent results about the automorphism groups and
twisted loop conformal superalgebras of the N=1,2,3, small N=4 and
large N=4 conformal superalgebras.

February 15, 2013 
TingYu Lee
The embedding of a twisted root datum into a reductive group
and the
corresponding arithmetic properties
First, I would like to describe the relation between the embedding
of an étale algebra into a central simple algebra and the embedding
of a twisted root datum into a reductive group. In the second part,
I will focus on the embedding of a twisted root datum into a reductive
group over local fields. In this case, the Tits index determines the
existence of the embedding. I would also like to explain the obstruction
to the localglobal principle for the embedding problem and provide
an example when the localglobal principle for the embedding problems
fail.

February 6, 2013 
Anastasia Stavrova
Isotropic reductive groups over rings
Let R be a unital commutative ring. The elementary subgroup E_n(R)
of the general linear group GL_n(R) is the subgroup generated by all
unipotent elementary matrices in GL_n(R). For any isotropic reductive
group scheme G over R, one can define analogs of unipotent elementary
matrices, and the respective elementary subgroup E(R) of the group
of Rpoints G(R). When G is a Chevalley group or R is a field, these
are the socalled elementary root unipotents in G, parametrized by
the roots in the root system of G, that were constructed in the 1950s
and 1960s in the work of C. Chevalley, A. Borel, J. Tits, M. Demazure
and A. Grothendieck. We will discuss the general case along the lines
of the joint papers of the speaker and V. Petrov, and provide some
applications to the study of the nonstable K_1functor, or the Whitehead
group, associated to G.

January 30, 2013 
Jeremy Jacobson
On derived Witt groups (Slides)
The Witt group of a scheme is a globalization to schemes of the Witt
group of a field. It is a part of a cohomology theory for schemes
called the derived Witt groups. After an introduction, we recall two
problems about the derived Witt groupsthe Gersten conjecture and
a finite generation question for arithmetic schemesand then explain
recent progress on them.

January 23, 2013

Changlong Zhong
Formal Group Algebras and Oriented Cohomology of Complete Flags
It is known that oriented cohomology of algebraic varieties generalizes
the notion of the Chow group and the Grothendieck group, and each
oriented cohomology determines a formal group law. On the other hand,
the classical characteristic map provides a combinatorial tool to
study the Chow group (and the Grothendieck group) of complete flags.
In this talk I will introduce the formal group algebras and the characteristic
maps introduced by CalmesPetrovZainoulline. Then I will talk about
applying it to study the gamma filtration of oriented cohomology of
complete flags. This is joint work with J. MalagonLopez and K. Zainoulline.

January 16, 2013

Wanshun Wong
An introduction to Essential Dimension
Informally speaking, essential dimension is the smallest number of
independent parameters needed to describe an algebraic object. In
this talk I will give the definition of essential dimension, and some
examples showing how essential dimension is connected to many problems
in algebra.


