Mini Course 2019


  1. Course of Nikita Karpenko:

    In the study of generic objects related to an algebraic group G, an important role is played by the classifying space of G. The Grothendieck ring of the classifying space is the representation ring R(G).
    We introduce and study the Chow filtration on R(G), which is an analogue of the coniveau filtration on the Grothendieck ring of a smooth variety. We compare it with two other filtrations on R(G) (the Chern filtration, also known under the name of gamma-filtration, and the augmentation filtration) and show that all three define the same topology on R(G).
    As an example and an application, for any n ≥ 1, we compute the Chow filtration on R(G) for the special orthogonal group G:=O+(2n+1). In particular, we show that the graded group associated with the filtration is torsion-free. On the other hand, the Chow ring of the classifying space of G over any field of characteristic ≠ 2 is known to contain non-zero torsion elements. As a consequence, any sufficiently good approximation of the classifying space yields an example of a smooth quasi-projective variety X such that its Chow ring is generated by Chern classes and at the same time contains non-zero elements vanishing under the canonical homomorphism onto the graded ring associated with the coniveau filtration on the Grothendieck ring of X. We relate the examples thus obtained with a conjecture on the structure of the Chow ring of a generic flag variety of a semisimple algebraic group.

  2. Course of Manfred Knebusch:

    An account of a new theory of quadratic forms over semirings, in particular supertropical semirings, developed by Zur Izhakian, Louis Rowen and me. The main sources are the following joint papers (I=Izhakian, K=me, R=Rowen).

    [1]=[IKR] Supertropical quadratic forms I, J. Pure Appl. Algebra 220 (2013), 61-93.

    [2]=[IKR] Minimal orderings and quadratic forms on a free module over a supertropical semiring, Linear Alg. Appl. 507 (2016), 420-461.

    [3]=[IKR] Quadratic and symmetric bilinear forms on modules with unique base over a semiring, Documenta Math. 21 (2016), 773-808.

    [4]=[IK] Basic operations on supertropical quadratic forms, Documenta Math. 22 (2017), 1661-1707.

    [5]=[IK] Quasilinear convexity and quasilinear stars in the ray space of a supertropical quadratic form, Linear and Multilin. Algebra, to appear.

    [6]=[IKR] Supertropical quadratic forms II: Tropical trigonometry and applications , Inter. J. of Alg. and Comput. 28 (2018), 1633-1675.

    Furthermore, background material on supertropical valuation theory and supertropical linear algebra:

    [7]=[IKR] Supertropical semirings and supervaluations, J. Pure Appl. Algebra 215(10) (2011), 2431-2463.

    [8]=[IKR] Monoid valuations and value ordered supervaluations, Commun. Alg. 43 (2015), 3207-3248.

    [9]=[IKR] Supertropical monoids: Basics and canonical factorization, J. Pure Appl. Algebra, 217 (2013), 2135-2162.

  3. Course of Sergey Shpectorov:

    Axial algebras are a new class of non-associative algebras related to groups. Examples include Jordan algebras for classical groups and groups of type G2 and F4. Matsuo algebras corresponding to groups of 3-transpositions, and the Griess algebra for the Monster sporadic simple group. The five-lecture course will cover the basics of axial algebras, examples, general structure, and several classification results, both theoretical and computational.

Short talks:

  1. 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 non-rationality 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.

  2. Talk of Rony Bitan:

    Let K=Fq(C) be the global function field of rational functions over a smooth and projective curve C defined over a finite field Fq. The ring of regular functions on C-S where S ≠ Ø is any finite set of closed points on C is a Dedekind domain OS of K. For a semisimple OS-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 GOS K is isotropic at S) in terms of finite abelian groups depending on OS and F only. This leads to a necessary and sufficient condition for the Hasse local-global principle to hold for certain G. We also use it to express the Tamagawa number τ(G) of a semisimple K-group G by the Euler--Poincar\'e invariant. If time permits we shall briefly discuss the classification of all twisted forms, i.e., including the outer ones.

  3. 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.

  4. 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 so-called canonical dimension of flag varieties and groups over non-algebraically-closed fields.

  5. 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.

  6. 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.

  7. 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 so-called {isospectral} and {medial} algebras. Given a finite-dimensional commutative maybe nonassociative algebra A over a field K and a semi-simple 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 two-dimensional 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]/(zn-1).
    The talk is based on a recent joint project with Yakov Krasnov (Bar-Ilan University)