Short talks:
 Talk of Sanghoon Baek:
For a connected group G over an algebraically closed field, the stable rationality of the
classifying variety BG is not known. A natural way of attacking the nonrationality is to
provide an unramified cohomological invariant of G in some degree. In this talk, we shall
discuss the degree three cohomological invariants of split semisimple groups of Dynkin type
D, which essentially come from the invariants of quadratic forms. As a result, we show the
triviality of their unramified cohomology groups. In addition, we present an application to the
invariants for exceptional groups.
 Talk of Rony Bitan:
Let K=F_{q}(C) be the global function field of rational functions over a smooth and projective curve C defined over a finite field F_{q}. The ring of regular functions on CS where S ≠ Ø is any finite set of closed points on C is a Dedekind domain O_{S} of K.
For a semisimple O_{S}group G with a smooth fundamental group F,
we aim to describe both the set of genera of G and its principal genus (the latter if G⊗_{OS} K is isotropic at S) in terms of finite abelian groups depending on O_{S} and F only.
This leads to a necessary and sufficient condition for the Hasse localglobal principle to hold for certain G.
We also use it to express the Tamagawa number τ(G) of a semisimple Kgroup G by the EulerPoincar\'e invariant.
If time permits we shall briefly discuss the classification of all twisted forms, i.e., including the outer ones.
 Talk of Vineeth Chintala:
In this talk we will examine the notion of an embedding a quadratic space in an associative
algebra. Familiar examples of embeddings are given by Composition algebras, Clifford Algebras
and Suslin Matrices.
We’ll look at two applications. First, we’ll study some fundamental properties that every
embedding should satisfy, by linking them to Clifford Algebras. Conversely, one can give an
explicit description of the Clifford Algebra in some cases and compute low dimensional Spin
Groups easily. (Published in Doc. Math. 23, 2018).
At the end, we will look at a generalization of the classical Hurwitz’s theorem that says that
Composition algebras exists only in dimensions 1, 2, 4 and 8.
 Talk of Rostislav Devyatov
Let G/B be a flag variety over C, where G is a simple algebraic group
with a simply laced Dynkin diagram, and B is a Borel subgroup. The
Bruhat decomposition of G defines subvarieties of G/B called Schubert
subvarieties. The codimension 1 Schubert subvarieties are called
Schubert divisors. We study the Chow ring of G/B (it is isomorphic to
the cohomology ring of G/B viewed as a manifold with classical
topology). This ring is generated as an abelian group by the classes
of all Schubert varieties, and is "almost" generated as a ring by the
classes of Schubert divisors. More precisely, an integer multiple of
each element of G/B can be written as a polynomial in Schubert
divisors with integer coefficients. In particular, each product of
Schubert divisors is a linear combination of Schubert varieties with
integer coefficients.
In the first part of my talk I am going to speak about the
coefficients of these linear combinations. In particular, I am going
to explain how to check if a coefficient of such a linear combination
is nonzero and give an idea how to check if such a coefficient equals
1. If there is time left, then in the second part of my talk, I will
say something about an application of my result, namely, how it makes
it possible estimate socalled canonical dimension of flag varieties
and groups over nonalgebraicallyclosed fields.
 Talk of Nicolas Garrel
A crucial feature of the theory of ordered fields, as initiated by Artin and Schreier,
is the correspondance between orderings of a field, signatures of quadratic forms, and
prime ideals of the Witt ring. Many efforts have been made to extend the theory of
signatures to hermitian forms over algebras with involution, notably by Astier and Unger,
but the lack of ring structure for Witt groups of hermitian forms prevents a correspondance
as elegant as for quadratic forms. We will present a natural "mixed" ring structure for such
Witt groups, and show that the recent results on signatures can be intepreted in a very satisfying
way through the study of its prime spectrum.
 Talk of Zur Izhakian
Tropical mathematics is carried out over idempotent semirings, a weak algebraic structure
that on one hand allows descriptions of objects
having a discrete nature, but on the other hand, its lack of additive inverse prevents the
access to basic mathematical notions. These drawbacks are overcome by the use of a
supertropical semiring  a ``cover'' semiring structure having a special distinguished ideal
which plays the role of the zero element in classical mathematics. This semiring structure is
rich enough to permit a systematic development of tropical
algebraic theory, yielding direct analogs to many important results and notions from classical
algebra. As well, it provides a suitable algebraic framework for natural representations of
semigroups, matroids, and simplicial complexes.
 Talk of Vladimir Tkachev
Algebras with a prescribed Peirce spectrum has been the object of extensive investigation in several contexts including automorphisms of finite simple groups (natural algebras of finite groups, axial algebras etc), combinatorics, quasigroups and differential geometry. I will talk about two new remarkable classes of commutative nonassociative algebras, the socalled {isospectral} and {medial} algebras.
Given a finitedimensional commutative maybe nonassociative algebra A over a field K and a semisimple idempotent, its spectrum is the spectrum of the corresponding multiplication operator L(c):x ↦ cx understood as the multiset of all eigenvalues of L(c) (counted with multiplicities). An algebra A is called isospectral if all its idempotents have the same spectrum. The simplest example of an isospectral algebra is the
twodimensional commutative nonassociative (Matsuo type) algebra generated by two distinct idempotents subject to the condition that the product of any idempotents is again an idempotent.
In my talk I will discuss structural theory of isospectral algebras. In particular, it turns out that any generic isospectral algebra is medial, i.e. satisfies the medial magma identity (xy)(zw)=(xz)(yw). We also establish that any such an algebra over K is an isotopic deformation of the commutative associative quotient algebra K[z]/(z^{n}1).
The talk is based on a recent joint project with Yakov Krasnov (BarIlan University)
