Course-I** Roman Fedorov**

*Introduction to principal bundles*

The course will be devoted to the basics in the theory of principal
bundles and the Grothendieck-Serre conjecture with an emphasis
on the recently found interplay between the conjecture and the
affine Grassmannians.

Here is the program of the course.

1. Reductive algebraic groups, reductive group schemes, and principal
bundles.

2. Principal bundles on curves and their moduli. Relation to loop
groups and affine Grassmannians.

3. Geometric Langlands correspondence. 4. Principal bundles on
families of rational curves parameterized by local schemes.

5. A proof of the geometric case of the conjecture of Grothendieck
and Serre on principal bundles.

6. Local purity conjecture.

Course-II **Kalle Karu**

*Toric varieties and equivariant cohomology*

These lectures are an introduction to combinatorial methods for
computing equivariant cohomology. We construct the ordinary and
intersection cohomologies of spaces such as toric varieties, Grassmannians
and flag varieties.

The three lectures cover roughly the following:

Lecture 1: Localization in equivariant cohomology and construction
of smooth varieties from its moment graph.

Lecture 2: Intersection cohomology of toric varieties using combinatorial
sheaves on fans.

Lecture 3: Intersection cohomology of Flag varieties using sheaves
on moment graphs.

**Alexander Duncan** (Michigan)

Twisted Forms of Toric Varieties

A Severi-Brauer variety is a twisted form of projective space.
I consider twisted forms of toric varieties as a natural generalization
of Severi-Brauer varieties. We associate to each form an element
of a non-abelian second Galois cohomology set which plays the
role of the "Brauer class." In addition, we determine
separable algebras which generalize the central simple algebras
appearing the Severi-Brauer case.

**Stefan Gille** (Alberta)

Geometrically rational surfaces with 0-dimensional motives

We will discuss criteria when the motive of a geometrically rational
surface S over a perfect field is 0-dimensional. It turns out
that this is the case if and only if the Picard group of S over
the algebraic closure is a permutation module and S has a 0-cycle
of degree 1.

**Mathieu Huruguen** (UBC)

Special reductive groups over an arbitrary field

A linear algebraic group G defined over a field k is called
special if every G-torsor over every field extension of k is trivial.
In 1958 Grothendieck classified special groups in the case where
the base field k is algebraically closed. In this talk I will
explain some recent progress towards the classification of special
reductive groups over an arbitrary field. In particular, I will
give the classification of special semisimple groups, special
reductive groups of inner type and special quasisplit reductive
groups over an arbitrary field k.

**Nikita Karpenko** (Alberta)

Incompressibility of Weil transfer of generalized Severi-Brauer
varieties

The result on incompressibility of the Weil transfer of generalized
Severi- Brauer varieties obtained by the speaker in 2009 for the
case of a quadratic separable field extension, is generalized
to p-primary separable field extensions with an arbitrary prime
p. While the old result has been motivated by the study of isotropy
of unitary involutions, its present generalization is motivated
by a recent work of Zinovy Reichstein on essential dimension of
representations of finite groups. The old proof does not seem
to work for the generalization and the new approach gives a simpler
proof of the old result.

**Nicole Lemire** (Western)

Four-dimensional algebraic tori

We investigate the rationality of four-dimensional algebraic
tori and the associated equivariant birational linearisation problem.
We connect these problems to the question of determining when
an algebraic group is (stably) Cayley - that is (stably) equivariantly
birationally isomorphic to its Lie algebra and earlier joint work
on the (stably) Cayley problem with Popov, Reichstein, Borovol
and Kunyavskii.

**Alexander Merkurjev** (UCLA)

Motivic decomposition of of certain group varieties

Let D be a central simple algebra of prime degree over a field
and let G be the algebraic group of norm 1 elements of D. We determine
the motivic decomposition of G.

A**lexander Neshitov** (Steklov Institute/Ottawa)

Invariants of degree 3 and torsion on Chow group of a versal flag.

In this talk we will discuss the connection between degree 3
cohomological invariants of a semisimple split group and the torsion
in the Chow group of codimension two cycles of the corresponding
versal flag variety. The talk is based on the joint project with
Alexander Merkurjev and Kirill Zainoulline

**Julia Pevtsova** (University of Washington)

Supports and tensor ideals for finite group schemes.

The problem of classifying thick subcategories in a given triangulated
category goes back to the seminal work of Devinatz-Hopkins-Smith
in stable homotopy theory and Hopkins, Neeman, and

Thomason in algebraic geometry. I'll discuss how comparing different
theories of supports leads to classification of tensor ideals
in a stable module category of a finite group (scheme). The talk
is based on joint work with E. Friedlander and a work in progress
with D.Benson-S.Iyengar-H.Krause.

**Andrei Rapinchuk** (University of Virginia)

On algebraic groups with the same tori (joint work with V. Chernousov
and I. Rapinchuk)

Let G be an absolutely almost simple simply connected algebraic
group over a field K. One defines the genus gen K(G) to be the
collection of K-isomorphism classes of K-forms G0 of G that have
the same isomorphism classes of maximal K-tori as G. We will discuss
conjectures and some recent results about the size of genK(G)
over finitely generated fields K of good characteristic, focusing
primarily on the following two questions:

(1) When does genK(G) reduce to a single element?, and

(2) When is genK(G) finite?

**Igor Rapinchuk** (Harvard University)

The genus of a division algebra

Let D be a finite-dimensional central division algebra over a
field K. The genus gen(D) is defined to be the collection of Brauer
classes [D'] in Br(K), where D' is a central division K-algebra
having the same maximal subfields as D. I will discuss a finiteness
result for the genus in the case that K is a finitely generated
field and also give some explicit estimations on the size of the
genus in several situations. This is joint work with V.~Chernousov
and A.~Rapinchuk.

**Zinovy Reichstein** (UBC)

An invariant of a representation of a finite group

Let K/k be fields of characteristic zero, G be a finite group
and \rho: G -> GL_n(K) be a linear representation of G whose
character takes values in k. A theorem of Brauer says that if
k contains a primitive e-th root of unity, where e is the exponent
of G, then \rho is defined over k, i.e., \rho has the same character
as some representation \rho' : G -> GL_n(k).

In general we would like to know ``how far" \rho is from
being defined over k. In the case, where \rho is absolutely irreducible,
a partial answer to this question is given by the Schur index
of \rho, which is defined as the smallest degree of a finite field
extension l/k such that \rho is defined over l.

In this talk based on joint work with Nikita Karpenko, I will
discuss a different invariant of \rho, which is based on considering
all (rather than just finite) field extensions l/k such that \rho
is defined over l. I will explain how this new invariant can be
expressed as the canonical dimension of a certain projective k-variety
related to the Schur algebra of \rho. In his talk Nikita will
show how canonical dimensions of these varieties can be computed
in some cases.

**Wanshun Wong** (Ottawa)

Formal group rings of toric varieties

For a smooth toric variety X, it is known that the equivariant
Chow ring, the equivariant K_0, and the equivariant cobordism
ring of X are isomorphic respectively to the ring of integral
piecewise polynomial functions, the ring on integral piecewise
exponential functions, and the ring of piecewise power series
with coefficients in the Lazard ring on the fan of X. In this
talk I will present a uniform construction of rings associated
to different formal group laws, such that specializing to the
additive, multiplicative, and universal formal group laws will
recover the above cases. This talk is based on joint work with
Kirill Zainoulline.

**Uladzimir Yahorau** (Alberta)

Conjugacy theorem for extended affine Lie algebras

An extended affine Lie algebra (EALA) is a generalization of
an affine Kac-Moody Lie algebra to higher nullity (in a sense
that can be made precise). It is a pair consisting of a Lie algebra
and its maximal adjoint-diagonalizable subalgebra (MAD), satisfying
certain axioms. It is natural to ask if a given Lie algebra admits
a unique structure of an extended affine Lie algebra, i.e. if
two MADs which are parts of two different structures are conjugate.
In a joint work with V. Chernousov, E. Neher and A. Pianzola we
proved that if the centreless core of an EALA (E,H) is a module
of finite type over its centroid then such MADs are conjugate,
thereby obtaining a positive answer to this question.

In this talk I will give the definition and construction of an
EALA. I will then discuss the proof of the conjugacy theorem for
EALAs.

**Changlong Zhong** (Alberta)

Equivariant cohomology and formal Demazure algebra.

In this talk I will introduce the construction of formal Demazure
algebra, which is considered as algebraic version of the T-equivariant
algebraic oriented cohomology of flag varieties. This is a joint
project with Baptiste Calmes and Kirill Zainoulline

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