# SCIENTIFIC PROGRAMS AND ACTIVITIES

October 18, 2017
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

### SPEAKER ABSTRACTS

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.

Alexander 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.