- Philippe Gille,
*University of Lyon* - Zinovy Reichstein,
*University of British Columbia* - Kirill Zainoulline,
*University of Ottawa*

This online seminar is focusing on new results and developments in the theory of quadratic forms, linear algebraic groups and related areas: Galois cohomology, cohomological invariants, torsors, group schemes, algebraic cycles and motives of twisted flag varieties...

**May 8:** Nikita Karpenko (University of Alberta) **Recorded Lecture**

*An ultimate proof of Hoffmann-Totaro's conjecture*

We prove the last open case of the conjecture on the possible values
of the first isotropy index of an anisotropic quadratic form over a
field. It was initially stated by Detlev Hoffmann for fields of
characteristic not 2 and then extended to arbitrary characteristic by
Burt Totaro. The initial statement was proven by the speaker in 2002.
In characteristic 2, the case of a totally singular quadratic form was
done by Stephen Scully in 2015 and the nonsingular case by Eric
Primozic in early 2019.

**May 15:** David Stewart (University of Newcastle) **Recorded Lecture**

*Irreducible modules for pseudo-reductive groups*

(Jt with Michael Bate) For any smooth connected group G over an arbitrary field k, its irreducible modules are in 1-1 correspondence with those of the pseudo-reductive quotient G/R_{u,k}(G) where R_{u,k}(G) is the k-defined unipotent radical of G. If k is imperfect, a pseudo-reductive group may not be reductive. That means that over the algebraic closure of k, one sees some unipotent radical which is not visible over k. If G has a split maximal torus, much of the theory of split reductive groups carries over and we give dimension formulae for irreducible G-modules which reduce the study to the split reductive case and commutative pseudo-reductive case.

**May 22:** Alexander Duncan (University of South Carolina) **Recorded Lecture**

*Cohomological invariants and separable algebras*

A separable algebra over a field k is a finite direct sum of central simple algebras over finite separable extensions of k. It is natural to attach separable algebras to k-forms of algebraic objects. The fundamental example is the central simple algebra corresponding to a Severi-Brauer variety. Blunk considered a pair of Azumaya algebras attached to a del Pezzo surface of degree 6. More generally, one can consider endomorphism algebras of exceptional objects in derived categories. Alternatively, one can view these constructions as cohomological invariants of degree 2 with values in quasitrivial tori.

In the case of Severi-Brauer varieties and Blunk's example of del Pezzo surfaces of degree 6, these invariants suffice to completely determine the isomorphism classes of the underlying objects. However, in general they are not sufficient. We characterize which k-forms can be distinguished from one another using the theory of coflasque resolutions of reductive algebraic groups. Moreover, we discuss connections to rationality questions and to the Tate-Shafarevich group for number fields.
This is based on joint work with Matthew Ballard, Alicia Lamarche, and Patrick McFaddin.

**May 29:** Federico Scavia (University of British Columbia) **Recorded Lecture**

*Codimension two cycles on classifying stacks of algebraic tori*

We give an example of an algebraic torus T such that the group
CH^2(BT)_{tors} is non-trivial. This answers a question of Blinstein and Merkurjev.

**June 5:** Danny Krashen (Rutgers University) **Recorded Lecture**

*Field patching, local-global principles and rationality*

This talk will describe local-global principles for
torsors for algebraic groups over a semiglobal field - that is, a one variable
function field over a complete discretely valued field.

In particular, I will describe recent joint work with Colliot-Thélène,
Harbater, Hartmann, Parimala and Suresh in which we connect this question in
certain cases to questions of R-equivalence for the group, and in some cases
are able to give finiteness results and combinatorial descriptions for the
obstruction to local-global principles.

**June 12:** Roberto Pirisi (KTH Royal Institute of Technology) **Recorded Lecture**

*Brauer groups of moduli of hyperelliptic curves, via cohomological invariants*

Given an algebraic variety X, the Brauer group of X is the group of Azumaya algebras over X, or equivalently the group of Severi-Brauer varieties over X, i.e. fibrations over X which are étale locally isomorphic to a projective space. It was first studied in the case where X is the spectrum of a field by Noether and Brauer, and has since became a central object in algebraic and arithmetic geometry, being for example one of the first obstructions to rationality used to produce counterexamples to Noether's problem of whether given a representation V of a finite group G the quotient V/G is rational. While the Brauer group has been widely studied for schemes, computations at the level of moduli stacks are relatively recent, the most prominent of them being the computations by Antieau and Meier of the Brauer group of the moduli stack of elliptic curves over a variety of bases, including Z, Q, and finite fields.

In a recent joint work with A. Di Lorenzo, we use the theory of cohomological invariants, and its extension to algebraic stacks, to completely describe the Brauer group of the moduli stacks of hyperelliptic curves over fields of characteristic zero, and the prime-to-char(k) part in positive characteristic. It turns out that the (non-trivial part of the) group is generated by cyclic algebras, by an element coming from a map to the classifying stack of étale algebras of degree 2g+2, and when g is odd by the Brauer-Severi fibration induced by taking the quotient of the universal curve by the hyperelliptic involution. This paints a richer picture than in the case of elliptic curves, where all non-trivial elements come from cyclic algebras.

**June 19:** Maike Gruchot (University of Aachen) **Recorded Lecture**

*Variations of G-complete reducibility*

In this talk we discuss variations of Serre's notion of
complete. Let G be reductive algebraic group and K be a reductive
subgroup. First we consider a relative version in the case of a subgroup
of the G which normalizes the identity component K^0 of K. It turns that
such a subgroup is relatively G-completely reducible with respect to K
if and only if its image in the automorphism group of K^0 is completely
reducible. This allows us to generalize a number of fundamental results
from the absolute to the relative setting.
By results of Serre and Bate-Martin-Röhrle, the usual notion of
G-complete reducibility can be re-framed as a property of an action of a
group on the spherical building of the identity component of G. We
discuss that other variations of this notion, such as relative complete
reducibility and σ-complete reducibility which can also be viewed as
special cases of this building-theoretic definition.
This is on joint work with A. Litterick and G. Röhrle.

**June 26:** Burt Totaro (University of California at Los-Angeles) **Recorded Lecture**

*Cohomological invariants in positive characteristic
*

We determine the mod p cohomological invariants
for several affine group schemes G in chararacteristic p.
These are invariants of G-torsors with values
in étale motivic cohomology, or equivalently in
Kato's version of Galois cohomology based on differential forms.
In particular, we find the mod 2 cohomological invariants
for the symmetric groups and the orthogonal groups
in characteristic 2, which Serre computed in characteristic not 2.

**July 3:** Maurice Chayet (ECAM-EPMI) Slides **Recorded Lecture**

*E8- and new class of commutative non-associative algebras with a continuous Pierce Spectrum*

T.A. Springer knew decades ago of the existence of a Group invariant commutative algebra structure on the 3875 dimensional representation of E8. It was recently shown by S. Garibaldi and R. Guralnick that the automorphism group of this unique commutative algebra coincides with E8. However a description of this algebra has been a lingering question, ever since it was noticed by T.A. Springer.

In this talk, based on joint work with Skip Garibaldi, we explain a correspondence which associates to each simple Lie algebra, a commutative non associative unital algebra, and provide an explicit closed form expression for the product. This correspondence encompasses the 3875 invariant algebra for E8 via the addition of a unit. These algebras turn out to be simple and are endowed with a non-degenerate associative bilinear invariant form. Unlike their closet cousins, the Jordan Algebras, these algebras are not power associative and share the unusual property of having the unit interval as part of their Pierce Spectrum.

**July 10:** Ben Williams (University of British Columbia) **Recorded Lecture**

*Algebras requiring many generators
*

A result of Forster says that if R is a noetherian ring of
(Krull) dimension d, then a rank-n projective module over R can be
generated by d+n elements, and results of Chase and Swan imply that this
bound is sharp - there exist examples that cannot be generated by fewer
than d+n elements. We view "projective modules" as forms of the most
trivial kind of non-unital R-algebra, i.e., where the multiplication is
identically 0. We take the results of Forster, Chase and Swan as a
starting point for investigations into forms of other algebras.

Fix a field k and a k-algebra B, not assumed unital or commutative. Let
G denote the automorphism group scheme of B as an algebra. Let U_r
denote the variety of r-tuples of elements that generate B as a
k-algebra. In favourable circumstances, U_r/G is a k-variety,
generalizing the Grassmannian, that classifies forms of the algebra B
equipped with r generators. In addition, as far as A1-invariant
cohomology theories are concerned U_r/G approximates the classifying
stack BG. By measuring the non-injectivity of the map of Chow rings
CH(BG)->CH(U_r/G), we can produce examples of algebras (over a ring R)
requiring many generators, generalizing the example of Chase and Swan. I
will tell a fuller version of this story, with emphasis on the case
where B is a matrix algebra, so that U_r/G classifies Azumaya algebras
with r generators. The majority of the talk concerns joint work with
Uriya First and Zinovy Reichstein, but I will mention some joint work
with Taeuk Nam & Cindy Tan and some independent work of Sebastian Gant.

**September 14:** Raman Parimala (Emory University)

*TBA
*

TBA

**September 21:** Mikhail Borovoi (Tel Aviv University)

*TBA
*

TBA

**September 28:** Alexandre Lourdeaux (University of Lyon)

*TBA
*

TBA

**October 5:** Igor Rapinchuk (Michigan State University)

*TBA
*

TBA