Conference "Linear algebraic groups and related structures"

11 Nov 2021, 10:00 → 12 Nov 2021, 19:30 Europe/Moscow

Online

Conference "Linear algebraic groups and related structures"
Date: 70 anniversary of Nikolai Gordeev

November 11 – 12, 2021

The goal of the conference is to discuss recent progress in the structure theory and classification of linear algebraic groups theory with applications to related problems of representation theory, invariant theory, algebraic geometry, finite group theory, asymptotic group theory, word maps, and the like. These subjects were in the focus of interests of Professor Nikolai Gordeev, and the purpose of the conference is to exchange ideas on the recent advances and further development in these fields.

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Speakers Eva Bayer-Fluckiger • EPFL Lausanne Vladimir Chernousov • University of Alberta Corrado De Concini • Sapienza University of Rome Pavel Gvozdevsky • SPbU Andrei Lavrenov • SPbU Alex Lubotzky • Weizmann Institute Alexander Merkuriev • UCLA

Ivan Panin • PDMI RAS Victor Petrov • SPbU Vladimir Popov • MI RAS Sergey Sinchuk • SPbU Anastasia Stavrova • SPbU Andreas Thom • TU Dresden Anatoly Vershik • PDMI RAS Egor Voronetsky • SPbU

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Program Committee Alexander Generalov • SPbU Ivan Panin (chair) • PDMI RAS Boris Plotkin • Hebrew University of Jerusalem Maxim Vsemirnov • PDMI RAS Anatoliy Yakovlev • SPbU Efim Zelmanov • UCSD

Organizing Committee Boris Kunyavskii • Bar-Ilan University Victor Petrov • SPbU and PDMI RAS Eugene Plotkin • Bar-Ilan University Anastasia Stavrova • SPbU Alexey Stepanov • SPbU Nikolai Vavilov (chair) • SPbU

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Local institutions participating in the organization of the event

• St. Petersburg Department of Steklov Mathematical Institute of the Russian Academy of Sciences
• Leonhard Euler International Mathematical Institute in St. Petersburg
• Department of Mathematics and Computer Science of St Petersburg University

Gordeev70-abstracts-new.pdf Gordeev70-schedule-new.pdf

Thursday, 11 November

10:45 → 11:00 Welcome

11:00 → 11:50 A new look at groups generated by involutions

We construct a general theory of finite groups generated by reflections based on the notion of numberings of partially ordered sets. The classical theory of Coxeter groups (for the case of the symmetric group) corresponds to a special choice of an ordered set --- the simplest Young diagram $(n,1)$. From a formal viewpoint, we replace the defining relation $(\sigma_i \cdot \sigma_{i + 1})^3 = Id$, where $\sigma_i, i = 1,2 ... k$, are involutions, by the relation $(\sigma_i \cdot \sigma_{i + 1})^6 = Id$ leaving the commutation condition unchanged: $(\sigma_i \cdot \sigma_j)^2 = Id$ for $|i-j|>1$, with the following extra condition: the group generated by the adjacent involutions $\sigma_i$ and $\sigma_{i+1}$ is a finite product of the groups of orders $2$ and $3$. Such symmetry groups (finite and countable) naturally arise in combinatorics and their classification does not seem hopeless. For example, if the ordered set is a finite Young diagram, then hypothetically we do not go beyond the Coxeter groups. The characteristic of our approach is that groups are considered as special subgroups of symmetric groups. Particularly interesting are infinite such groups and their representations.

Speaker: Anatoly Vershik (PDMI RAS)

12:00 → 12:50 Group varieties and group structures of algebraic groups

The talk is aimed to discuss to what extent the group variety of a connected algebraic group or the group manifold of a connected real Lie group determines its group structure.

Speaker: Vladimir Popov (MI RAS)

13:00 → 13:50 Paving Springer fibers. The Case of E7

In the paper De Concini, C.; Lusztig, G.; Procesi, C. Homology of the zero-set of a nilpotent vector field on a flag manifold. J. Amer. Math. Soc. 1 (1988), no. 1, 15-34, it was proven the so-called Springer fiber $B_n$ for any nilpotent n element in a complex simple Lie algebra $g$ has homological properties that suggest that $B_n$ should have a paving by affine spaces. This last statement was proved to hold in the case in which $g$ is classical but remained open for exceptional groups in types $E_7$ and $E_8$. In a joint project with Maffei we are trying to fill this gap. At this point our efforts has been successful in type $E_7$ and "almost" in type $E_8$ where one is reduced to show it only in one case.
The goal of the talk is to survey the problem and give an idea on how to show our new results.

Speaker: Corrado De Concini (Univ. Roma I)

14:00 → 16:00 Break

16:00 → 16:50 Automorphisms of K3 surfaces and isometries of lattices

We show that every Salem number of degree $4, 6, 8, 12, 14, 16$ and $18$ is the dynamical degree of an automorphism of a complex K3 surface, and give necessary and sufficient conditions for this in degrees $10$ and $18$.

Speaker: Eva Bayer-Fluckiger (EPF Lausanne)

17:00 → 17:50 Stability, non-approximated groups, and high-dimensional expanders

Several well-known open questions, such as: "are all groups sofic or hyperlinear?", have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, some of these versions, showing that there exist finitely presented groups that are  not approximated by U(n) with respect to the Frobenius (=L_2) norm and many other norms. The strategy is via the notion of "stability": Some higher dimensional cohomology vanishing phenomenon is proven to imply stability. Using the Garland method  (a.k.a. high dimensional expanders as quotients of Bruhat-Tits buildings), it is shown that some non-residually-finite groups are stable and hence cannot be approximated. These groups are central extensions of some lattices in p-adic Lie groups (constructed via a p-adic version of a result of Deligne).

All notions will be explained. Based on joint works with M. De Chiffre, L. Glebsky and A. Thom and with I. Oppenheim .

Speaker: Alex Lubotzky (Weizmann Inst.)

18:00 → 18:50 On conjugacy of Cartan subalgebras in extended affine Lie algebras and classification of torsors over Laurent polynomial rings

In the talk we will present results on a problem of conjugacy of Cartan subalgebras for a class of infinite dimensional Lie algebras called extended affine Lie algebras and how this problem intertwinds with the classification of torsors over Laurent polynomial rings. Joint work with P. Gille, E. Neher, A. Pianzola, U. Yahorau.

Speaker: Vladimir Chernousov (Univ. Alberta)

19:15 → 21:15 Informal meeting, greetings, virtual refreshments

Friday, 12 November

09:00 → 09:50 Degree two negligible cohomology of finite groups

Let $G$ be a finite group and let M be an abelian group viewed as a $G$-module with trivial $G$-action. Fix a field $F$. A cohomology class $c$ in $H^n(G,M)$ is called negligible over $F$ if for every field extension $L/F$ and every continuous group homomorphism of the absolute Galois group of $L$ to $G$ the class $c$ belongs to the kernel of the induced homomorphism $H^n(G,M) \to H^n(L,M)$. We determine all negligible cohomology classes $c$ in $H^2(G,M)$.
This is a joint work with M.Gherman.

Speaker: Alexander Merkurjev (UCLA)

09:50 → 10:30 Short break

10:30 → 11:20 Dense images of word maps

Let $w$ be a non-trivial element of the free group. For $\varepsilon >0$, we prove that there exists an integer $N(\varepsilon,w)$ such that $w(G)$ is $\varepsilon$-dense in $G$, where $G$ is a finite simple group or compact Lie group of rank $N(\varepsilon,w)$ endowed with its natural bi-invariant metric. This confirms metric versions of a conjectures by Shalev and Larsen.

Speaker: Andreas Thom (TU Dresden)

11:30 → 12:20 On Grothendieck---Serre conjecture in mixed characteristic.

We prove the conjecture for the group $SL_{1,D}$ in the mixed characteristic smooth case.

Speaker: Ivan Panin (PDMI RAS)

12:30 → 12:55 Isotropy of Tits construction

Tits construction produces a Lie algebra out of a composition algebra and an exceptional Jordan algebra. The type of the result is $F_4$, ${}^2E_6$, $E_7$ or $E_8$ when the composition algebra has dimension 1,2,4 or 8 respectively. Garibaldi and Petersson noted that the Tits index ${}^2E_6^{35}$ cannot occur as a result of Tits construction. Recently Alex Henke proved that the Tits index $E_7^{66}$ is also not possible. We push the analogy further and show that Lie algebras of Tits index $E_8^{133}$ don't lie in the image of the Tits construction. The proof relies on basic facts about symmetric spaces and our joint result with Garibaldi and Semenov about isotropy of groups of type $E_7$ in terms of the Rost invariant. This is a part of a work joint with Simon Rigby.

Speaker: Victor Petrov (SPbU)

13:00 → 13:25 R-equivalence on reductive algebraic groups

We generalize Manin’s notion of R-equivalence for algebraic varieties to schemes and use this generalization to solve the specialization problem for R-equivalence class groups of reductive groups in the equicharacteristic case. The talk is based on the joint work with Philippe Gille https://arxiv.org/abs/2107.01950.

Speaker: Anastasia Stavrova (SPbU)

13:30 → 15:30 Break

15:30 → 15:55 On the A1-fundamental groups of Chevalley groups

Let $k$ be an arbitrary field. The aim of the talk is to give on overview of a recent preprint, in which it is shown that in the linear case ($\Phi=\mathrm{A}_\ell$, $\ell \geq 4$) and even orthogonal case ($\Phi = \mathrm{D}_\ell$, $\ell\geq 7$, $\mathrm{char}(k)\neq 2$) the unstable functor $\mathrm{K}_2(\Phi, R)$ possesses the $\mathbb{A}^1$-invariance property and therefore can be represented in the unstable $\mathbb{A}^1$-homotopy category $\mathcal{H}^{\mathbb{A}^1}_{k}$ as the fundamental group of the simply-connected Chevalley--Demazure group scheme $\mathrm{G}_\mathrm{sc}(\Phi,-)$. This invariance result can be considered as the $\mathrm{K}_2$-analogue of the geometric case of Bass--Quillen conjecture. With our technique we are also able to show for a semilocal regular $k$-algebra $A$ that $\mathrm{K}_2(\Phi, A)$ embeds as a subgroup into the Milnor group $\mathrm{K}^\mathrm{M}_2(\mathrm{Frac}(A))$.

Speaker: Sergey Sinchuk (SPbU)

16:00 → 16:25 Morava K-theory of orthogonal groups

Let $G$ be a Chevalley group, and $A^*$ denote an oriented cohomology theory in the sense of Levine-Morel (e.g., Grothendieck's $K$-theory, Chow group, etc.) Then the ring $A^*(G)$ is an interesting invariant of the group. In the talk I will explain how to compute this ring for the special orthogonal group $G = SO_m$ and the Morava $K$-theory $A^* = K(n)^*$. The talk is based on a joint work with V. Petrov, P. Sechin, and N. Semenov.

Speaker: Andrei Lavrenov (SPbU)

16:30 → 16:55 Bounded reduction of orthogonal matrices over polynomial rings

The talk is based on the author’s paper [1].In the paper [2], Vaserstein showed that for any coefficient ring $C$ of finite Krull dimension and any $r \ge \max(3, \dim C +2)$, an arbitrary matrix from the special linear group over a polynomial ring $g \in SL_r(C[x_1, . . . , x_n])$ can be reduced to the diagonal shape $diag(g’,1)$ where $g’ \in SL_{r−1}(C[x_1, . . . , x_n])$, by a bounded number (namely $n(21n − 79)/2 + 33nr + 4r − 4)$ of elementary operations. He also deduced from this the similar result for the symplectic group.
In the talk we state the similar result recently obtained by the author for the split orthogonal group. That is the last remaining case among the split classical groups. This result can be viewed as the effective version of the early surjective $K_1$-functor stability, proved by Suslin and Kopeiko in [3].
We also discuss the connection of such theorems with the proof of the Kazhdan property (T) for split groups over finitely generated rings.

Speaker: Pavel Gvozdevsky (SPbU)

17:00 → 17:25 Explicit presentation of relative Steinberg groups

Relative Steinberg groups are defined as crossed modules over the absolute Steinberg group with some generators and relations. We find their presentations as abstract groups.

Speaker: Egor Voronetsky (SPbU)

17:30 → 19:30 Free time, greetings, closing