### Abstracts

**Yuri Bahturin**

*Group gradings on infinite-dimensional algebras*

One of the uses of affine group schemes is to establish connections between
group gradings on finite-dimensional Lie algebras and associative algebras.
In this talk I would like to discuss what can be done if the dimension is
not necessarily finite.

**Mohammad Bardestani**

*Howe-Kirillov's orbit method and faithful representation of finite p-groups*

A recent result of Karpenko and Merkurjev states that the essential dimension
of a p-group G over a field K containing a primitive pth root of unity is
equal to the minimal dimension of faithful representations of G over K.
Motivated by this result, it is then interesting to compute the minimal
dimension of complex faithful representations of a given finite p-group.
In this talk I will explain how Lie algebraic method, namely Howe-Kirillov's
orbit method, can be applied to answer this question. This is a joint work
with Keivan Mallahi-Karai and Hadi Salmasian.

**John Binder**

*Cusp Forms, Fields of Rationality, and Plancherel Equidistribution*

I will discuss two questions. First: given a family of (classical) cusp
forms, how many of them have Fourier coefficients which generate a "small"
number field? Second: given a family of discrete automorphic representations
{pi} of a reductive group G, how are the local components {pi_p} distributed?
I'll explain the answer to the second question, at least for G = GL_2, and
show how this gives an answer to the first question. I'll then discuss what
is known (and what is expected!) regarding question 2 for more general groups,
and, time permitting explain parts of the proof.

**Philippe Gille**

*Parabolic subgroups of reductive group schemes*

We come back to Demazure-Grothendieck's definition of parabolic group schemes
for reductive group schemes (SGA3) and will show how this notion is capital
for the classification of reductive group schemes over an affine base. At
the end we will discuss the special case of Laurent polynomial rings focusing
on the work on collaboration with V. Chernousov and A. Pianzola in relation
with infinite dimensional Lie theory.

**Stefan Gille**

*Milnor-Witt K-Theory of local rings*

Milnor-Witt K-Groups play a prominent role in recent work on the splitting
properties of vector bundles over a smooth affine schemes over a field.
These groups has been introduced (although not so named) by Barge and Morel
some 15 years ago. Morel in collaboration with Hopkins found a presentation
of these Milnor-Witt groups of a field of characteristic not 2. In this
talk I will present a generalization of this result to local rings which
contain an infinite field of characteristic not 2, which has been proven
in collaboration with Stephen Scully and Changlong Zhong.

**Nikita Karpenko**

*Incompressibility of products*

We show that the conjectural criterion of p-incompressibility for products
of projective homogeneous varieties in terms of the factors, previously
known in a few special cases only, holds in general. We identify the properties
of projective homogeneous varieties actually needed for the proof to go
through. For instance, generically split (non-homogeneous) varieties also
satisfy these properties.

**Mikhail Kochetov**

*Affine group schemes and duality between gradings and actions*

We introduce the concept of affine group scheme and its representing object
(a commutative Hopf algebra). Then we explore to what extent the basic results
about groups are valid for affine group schemes. In the end, we give one
application: an extension of duality between gradings and actions over an
algebraically closed field of characteristic zero to the case of an arbitrary
field.

**Jie Sun**

*Universal central extensions of twisted current algebras*

Twisted current algebras are fixed point subalgebras of tensor products
of Lie algebras and associative algebras under finite group actions. Examples
of twisted current algebras include equivariant map algebras and twisted
forms. In this talk, central extensions of twisted current algebras are
constructed and conditions are found under which the construction gives
universal central extensions of twisted current algebras.

**Qiao Zhou**

*Affine Grassmannian, Affine Flag Variety, and Their Global Counterparts*

I would like to introduce the local and global affine Grassmannian and
affine flag variety for a reductive algebraic group $G$. Then I will discuss
the relations between the geometry of certain objects in the Iwahori orbits
in the affine Grassmannian and some representation-theoretic data. Moreover,
I will discuss some results related to a degeneration process from a trivial
flag variety bundle on the affine Grassmannian to the affine flag variety.

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