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BEGIN:VEVENT
SUMMARY:Agatha Atkarskaya (Bar Ilan University)
DTSTART:20200428T100000Z
DTEND:20200428T120000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/1
DESCRIPTION:Title: Group-like small cancellation theory for rings (joint work with
A.Kanel-Belov\, E.Plotkin\, E.Rips)\nby Agatha Atkarskaya (Bar Ilan U
niversity) as part of St. Petersburg algebraic groups seminar\n\nLecture h
eld in Zoom 675-315-555.\n\nAbstract\nIt is well known that small cancella
tion groups play a crucial role in the solution of long-lasting problems.
Namely Burnside problem\, Tarski monster problem and so on. In the talk I
will present a construction of a similar object for associative rings\, th
at is a small cancellation associative ring. I am going to give a brief ov
erview of small cancellation groups and then explain how the theory works
for the case of rings.\n\nIn more details\, let $F$ be a free group of a f
inite rank\, and $k$ be a field. Let $I$ be an ideal of $kF$ generated as
an ideal by a set of generators $R$. We impose special conditions on $R$ t
hat are similar to small cancellation conditions for groups. We study the
quotient algebra $kF / I$. We prove that $kF / I$ is non-trivial and expli
citly construct its linear basis. Moreover\, we show that the ideal member
ship problem for the ideal $I$ is solvable.\n \nIt is well-known that fini
tely presented small cancellation groups are word-hyperbolic. So\, our wor
k is an attempt to express an idea of negative curvature for rings. For gr
oups we have a naturally corresponding geometric object\, namely\, its Cay
ley graph. For rings we do not have such object\, so\, we are working usin
g purely combinatorial methods. That is\, the relation to geometry is only
indirect. Nevertheless\, we feel that\nnegative curvature is an important
underlying force in our study.\n\nOn the one hand\, our algorithmic appro
ach can be considered as an extension of Dehn's algorithm\, which we have
in hyperbolic groups. On the other hand\, the circle of ideas that we are
using in our proof has a very clear analogy with the notion of a Gr\\"obne
r Basis. So\, our work is also an extension of this notion for a complicat
ed ordering of monomials.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Smolensky (St. Petersburg State University)
DTSTART:20200505T100000Z
DTEND:20200505T120000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/2
DESCRIPTION:Title: Root subgroup centralizers in Kac—Moody groups\nby Andrei
Smolensky (St. Petersburg State University) as part of St. Petersburg alg
ebraic groups seminar\n\nLecture held in Zoom 675-315-555.\n\nAbstract\nWh
ether two root subgroups of a Chevalley group or\, more generally\, of a K
ac—Moody group commute is determined by the configuration of the corresp
onding elements of the root system. In the finite case the structure of th
e root subgroups centralizers can be seen on the Dynkin diagram\, while in
the hyperbolic case the answer is much more complicated. I will tell how
one can compute these centralizers (for the straightforward enumeration\,
which is done for the exceptional groups\, can no longer be applied). Alon
g the way we will discuss the geometry of the root systems of rank 2 and 3
and why one should distinguish between the rank of a root system and its
dimension.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Egor Voronetsky (St. Petersburg State University)
DTSTART:20200512T100000Z
DTEND:20200512T120000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/3
DESCRIPTION:Title: A new approach to centrality of $K_2$\nby Egor Voronetsky (
St. Petersburg State University) as part of St. Petersburg algebraic group
s seminar\n\nLecture held in Zoom 675-315-555.\n\nAbstract\nIt is known th
at the Steinberg group $\\mathrm{St}(n\, A)$ is a crossed module over the
linear group $\\GL(n\, A)$ for any almost commutative ring $A$ if $n\\ge 4
$. I generalized this in two directions: for rings $A$ satisfying a local
stable rank condition and for isotropic case. The proof uses a new object\
, Steinberg pro-group\, instead of van der Kallen's another presentation.
In the talk I will tell how the proof works.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Jaikin-Zapirain (Universidad Autonoma de Madrid and Institu
to de Ciencias Matematicas)
DTSTART:20200519T100000Z
DTEND:20200519T120000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/4
DESCRIPTION:Title: Free ${\\mathbb{Q}}$-groups are residually torsion-free nilpot
ent\nby Andrei Jaikin-Zapirain (Universidad Autonoma de Madrid and Ins
tituto de Ciencias Matematicas) as part of St. Petersburg algebraic groups
seminar\n\nLecture held in Zoom 675-315-555.\n\nAbstract\nA group $G$ is
called a ${\\mathbb{Q}}$-group if for any $n\\in {\\mathbb{N}}$ and $ g
\\in G$ there exists exactly one $h\\in G$ satisfying $h^n=g$. These grou
ps were introduced by G. Baumslag in the sixties under the name of $\\math
cal{D}$-groups. The free ${\\mathbb{Q}}$-group $F^{{\\mathbb{Q}}}(X)$ ca
n be constructed from the free group $F(X)$ by applying an infinite numbe
r of amalgamations over cyclic subgroups. In this talk I will explain how
to show that the group $F^{{\\mathbb{Q}}}(X)$ is residually torsion-free
nilpotent. This solves a problem raised by G. Baumslag. A key ingredient
of our argument is the proof of the L\\"uck approximation in characteristi
c $p$ corresponding to an embedding of a group into a free pro-$p$ group.
\n\nSee\nhttp://matematicas.uam.es/~andrei.jaikin/preprints/baumslag.pdf\
nand\nhttp://matematicas.uam.es/~andrei.jaikin/preprints/slidesbaumslag.pd
f\nfor the details.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katrin Tent (University of Muenster)
DTSTART:20200623T100000Z
DTEND:20200623T120000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/5
DESCRIPTION:Title: Defining R and G(R)\nby Katrin Tent (University of Muenster
) as part of St. Petersburg algebraic groups seminar\n\n\nAbstract\nIn joi
nt work with Segal we use the fact that for Chevalley groups G(R)\nof rank
at least 2 over a ring R the root subgroups are (nearly always)\nthe doub
le centralizer of a corresponding root element to show for many\nimportant
classes of rings and fields that R and G(R) are\nbi-interpretable. For su
ch groups it then follows that the group G(R) is\nfinitely axiomatizable i
n the appropriate class of groups provided R is\nfinitely axiomatizable in
the corresponding class of rings. We will also\nmention and explain earli
er results obtained in joint work with Nies and\nSegal.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Sinchuk (St. Petersburg State University)
DTSTART:20200914T100000Z
DTEND:20200914T120000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/6
DESCRIPTION:Title: A pro-group approach to the centrality of K_2 (joint work with
A. Lavrenov and E. Voronetsky)\nby Sergey Sinchuk (St. Petersburg Stat
e University) as part of St. Petersburg algebraic groups seminar\n\n\nAbst
ract\nThe aim of the talk is to present an overview of the recent preprint
https://arxiv.org/abs/2009.03999\, where the centrality of K_2 is proved
for all Chevalley groups of rank >= 3. We will discuss the history of the
problem and the motivation behind it. Also we will focus on the novel pro-
group technique introduced by Voronetsky.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anastasia Stavrova (St. Petersburg State University)
DTSTART:20200921T100000Z
DTEND:20200921T120000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/7
DESCRIPTION:Title: Isotropic reductive groups over Laurent polynomials\nby Ana
stasia Stavrova (St. Petersburg State University) as part of St. Petersbur
g algebraic groups seminar\n\n\nAbstract\nLet $k$ be a field of characteri
stic 0. Let $G$ be a reductive group over the ring of Laurent polynomials
$R=k[x_1^{\\pm 1}\,...\,x_n^{\\pm 1}]$. We say that $G$ is isotropic\, if
every semisimple normal subgroup of $G$ contains $\\mathbf{G}_{m\,R}$. We
settle in positive the conjecture\nof V. Chernousov\, P. Gille\, and A. Pi
anzola that $H^1_{Zar}(R\,G)=1$ for isotropic loop reductive groups\, and
we conclude that every isotropic reductive $R$-group is loop reductive\,
i.e. contains a maximal $R$-torus. These results are proved in arXiv:1909.
01984.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raimund Preusser (St. Petersburg State University)
DTSTART:20200928T100000Z
DTEND:20200928T120000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/8
DESCRIPTION:Title: The subnormal structure of classical-like groups over commutati
ve rings\nby Raimund Preusser (St. Petersburg State University) as par
t of St. Petersburg algebraic groups seminar\n\n\nAbstract\nLet n>2 and (R
\,\\Delta) a Hermitian form ring where R is commutative. We prove that if
H is a subgroup of the odd-dimensional unitary group U_{2n+1}(R\,\\Delta)
normalised by a relative elementary subgroup EU_{2n+1}((R\,\\Delta)\,(I\,\
\Omega))\, then there is an odd form ideal (J\,\\Sigma) such that \nEU_{2n
+1}((R\,\\Delta)\,(J\,\\Sigma)*I^{k}) < H < CU_{2n+1}((R\,\\Delta)\,(J\,\\
Sigma))\nwhere k=12 if n=3 respectively k=10 if n>3. As a consequence of t
his result we obtain a sandwich theorem for subnormal subgroups of odd-dim
ensional unitary groups.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Sosnilo (St. Petersburg State University)
DTSTART:20201005T100000Z
DTEND:20201005T120000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/9
DESCRIPTION:Title: Comparing Nisnevich descent\, Milnor excision\, and the pro-cdh
excision\nby Vladimir Sosnilo (St. Petersburg State University) as pa
rt of St. Petersburg algebraic groups seminar\n\n\nAbstract\nVoevodsky int
roduced the notion of cdh-topology in the 90s for the sake of developing m
otivic homotopy theory. One can ask whether K-theory satisfies descent wit
h respect to this topology and it turns out to be ultimately related to th
e Milnor excision property. In the talk we compare the aforementioned exci
sion properties for the G-equivariant K-theory where G is a linearly reduc
tive group. In the end we'll also try to explain how these excision result
s can be used to prove vanishing of the equivariant negative K-theory.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Rigby (Ghent University)
DTSTART:20201019T100000Z
DTEND:20201019T120000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/11
DESCRIPTION:Title: Bi-octonion algebras\, algebraic groups\, and cohomological in
variants\nby Simon Rigby (Ghent University) as part of St. Petersburg
algebraic groups seminar\n\n\nAbstract\nA bi-octonion algebra is a central
simple nonassociative algebra that becomes isomorphic over some field ext
ension to a tensor product of two octonion algebras. We look at various re
ductive algebraic groups\, quadratic forms\, and higher-degree forms invol
ved with these algebras and discuss some consequences of their Galois coho
mology. For instance\, we get a different proof of Rost's Theorem on 14-di
mensional quadratic forms with trivial Clifford invariant. Finally\, we cl
assify the cohomological invariants of bi-octonion algebras and give eleme
ntary descriptions of all the invariants.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anand Sawant (TIFR)
DTSTART:20201102T100000Z
DTEND:20201102T120000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/12
DESCRIPTION:Title: Motivic version of Matsumoto’s theorem\nby Anand Sawant
(TIFR) as part of St. Petersburg algebraic groups seminar\n\n\nAbstract\nI
will describe the motivic version of Matsumoto’s theorem about central
extensions of split\, semisimple\, simply connected algebraic groups. I w
ill give an overview of the proof and will also describe a topological app
roach to the description of central extensions of split reductive groups.
The talk is based on joint work with Fabien Morel.\n\nAttention! This tal
k will take place in Zoom 384-956-974 (different form the usual one). The
password is the same as usual.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Merkurjev (UCLA)
DTSTART:20201110T163000Z
DTEND:20201110T183000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/13
DESCRIPTION:Title: Classification of special reductive groups\nby Alexander M
erkurjev (UCLA) as part of St. Petersburg algebraic groups seminar\n\n\nAb
stract\nAn algebraic group $G$ over a field $F$ is called \\emph{special}
if for every field extension $K/F$\nall $G$-torsors (principal homogeneous
$G$-spaces) over $K$ are trivial. Examples of special groups are\nspecial
and general linear groups\, symplectic groups. A.~Grothendieck classified
special groups\nover an algebraically closed field. In 2016\, M.~Huruguen
classified special reductive groups over arbitrary fields. We improve the
classification given by Huruguen.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Wertheim (UCLA)
DTSTART:20201124T163000Z
DTEND:20201124T183000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/14
DESCRIPTION:Title: Degree One Milnor K-Invariants of Groups of Multiplicative Typ
e\nby Alex Wertheim (UCLA) as part of St. Petersburg algebraic groups
seminar\n\n\nAbstract\nMany important algebraic objects can be viewed as G
-torsors over a field F\, where G is an algebraic group over F. For exampl
e\, there is a natural bijection between F-isomorphism classes of central
simple F-algebras of degree n and PGL_n(F)-torsors over Spec(F). Much as o
ne may study principal bundles on a manifold via characteristic classes\,
one may likewise study G-torsors over a field via certain associated Galoi
s cohomology classes. This principle is made precise by the notion of a co
homological invariant\, which was first introduced by Serre. \n\nIn this t
alk\, we will determine the cohomological invariants for algebraic groups
of multiplicative type with values in H^{1}(-\, Q/Z(1)). Our main technica
l analysis will center around a careful examination of mu_n-torsors over a
smooth\, connected\, reductive algebraic group. Along the way\, we will c
ompute a related group of invariants for smooth\, connected\, reductive gr
oups.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victor Petrov (St. Petersburg State University)
DTSTART:20201130T100000Z
DTEND:20201130T120000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/15
DESCRIPTION:Title: Isotropy of Tits construction\nby Victor Petrov (St. Peter
sburg State University) as part of St. Petersburg algebraic groups seminar
\n\n\nAbstract\nTits construction produces a Lie algebra out of a composit
ion 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 dimensi
on 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. Recent
ly 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 o
n basic facts about symmetric spaces and our joint result with Garibaldi a
nd Semenov about isotropy of groups of type $E_7$ in terms of the Rost inv
ariant. This is a part of a work in progress joint with Simon Rigby.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ning Guo (Institut de Mathématique d’Orsay)
DTSTART:20201208T163000Z
DTEND:20201208T183000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/16
DESCRIPTION:Title: The Grothendieck--Serre conjecture over valuation rings\nb
y Ning Guo (Institut de Mathématique d’Orsay) as part of St. Petersburg
algebraic groups seminar\n\n\nAbstract\nWe establish the Grothendieck–S
erre conjecture over valuation rings: for a reductive group scheme G over
a valuation ring V with fraction field K\, a G-torsor over V is trivial if
it is trivial over K. This result is predicted by the original Grothendie
ck–Serre conjecture and the resolution of singularities. The novelty of
our proof lies in overcoming subtleties brought by general nondiscrete val
uation rings. By using flasque resolutions and inducting with local cohomo
logy\, we prove a non-Noetherian counterpart of Colliot-Thélène– Sansu
c’s case of tori. Then\, taking advantage of techniques in algebraizatio
n\, we obtain the passage to the Henselian rank one case. Finally\, we ind
uct on Levi subgroups and use the integrality of rational points of anisot
ropic groups to reduce to the semisimple anisotropic case\, in which we ap
peal to properties of parahoric subgroups in Bruhat–Tits theory to concl
ude.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeroen Meulewaeter (Ghent University)
DTSTART:20201215T163000Z
DTEND:20201215T183000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/17
DESCRIPTION:Title: Structurable algebras and inner ideals: Moufang sets\, triangl
es and hexagons\nby Jeroen Meulewaeter (Ghent University) as part of S
t. Petersburg algebraic groups seminar\n\n\nAbstract\nStructurable algebra
s are a class of non-associative algebras introduced by Bruce Allison\, wh
ich includes the class of Jordan algebras.\n In earlier work of Lien Boela
ert\, Tom De Medts and Anastasia Stavrova on low rank incidence geometries
related to exceptional groups it became clear that structurable algebras
play an important role in their description.\nThe natural question arose t
o what extent it would be possible to recover those geometries directly fr
om the structurable algebras and their associated Tits-Kantor-Koecher Lie
algebra (which are Lie algebras of algebraic groups). It turns out that th
e notion of an inner ideal is essential. We have been able to recover many
geometries of rank one and two directly from the algebras in a surprising
ly direct fashion. More precisely\, we describe the so-called Moufang sets
\, triangles and hexagons.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrey Smolensky (St. Petersburg State University)
DTSTART:20210126T140000Z
DTEND:20210126T160000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/18
DESCRIPTION:Title: Root systems of type E_{k\,n}\nby Andrey Smolensky (St. Pe
tersburg State University) as part of St. Petersburg algebraic groups semi
nar\n\n\nAbstract\nWe will discuss a construction of root systems of type
$E_{k\,n}$ (including all finite simply-laced systems). This construction
provides a simpler description for many objects related to these root syst
ems (fundamental weights\, affine roots\, etc.) and explains various obser
ved phenomena (monotonicity of root coefficients\, some of the branching r
ules). We will discuss the ways one can arrive at this construction\, in p
articular\, we will relate it to Manin`s "hyperbolic construction" of $E_8
$.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kuntal Chakraborty (IISER Pune)
DTSTART:20210205T140000Z
DTEND:20210205T160000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/19
DESCRIPTION:Title: NK1 of Bak’s unitary groups over graded rings\nby Kunta
l Chakraborty (IISER Pune) as part of St. Petersburg algebraic groups semi
nar\n\n\nAbstract\nIn 1966-67\, A Bak introduced the concept of “form ri
ngs” and “form parameter”\nto give a uniform definition of classical
groups. This group is known as Bak’s Unitary\ngroup or general quadrati
c group. In this talk we recall the definition of Bak’s group\nand its e
lementary subgroups. After recalling the notion of Bak’s Unitary group\,
we\nhave deduced the graded Local-Global principle for this group. The ke
rnel of the\ngroup homomorphism $K_1GQ^λ(R[X]\,Λ[X])\\to K_1GQ^λ(R\,Λ)
$ induced from the form\nring homomorphism $(R[X]\,Λ[X])\\to (R\,Λ)\\ \\
colon X\\mapsto0$ is defined by $NK_1Q^λ(R\,Λ)$. We\noften say it as Bas
s’s nilpotent unitary $K_1$-group of $R$. We have proved that Bass’s\n
nil group has no k-torsion when $kR = R$. Using graded Local-Global princi
ple of\nUnitary group\, we also deduce the analog result for the graded ri
ngs.\nThis is a joint work with R. Basu.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Waldemar Holubowski (Silesian University of Technology)
DTSTART:20210216T141500Z
DTEND:20210216T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/20
DESCRIPTION:Title: On normal subgroups in infinite dimensional linear groups\
nby Waldemar Holubowski (Silesian University of Technology) as part of St.
Petersburg algebraic groups seminar\n\n\nAbstract\nI will give a survey o
f old and new results on normal structure of subgroups in GL(V) where V i
s infinite dimensional vector space and some similar results on Lie algeb
ras.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Rapinchuk (Michigan State University)
DTSTART:20210223T141500Z
DTEND:20210223T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/21
DESCRIPTION:Title: Abstract homomorphisms of algebraic groups and applications\nby Igor Rapinchuk (Michigan State University) as part of St. Petersburg
algebraic groups seminar\n\n\nAbstract\nI will discuss several results on
abstract homomorphisms between the groups of rational points of algebraic
groups. The main focus will be on a conjecture of Borel and Tits formulat
ed in their landmark 1973 paper. Our results settle this conjecture in sev
eral cases\; the proofs make use of the notion of an algebraic ring. I wil
l mention several applications to character varieties of finitely generate
d groups and representations of some non-arithmetic groups.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Lavrenov (St. Petersburg State University)
DTSTART:20210326T141500Z
DTEND:20210326T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/22
DESCRIPTION:Title: Morava motives of projective quadrics\nby Andrei Lavrenov
(St. Petersburg State University) as part of St. Petersburg algebraic grou
ps seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maneesh Thakur (Indian Statistical Institute\, Bangalore)
DTSTART:20210316T141500Z
DTEND:20210316T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/23
DESCRIPTION:Title: The Albert problem on Cyclicity of Albert division algebras\nby Maneesh Thakur (Indian Statistical Institute\, Bangalore) as part of
St. Petersburg algebraic groups seminar\n\n\nAbstract\nIn 1950's Adrian A
lbert\, inspired perhaps by Wedderburn's cyclicity theorem for degree 3 ce
ntral division algebras\, raised the question whether every Exceptional ce
ntral simple Jordan algebra (now called an Albert division algebra) always
contains a cubic cyclic subfield. \nThe first progress on this problem i
s due to Holger Petersson and Michel Racine. They proved that the question
has an affirmative answer when the base field contains a primitive cube r
oot of unity. \nRecently\, while attempting a proof of the Tits-Weiss c
onjecture for Albert division algebras\, we proved that every Albert divis
ion algebra has an isotope that is cyclic\, i.e. contains a cubic cyclic s
ubfield\, with no assumptions on the base field. \nThis result has intere
sting consequences for algebraic groups\, as well as Albert algebras. We w
ill discuss a few of these in the seminar.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raimund Preusser (St. Petersburg State University)
DTSTART:20210402T141500Z
DTEND:20210402T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/24
DESCRIPTION:Title: Irreducible representations of Leavitt algebras\nby Raimun
d Preusser (St. Petersburg State University) as part of St. Petersburg alg
ebraic groups seminar\n\n\nAbstract\nI will talk about the paper "Irreduci
ble representations of Leavitt algebras"\, a joint work with Roozbeh Hazra
t and Alexander Shchegolev. In the paper we investigate representations o
f weighted Leavitt path algebras L(E) defined by so-called representation
graphs F. We characterise the representation graphs F that yield irreducib
le representations of L(E). Specialising to weighted graphs E with one ver
tex and m loops of weight n\, we obtain irreducible representations for th
e celebrated Leavitt algebras L(m\,n).\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Evgeny Plotkin (Bar-Ilan University)
DTSTART:20210430T141500Z
DTEND:20210430T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/25
DESCRIPTION:Title: Logical equivalences of Chevalley and Kac-Moody groups\nby
Evgeny Plotkin (Bar-Ilan University) as part of St. Petersburg algebraic
groups seminar\n\n\nAbstract\nWe will survey a series of recent developmen
ts in the area of first-order descriptions of groups. The goal is to illum
inate the known results and to pose new problems relevant to logical chara
cterizations of Chevalley and Kac-Moody groups. We describe three types of
logical equivalences: geometric similarity\, elementary equivalence and i
sotipicity. We also dwell on the principal problem of isotipicity of finit
ely generated groups\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Rapinchuk (University of Virginia)
DTSTART:20210514T141500Z
DTEND:20210514T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/26
DESCRIPTION:Title: Groups with bounded generation: old and new\nby Andrei Rap
inchuk (University of Virginia) as part of St. Petersburg algebraic groups
seminar\n\n\nAbstract\nA group is said to have bounded generation (BG) if
it is a finite product of cyclic subgroups. We will survey the known exam
ples of groups with (BG) and their properties. We will then report on a re
cent result (joint with P. Corvaja\, J. Ren and U. Zannier) that non-virtu
ally abelian anisotropic linear groups (i. e. those consisting entirely of
semi-simple elements) are not boundedly generated. The proofs rely on num
ber-theoretic techniques.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Egor Voronetsky (St. Petersburg State University)
DTSTART:20210423T150000Z
DTEND:20210423T170000Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/27
DESCRIPTION:Title: Explicit presentation of relative Steinberg groups\nby Ego
r Voronetsky (St. Petersburg State University) as part of St. Petersburg a
lgebraic groups seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikolai Vavilov (St. Petersburg State University)
DTSTART:20210507T141500Z
DTEND:20210507T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/28
DESCRIPTION:Title: Exceptional uniform polytopes and conjugacy classes of the Wey
l groups (joint with V. Migrin)\nby Nikolai Vavilov (St. Petersburg St
ate University) as part of St. Petersburg algebraic groups seminar\n\n\nAb
stract\nThe present talk is mostly of expository nature\, but contains\nal
so some new results. We revisit the combinatorial\nstructure of the semire
gular and other uniform Gosset--Elte\npolytopes of exceptional symmetry ty
pes E_6\, E_7 and E_8.\n We show that the results by Coxeter\, Conway\
, Sloane\,\nMoody and Patera on the types\, number\, and adjacency of\nfac
es of these polytopes can be easily regained by using\nthe known descripti
on of root subsystems and conjugacy\nclasses of the Weyl groups\, and vers
ions of the familiar\ngraphic means such as Schreier diagrams\, weight dia
grams\nor the like.\n In particular\, we calculate cycle indices for [s
ome of] these\npolytopes.\n As an interesting byproduct\, we noticed tha
t the Carter\ndiagrams and Stekolshchik diagrams for cuspidal conjugacy\nc
lasses of the Weyl groups are uniformly explained within the\nENHANCED Dyn
kin diagrams introduced by Dynkin and\nMinchenko.\n The present work is
part of the Diploma paper of the\nfirst-named author under the supervision
of the second-named\nauthor.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philippe Gille (Université Claude Bernard Lyon 1)
DTSTART:20210521T141500Z
DTEND:20210521T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/29
DESCRIPTION:Title: Local triviality for G-torsors\nby Philippe Gille (Univers
ité Claude Bernard Lyon 1) as part of St. Petersburg algebraic groups sem
inar\n\n\nAbstract\nThis is a report on joint work with Parimala and Sures
h motivated by local-global principles \nfor function fields of p-adic cur
ves. For a torsor E over a smooth projective curve X over the ring of p-ad
ic integers under a reductive X-group scheme G\, we provide a criterion f
or the local triviality of E with respect to the Zariski topology.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Trost (Bochum University)
DTSTART:20210528T141500Z
DTEND:20210528T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/30
DESCRIPTION:Title: Quantitative aspects of normal generation of ${\\rm SL}_2(R)$<
/a>\nby Alexander Trost (Bochum University) as part of St. Petersburg alge
braic groups seminar\n\n\nAbstract\nIt has been known by work of Carter-Ke
ller and Tavgen since the 90s that split Chevalley groups $G(\\Phi\,R)=:G$
defined using rings $R$ of S-algebraic integers and irreducible root syst
ems $\\Phi$ of rank two are boundedly generated by root elements. Work by
Kedra-Gal has further shown that if a finite collection of conjugacy class
es generates $G(\\Phi\,R)$\, then it boundedly generates $G(\\Phi\,R)$. Al
so\, it was shown in the case of $G={\\rm SL}_n(R)$ for $n\\geq 3$ by Morr
is that there is a bound (for bounded generation) only depending on the nu
mber of finitely many conjugacy classes (rather than the classes themselve
s) that are taken as a generating set and by Kedra-Libman-Martin that the
bound is actually linear in the number of conjugacy classes\, if $R$ is a
principal ideal domain. A group with this property is called \\textit{stro
ngly bounded.}\n In this talk\, I will explain a method to generalize stro
ng boundedness results to other $G(\\Phi\,R)$ for arbitrary rings of algeb
raic integers and all split Chevalley groups groups by using G\\"odels Com
pactness theorem together with classical Sandwich Classification Theorems
of split Chevalley groups. I will demonstrate this method in the case of $
{\\rm SL}_2(R)$ for $R$ a ring of S-algebraic integers with infinitely man
y units. I will also\, if time allows\, talk about the existence of small
normally generating subsets of $G(\\Phi\,R)$ and explain how the existence
or non-existence of small normally generating sets distinguish ${\\rm Sp}
_4(R)\, G_2(R)$ and ${\\rm SL}_2(R)$ from the other $G(\\Phi\,R)$ in regar
ds to strong boundedness.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rostislav Devyatov (Max-Planck Institute for Mathematics)
DTSTART:20210611T141500Z
DTEND:20210611T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/31
DESCRIPTION:Title: Multiplicity-free products of Schubert divisors and an applica
tion to canonical dimension of torsors\nby Rostislav Devyatov (Max-Pla
nck Institute for Mathematics) as part of St. Petersburg algebraic groups
seminar\n\n\nAbstract\nIn the first part of my talk I am going to speak ab
out Schubert calculus. Let G/B be a flag variety\, where G is a linear sim
ple algebraic group\, and B is a Borel subgroup. Schubert calculus studies
(in classical terms) multiplication in the cohomology ring of a flag vari
ety over the complex numbers\, or (in more algebraic terms) the Chow ring
of the flag variety. This ring is generated as a group by the classes of s
o-called Schubert varieties (or their Poincare duals\, if we speak about t
he classical cohomology ring)\, i. e. of the varieties of the form BwB/B\,
where w is an element of the Weyl group. As a ring\, it is almost generat
ed by the classes of Schubert varieties of codimension 1\, called Schubert
divisors. More precisely\, the subring generated by Schubert divisors is
a subgroup of finite index. These two facts lead to the following general
question: how to decompose a product of Schubert divisors into a linear co
mbination of Schubert varieties. In my talk\, I am going to address (and a
nswer if I have time) two more particular versions of this question: If G
is of type A\, D\, or E\, when does a coefficient in such a linear combina
tion equal 0? When does it equal 1?\n\nIn the second part of my talk I am
going to say how to apply these results to theory of torsors and their can
onical dimensions. A torsor of an algebraic group G (over an arbitrary fie
ld\, here this is important) is a scheme E with an action of G such that o
ver a certain extension of the base field E becomes isomorphic to G\, and
the action becomes the action by left shifts of G on itself. The canonical
dimension of a scheme X understood as a scheme is the minimal dimension o
f a subscheme Y of X such that there exists a rational map from X to Y. An
d the canonical dimension of an algebraic group G understood as a group is
the maximum over all field extensions L of the base field of G of the can
onical dimensions of all G_L-torsors. In my talk I am going to explain how
to get estimates on canonical dimension of certain groups understood as g
roups using the result from the first part.\n\nAttention! This talk will n
ot be in Zoom! To attend the talk\, go to https://bbb.mpim-bonn.mpg.de/b/
ros-z2x-mm6 and enter the password communicated by the organizers.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anastasia Stavrova (St. Petersburg State University)
DTSTART:20210709T141500Z
DTEND:20210709T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/32
DESCRIPTION:Title: Non-stable K1-functors\, R-equivalence\, and A1-equivalence\nby Anastasia Stavrova (St. Petersburg State University) as part of St.
Petersburg algebraic groups seminar\n\n\nAbstract\nI will speak about the
relations between the non-stable $K_1$-functors (also called Whitehead gro
ups)\, the (generalized) R-equivalence class groups\, and $A^1$-equivalenc
e class groups for isotropic reductive groups over regular rings. This is
part of the joint work with Philippe Gille https://arxiv.org/abs/2107.0195
0.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anastasia Stavrova (St. Petersburg State University)
DTSTART:20210913T141500Z
DTEND:20210913T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/33
DESCRIPTION:Title: Non-stable K1-functors\, R-equivalence\, and A1-equivalence: t
he case of henselian pairs\nby Anastasia Stavrova (St. Petersburg Stat
e University) as part of St. Petersburg algebraic groups seminar\n\n\nAbst
ract\nI will speak about the non-stable $K_1$-functors and R-equivalence c
lass groups for reductive groups over henselian pairs. In case of an equic
haracteristic henselian regular local ring $A$\, we show the existence of
a specialization isomorphism $G(K)/R\\cong G(k)/R$ between the R-equivalen
ce class groups of the fraction field $K$ and the residue field $k$ of $A$
. This is a second talk on our joint work with Philippe Gille arxiv.org/ab
s/2107.01950.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Tsvetkov (St. Petersburg State University)
DTSTART:20210920T141500Z
DTEND:20210920T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/34
DESCRIPTION:Title: Unipotent elements in microweight representations of exception
al groups\nby Konstantin Tsvetkov (St. Petersburg State University) as
part of St. Petersburg algebraic groups seminar\n\n\nAbstract\nI will spe
ak about unipotent elements in Chevalley groups of types E_6 and E_7 which
are analogues of linear transvections. These elements are also called tra
nsvections.\n\nIn the first part of the talk we consider the group of type
E_6. We prove relations between transvections. Also we show that every tr
ansvection T(u\, v) belongs to the elementary subgroup E(E_7\, R) and if u
is unimodular\, then T(u\, v) belongs to E(E_6\, R). These statements are
analogues of Whitehead lemma and Suslin's normality theorem.\n\nIn the se
cond part we consider the group of type E_7. Namely\, we define transvecti
ons and prove relations between them.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joshua Ruiter (Michigan State University)
DTSTART:20210927T141500Z
DTEND:20210927T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/35
DESCRIPTION:Title: Abstract homomorphisms of some special unitary groups\nby
Joshua Ruiter (Michigan State University) as part of St. Petersburg algebr
aic groups seminar\n\n\nAbstract\nA conjecture of Borel and Tits says that
an abstract group homomorphism between the groups of rational points of a
lgebraic groups (over infinite fields) should admit a "standard descriptio
n\," which is a factorization as a composition of maps in which one of tho
se maps is regular. For existing results on split groups and groups of the
form SL_{n\,D}\, they key object in analyzing such a homomorphism is an a
lgebraic ring structure on the Zariski-closure of the image of a one-dimen
sional (unipotent) root group. We prove a case of the conjecture for a cla
ss of quasi-split special unitary groups by extending this method to treat
two-dimensional root groups and their associated algebraic rings.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Semenov (St. Petersburg State University)
DTSTART:20211004T141500Z
DTEND:20211004T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/36
DESCRIPTION:Title: Twisted forms of commutative monoid structures on affine space
s\nby Andrei Semenov (St. Petersburg State University) as part of St.
Petersburg algebraic groups seminar\n\n\nAbstract\nThe talk is based on a
joint work with Pavel Gvozdevsky.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roman Isaev (Moscow State University)
DTSTART:20211011T141500Z
DTEND:20211011T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/37
DESCRIPTION:Title: Полная система инвариантов много
мерного Кубика Рубика\nby Roman Isaev (Moscow State
University) as part of St. Petersburg algebraic groups seminar\n\n\nAbstr
act\nСистемой инвариантов Кубика Рубика н
азывают набор величин\, зависящих от пол
ожений и ориентаций его кубиков и сохран
яющихся при вращении слоёв головоломки.
Инварианты естественным образом возник
ают при исследовании связанных состояни
й Кубика Рубика. Совокупность всех возмо
жных инвариантов образует так называему
ю полную систему. Каждая из орбит\, на кот
орые разбиваются состояния головоломки
при действии группы поворотов\, характер
изуется уникальным значением полной сис
темы. Таким образом\, описание всех инвар
иантов позволяет сформулировать критер
ий связанности состояний. На семинаре мы
рассмотрим естественное обобщение голо
воломки на многомерный случай\, опишем п
олную систему и найдём количество орбит\
, возникающих при действии группы Кубика
Рубика.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roman Fedorov (University of Pittsburgh)
DTSTART:20211025T161500Z
DTEND:20211025T181500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/38
DESCRIPTION:Title: On the purity conjecture of Nisnevich for torsors under reduct
ive group schemes\nby Roman Fedorov (University of Pittsburgh) as part
of St. Petersburg algebraic groups seminar\n\n\nAbstract\nLet R be a regu
lar semilocal integral domain containing an infinite\nfield k. Let f be an
element of R that does not belong to the square\nof any maximal ideal of
R (equivalently\, the hypersurface {f=0} is\nregular). Let G be a reductiv
e group scheme over R. Under an isotropy\nassumption on G we show that a G
-torsor over the localization R_f is\ntrivial\, provided it is rationally
trivial. (In fact\, the isotropy\nassumption is necessary.)\n\nThe stateme
nt is derived from its abstract version concerning\nNisnevich sheaves sati
sfying some properties. Note that if f=1\, then\nwe recover the conjecture
of Grothendieck and Serre (already known for\nregular semilocal rings con
taining fields). The proof of Nisnevich\nconjecture follows the same strat
egy except that one needs an\nadditional statement concerning G-torsors de
fined on the complement of\na subscheme of A^1_R that is etale and finite
over R.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victor Petrov (St. Petersburg State University)
DTSTART:20211018T141500Z
DTEND:20211018T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/39
DESCRIPTION:Title: Geometry of symmetric spaces of types EIII and EVI\nby Vic
tor Petrov (St. Petersburg State University) as part of St. Petersburg alg
ebraic groups seminar\n\n\nAbstract\nWe generalize results of Atsuyama and
Vinberg on Rozenfeld planes to the case of arbitrary fields of characteri
stic not 2 and 3. Namely\, we show that two "lines" in EIII (resp. EVI) in
general position meet at 1 (resp. 3) points\, while the variety of common
points of lines in special positions is itself a smaller symmetric space
from Atsuyama's list. We show the connection with the classification of Fr
eudenthal triple systems (or\, equivalently\, structurable algebras of ske
w dimension 1) and use a recent result of Garibaldi and Gross on minuscule
embeddings.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Erhard Neher (University of Ottawa)
DTSTART:20211115T141500Z
DTEND:20211115T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/40
DESCRIPTION:Title: Steinberg groups and Jordan pairs\nby Erhard Neher (Univer
sity of Ottawa) as part of St. Petersburg algebraic groups seminar\n\n\nAb
stract\nLinear and unitary Steinberg groups are special cases of Steinberg
groups associated with Jordan pairs. I will give an introduction to these
types of Steinberg groups\, following my recent work with Ottmar Loos. As
an example\, I will describe the universal central extension of the proje
ctive elementary group of the Jordan pair of 1 x 2 matrices over a divisio
n octonion algebra\, i.e.\, the automorphism group of an octonion plane\,
a group of absolute type E_6.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Meirav Topol (Shamoon College of Engineering)
DTSTART:20211122T141500Z
DTEND:20211122T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/41
DESCRIPTION:Title: Algebra and geometry meet on the way - fundamental groups\
nby Meirav Topol (Shamoon College of Engineering) as part of St. Petersbur
g algebraic groups seminar\n\n\nAbstract\nI consider an algebraic surface
X embedded in some projective space and project it onto the projective pla
ne CP^2\, using a generic projection\, and get the branch curve S. The cur
ve S is cuspidal with nodes and branch points\, and it can tell a lot abou
t X. I calculate the fundamental group G of the complement of S in CP^2. G
roup G does not change when the complex structure of X changes continuousl
y\, and this is the motivation for me to try and classify algebraic surfac
es in the moduli space. Because it is not easy to determine G\, I can calc
ulate the fundamental group of the Galois cover of X\, it is a quotient of
G and is considered also as an invariant of classification. The curve S i
s usually hard to describe so I use the algorithm of degeneration of X to
ease calculations for G. At the end of the talk I will present an output o
f a new computer algorithm\, developed\njointly with Uriel Sinichkin (TAU\
, Israel)\, which gives a presentation of G.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pavel Gvozdevsky (St. Petersburg State University)
DTSTART:20211129T141500Z
DTEND:20211129T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/42
DESCRIPTION:Title: Bounded reduction of orthogonal matrices over polynomial rings
\nby Pavel Gvozdevsky (St. Petersburg State University) as part of St.
Petersburg algebraic groups seminar\n\n\nAbstract\nWe prove that a matrix
from the split orthogonal group over a polynomial ring with coefficients
in a small-dimensional ring can be reduced to a smaller matrix by a bounde
d number of elementary orthogonal transformations.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tariq Syed (Euler Institute in Saint Petersburg)
DTSTART:20220117T141500Z
DTEND:20220117T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/43
DESCRIPTION:Title: Spin-orbits of unimodular rows\nby Tariq Syed (Euler Insti
tute in Saint Petersburg) as part of St. Petersburg algebraic groups semin
ar\n\n\nAbstract\nMotivated by cancellation questions for projective modul
es\, I will discuss the degree maps for unimodular rows as defined by Susl
in. I will explain some connections to Spin-orbits of unimodular rows and
to Hermitian K-theory. Throughout the talk\, I will pose some related ques
tions which allow for open discussion.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cameron Ruether (University of Ottawa)
DTSTART:20220125T141500Z
DTEND:20220125T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/44
DESCRIPTION:Title: Cohomological Invariants of Half-Spin Groups\nby Cameron R
uether (University of Ottawa) as part of St. Petersburg algebraic groups s
eminar\n\n\nAbstract\nCohomological invariants of a linear algebraic group
are a tool introduced by Serre aimed at studying the first Galois cohomol
ogy set of the group. So called degree three invariants form a group\, and
the structure of this group is known in many\, but not all\, cases. In pa
rticular\, linear algebraic groups which are neither simply connected nor
adjoint have received less attention. We will discuss a recent computation
of these invariant for one such group\, the split half-spin group. The co
mputation exploits the functoriality of cohomological invariants by using
newly constructed homomorphisms into half-spin. Furthermore\, one can ask
the same question about non-split half-spin groups. We will discuss how ma
ny of the ingredients for the split computation can be adapted to the non-
split setting using Galois descent. In particular\, we show how Galois des
cent is compatible with the new morphisms used in the split case.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikolai Vavilov (St. Petersburg State University)
DTSTART:20220315T141500Z
DTEND:20220315T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/45
DESCRIPTION:Title: Bounded generation of Chevalley groups\, and around (joint wit
h Inna Capdeboscq\, Boris Kunyavskii\, Eugene Plotkin)\nby Nikolai Vav
ilov (St. Petersburg State University) as part of St. Petersburg algebraic
groups seminar\n\n\nAbstract\nWe discuss several results on bounded eleme
ntary generation and\nbounded commutator width for Chevalley groups over D
edekind\nrings of arithmetic type in positive characteristic. In particula
r\,\nChevalley groups of rank \\ge 2 over polynomial rings F_q[t] and\nChe
valley groups of rank \\ge 1 over Laurent polynomial F_q[t\,t^{-1}]\nrings
\, where F_q is a finite field of q elements\, are boundedly\nelementarily
generated. We sketch several proofs\, and establish\nrather plausible exp
licit bounds\, which are better than the known\nones even in the number ca
se. Using these bounds we can also\nproduce sharp bounds of the commutator
width of these groups.\nWe also mention several applications and possible
generalisations.\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Sivatski (St. Petersburg State University)
DTSTART:20220517T141500Z
DTEND:20220517T161500Z
DTSTAMP:20260404T083422Z
UID:SPbAlgebraicGroups/46
DESCRIPTION:Title: Applications of conics in the theory of quadratic forms and ce
ntral simple algebras\nby Alexander Sivatski (St. Petersburg State Uni
versity) as part of St. Petersburg algebraic groups seminar\n\n\nAbstract\
nWe discuss several interrelated problems from the theory of quadratic for
ms and central simple algebras over fields. Their proofs have a common\nfe
ature\, namely\, one of the main tools in the argument is a conic and its
function field.\n\\smallskip\n$\\bullet$ We construct an indecomposable d
ivision algebra in the relative Brauer group\n${}_2 Br(F)(\\sqrt{a_1}\, \\
sqrt{a_2}\\dots \\sqrt{a_n})/F$\, where $k$ is a field\, $n\\ge 3$\, $a_1\
,\\dots \,a_n\\in k^*$ are given elements\, and\n$F/k$ is a suitable field
extension. This problem is related to nonexcellence of $2$-primary field
extensions and the homology\ngroups of the Brauer complex for a triquadrat
ic extension\, which are of their own interest\, and will be also discusse
d.\n\n$\\bullet$ We consider the extension $k(X_1\\times X_2)/k$\, where $
X_i$ is the conic corresponding to the quaternion algebra $Q_i$\nover a fi
eld $k$.\nWe prove that this extension becomes nonexcellent after replacem
ent of the ground field $k$ by a suitable extension $F$\, provided that $i
nd (Q_1\\otimes_k Q_2)=4$. This result permits to show that\nthe torsion
of the Chow group $CH^2(X_1\\times X_2\\times X_3 )$ over $F$ equals $\\B
bb Z/2\\Bbb Z$ for a suitable conic $X_3$\nover $F$ such that $ind (Q_1\\
otimes_k Q_2\\otimes_k Q_3)=4$.\n This means that there is no such an exte
nsion $L/ F$ of degree $4m$ with an odd $m$ that all three conics ${X_i}_
L$ split.\n\n$\\bullet$ The method in the previous problem permits to cons
truct biquaternion algebras $D_1$ and $D_2$ over a field $k$\nwithout a co
mmon biquadratic splitting field with\n$ind (D_1\\otimes_k D_2)=2$.\n\n$\
\bullet$ Let $C_1\,C_2\,C_3$ be conics with the corresponding quaternion a
lgebras $Q_i$ over a field $F$. Assume that $ind(\\alpha)\\le 2$ for any\n
$\\alpha$ in the subgroup of ${}_2 Br(F)$ generated by all $Q_i$. We give
a necessary condition on the level of quadratic forms for triviality of th
e torsion of\n$CH^2(C_1\\times C_2\\times C_3 )$\, which is equivalent to
existence of a common slot for algebras $Q_i$.\n\n$\\bullet$ Applying the
excellence property of conics\, we prove that for any quadratic forms $\\v
arphi_1$ and $\\varphi_2$ over a field $F$\, and\n$d\\in F^*$\, the anisot
ropic part of the form $\\varphi_1\\perp (t^2-d)\\varphi_2$ over $F(t)$ ha
s a similar type\, i.e. there are forms $\\tau_1$ and\n$\\tau_2$ over $F$
such that $(\\varphi_1\\perp (t^2-d)\\varphi_2)_{an}\\simeq\\tau_1\\perp
(t^2-d)\\tau_2$. Further\, let\n$\\varphi$ be a $4$-dimensional form over
$F$\, $a\,b\\in F^*$.\nUsing the same method\, we give a criterion for the
anisotropic part of the form $\\varphi_{F(\\sqrt a\,\\sqrt b)}$ to be def
ined over $F$.\n\nThe talk will take place in room 201 of the M&CS buildin
g\n
LOCATION:https://researchseminars.org/talk/SPbAlgebraicGroups/46/
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