BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Agatha Atkarskaya (Bar Ilan University)
DTSTART:20200428T100000Z
DTEND:20200428T120000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/1
DESCRIPTION:Title: Group-like small cancellation theory for rings (joint wo
rk with A.Kanel-Belov\, E.Plotkin\, E.Rips)\nby Agatha Atkarskaya (Bar
Ilan University) as part of St. Petersburg algebraic groups seminar\n\nLe
cture held in Zoom 675-315-555.\n\nAbstract\nIt is well known that small c
ancellation groups play a crucial role in the solution of long-lasting pro
blems. Namely Burnside problem\, Tarski monster problem and so on. In the
talk I will present a construction of a similar object for associative rin
gs\, that is a small cancellation associative ring. I am going to give a b
rief overview 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 finite rank\, and $k$ be a field. Let $I$ be an ideal of $kF$ genera
ted as an ideal by a set of generators $R$. We impose special conditions o
n $R$ that are similar to small cancellation conditions for groups. We stu
dy the quotient algebra $kF / I$. We prove that $kF / I$ is non-trivial an
d explicitly construct its linear basis. Moreover\, we show that the ideal
membership problem for the ideal $I$ is solvable.\n \nIt is well-known th
at finitely presented small cancellation groups are word-hyperbolic. So\,
our work is an attempt to express an idea of negative curvature for rings.
For groups we have a naturally corresponding geometric object\, namely\,
its Cayley graph. For rings we do not have such object\, so\, we are worki
ng using purely combinatorial methods. That is\, the relation to geometry
is only indirect. Nevertheless\, we feel that\nnegative curvature is an im
portant underlying force in our study.\n\nOn the one hand\, our algorithmi
c approach can be considered as an extension of Dehn's algorithm\, which w
e 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
\\"obner Basis. So\, our work is also an extension of this notion for a co
mplicated ordering of monomials.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Smolensky (St. Petersburg State University)
DTSTART:20200505T100000Z
DTEND:20200505T120000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/2
DESCRIPTION:Title: Root subgroup centralizers in Kac—Moody groups\nby
Andrei Smolensky (St. Petersburg State University) as part of St. Petersb
urg algebraic groups seminar\n\nLecture held in Zoom 675-315-555.\n\nAbstr
act\nWhether two root subgroups of a Chevalley group or\, more generally\,
of a Kac—Moody group commute is determined by the configuration of the
corresponding elements of the root system. In the finite case the structur
e of the root subgroups centralizers can be seen on the Dynkin diagram\, w
hile in the hyperbolic case the answer is much more complicated. I will te
ll how one can compute these centralizers (for the straightforward enumera
tion\, which is done for the exceptional groups\, can no longer be applied
). Along 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 a
nd its dimension.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Egor Voronetsky (St. Petersburg State University)
DTSTART:20200512T100000Z
DTEND:20200512T120000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/3
DESCRIPTION:Title: A new approach to centrality of $K_2$\nby Egor Voron
etsky (St. Petersburg State University) as part of St. Petersburg algebrai
c groups seminar\n\nLecture held in Zoom 675-315-555.\n\nAbstract\nIt is k
nown that the Steinberg group $\\mathrm{St}(n\, A)$ is a crossed module ov
er 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 present
ation. In the talk I will tell how the proof works.\n
LOCATION:https://stable.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:20260404T110826Z
UID:SPbAlgebraicGroups/4
DESCRIPTION:Title: Free ${\\mathbb{Q}}$-groups are residually torsion-free
nilpotent\nby Andrei Jaikin-Zapirain (Universidad Autonoma de Madrid
and Instituto 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 $h^n=g$. The
se groups were introduced by G. Baumslag in the sixties under the name of
$\\mathcal{D}$-groups. The free ${\\mathbb{Q}}$-group $F^{{\\mathbb{Q}}}(X
)$ can be constructed from the free group $F(X)$ by applying an infinit
e number of amalgamations over cyclic subgroups. In this talk I will expla
in how to show that the group $F^{{\\mathbb{Q}}}(X)$ is residually torsio
n-free nilpotent. This solves a problem raised by G. Baumslag. A key ingr
edient of our argument is the proof of the L\\"uck approximation in charac
teristic $p$ corresponding to an embedding of a group into a free pro-$p$
group. \n\nSee\nhttp://matematicas.uam.es/~andrei.jaikin/preprints/baumsl
ag.pdf\nand\nhttp://matematicas.uam.es/~andrei.jaikin/preprints/slidesbaum
slag.pdf\nfor the details.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katrin Tent (University of Muenster)
DTSTART:20200623T100000Z
DTEND:20200623T120000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/5
DESCRIPTION:Title: Defining R and G(R)\nby Katrin Tent (University of M
uenster) as part of St. Petersburg algebraic groups seminar\n\n\nAbstract\
nIn joint work with Segal we use the fact that for Chevalley groups G(R)\n
of rank at least 2 over a ring R the root subgroups are (nearly always)\nt
he double centralizer of a corresponding root element to show for many\nim
portant classes of rings and fields that R and G(R) are\nbi-interpretable.
For such groups it then follows that the group G(R) is\nfinitely axiomati
zable in the appropriate class of groups provided R is\nfinitely axiomatiz
able in the corresponding class of rings. We will also\nmention and explai
n earlier results obtained in joint work with Nies and\nSegal.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Sinchuk (St. Petersburg State University)
DTSTART:20200914T100000Z
DTEND:20200914T120000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/6
DESCRIPTION:Title: A pro-group approach to the centrality of K_2 (joint wor
k with A. Lavrenov and E. Voronetsky)\nby Sergey Sinchuk (St. Petersbu
rg State University) as part of St. Petersburg algebraic groups seminar\n\
n\nAbstract\nThe aim of the talk is to present an overview of the recent p
reprint 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 nov
el pro-group technique introduced by Voronetsky.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anastasia Stavrova (St. Petersburg State University)
DTSTART:20200921T100000Z
DTEND:20200921T120000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/7
DESCRIPTION:Title: Isotropic reductive groups over Laurent polynomials\
nby Anastasia Stavrova (St. Petersburg State University) as part of St. Pe
tersburg algebraic groups seminar\n\n\nAbstract\nLet $k$ be a field of cha
racteristic 0. Let $G$ be a reductive group over the ring of Laurent polyn
omials $R=k[x_1^{\\pm 1}\,...\,x_n^{\\pm 1}]$. We say that $G$ is isotropi
c\, if every semisimple normal subgroup of $G$ contains $\\mathbf{G}_{m\,R
}$. We settle in positive the conjecture\nof V. Chernousov\, P. Gille\, an
d A. Pianzola that $H^1_{Zar}(R\,G)=1$ for isotropic loop reductive groups
\, and we conclude that every isotropic reductive $R$-group is loop reduc
tive\, i.e. contains a maximal $R$-torus. These results are proved in arXi
v:1909.01984.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raimund Preusser (St. Petersburg State University)
DTSTART:20200928T100000Z
DTEND:20200928T120000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/8
DESCRIPTION:Title: The subnormal structure of classical-like groups over co
mmutative rings\nby Raimund Preusser (St. Petersburg State University)
as part 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 t
hat 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 consequen
ce of this result we obtain a sandwich theorem for subnormal subgroups of
odd-dimensional unitary groups.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Sosnilo (St. Petersburg State University)
DTSTART:20201005T100000Z
DTEND:20201005T120000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/9
DESCRIPTION:Title: Comparing Nisnevich descent\, Milnor excision\, and the
pro-cdh excision\nby Vladimir Sosnilo (St. Petersburg State University
) as part of St. Petersburg algebraic groups seminar\n\n\nAbstract\nVoevod
sky introduced the notion of cdh-topology in the 90s for the sake of devel
oping motivic homotopy theory. One can ask whether K-theory satisfies desc
ent with respect to this topology and it turns out to be ultimately relate
d to the Milnor excision property. In the talk we compare the aforemention
ed excision properties for the G-equivariant K-theory where G is a linearl
y reductive group. In the end we'll also try to explain how these excision
results can be used to prove vanishing of the equivariant negative K-theo
ry.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Rigby (Ghent University)
DTSTART:20201019T100000Z
DTEND:20201019T120000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/11
DESCRIPTION:Title: Bi-octonion algebras\, algebraic groups\, and cohomolog
ical invariants\nby Simon Rigby (Ghent University) as part of St. Pete
rsburg algebraic groups seminar\n\n\nAbstract\nA bi-octonion algebra is a
central simple nonassociative algebra that becomes isomorphic over some fi
eld extension to a tensor product of two octonion algebras. We look at var
ious reductive algebraic groups\, quadratic forms\, and higher-degree form
s involved with these algebras and discuss some consequences of their Galo
is cohomology. For instance\, we get a different proof of Rost's Theorem o
n 14-dimensional quadratic forms with trivial Clifford invariant. Finally\
, we classify the cohomological invariants of bi-octonion algebras and giv
e elementary descriptions of all the invariants.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anand Sawant (TIFR)
DTSTART:20201102T100000Z
DTEND:20201102T120000Z
DTSTAMP:20260404T110826Z
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\nAbst
ract\nI will describe the motivic version of Matsumoto’s theorem about c
entral extensions of split\, semisimple\, simply connected algebraic group
s. I will give an overview of the proof and will also describe a topologi
cal approach to the description of central extensions of split reductive g
roups. The talk is based on joint work with Fabien Morel.\n\nAttention! T
his talk will take place in Zoom 384-956-974 (different form the usual one
). The password is the same as usual.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Merkurjev (UCLA)
DTSTART:20201110T163000Z
DTEND:20201110T183000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/13
DESCRIPTION:Title: Classification of special reductive groups\nby Alex
ander Merkurjev (UCLA) as part of St. Petersburg algebraic groups seminar\
n\n\nAbstract\nAn algebraic group $G$ over a field $F$ is called \\emph{sp
ecial} if for every field extension $K/F$\nall $G$-torsors (principal homo
geneous $G$-spaces) over $K$ are trivial. Examples of special groups are\n
special and general linear groups\, symplectic groups. A.~Grothendieck cla
ssified special groups\nover an algebraically closed field. In 2016\, M.~H
uruguen classified special reductive groups over arbitrary fields. We impr
ove the classification given by Huruguen.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Wertheim (UCLA)
DTSTART:20201124T163000Z
DTEND:20201124T183000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/14
DESCRIPTION:Title: Degree One Milnor K-Invariants of Groups of Multiplicat
ive Type\nby Alex Wertheim (UCLA) as part of St. Petersburg algebraic
groups seminar\n\n\nAbstract\nMany important algebraic objects can be view
ed as G-torsors over a field F\, where G is an algebraic group over F. For
example\, there is a natural bijection between F-isomorphism classes of c
entral simple F-algebras of degree n and PGL_n(F)-torsors over Spec(F). Mu
ch as one may study principal bundles on a manifold via characteristic cla
sses\, one may likewise study G-torsors over a field via certain associate
d Galois cohomology classes. This principle is made precise by the notion
of a cohomological invariant\, which was first introduced by Serre. \n\nIn
this talk\, we will determine the cohomological invariants for algebraic
groups of multiplicative type with values in H^{1}(-\, Q/Z(1)). Our main t
echnical analysis will center around a careful examination of mu_n-torsors
over a smooth\, connected\, reductive algebraic group. Along the way\, we
will compute a related group of invariants for smooth\, connected\, reduc
tive groups.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victor Petrov (St. Petersburg State University)
DTSTART:20201130T100000Z
DTEND:20201130T120000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/15
DESCRIPTION:Title: Isotropy of Tits construction\nby Victor Petrov (St
. Petersburg State University) as part of St. Petersburg algebraic groups
seminar\n\n\nAbstract\nTits construction produces a Lie algebra out of a c
omposition algebra and an exceptional Jordan algebra. The type of the resu
lt 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 th
e 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 pos
sible. We push the analogy further and show that Lie algebras of Tits inde
x $E_8^{133}$ don't lie in the image of the Tits construction. The proof r
elies on basic facts about symmetric spaces and our joint result with Gari
baldi and Semenov about isotropy of groups of type $E_7$ in terms of the R
ost invariant. This is a part of a work in progress joint with Simon Rigby
.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ning Guo (Institut de Mathématique d’Orsay)
DTSTART:20201208T163000Z
DTEND:20201208T183000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/16
DESCRIPTION:Title: The Grothendieck--Serre conjecture over valuation rings
\nby Ning Guo (Institut de Mathématique d’Orsay) as part of St. Pet
ersburg algebraic groups seminar\n\n\nAbstract\nWe establish the Grothendi
eck–Serre 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 tri
vial if it is trivial over K. This result is predicted by the original Gro
thendieck–Serre conjecture and the resolution of singularities. The nove
lty of our proof lies in overcoming subtleties brought by general nondiscr
ete valuation rings. By using flasque resolutions and inducting with local
cohomology\, we prove a non-Noetherian counterpart of Colliot-Thélène
– Sansuc’s case of tori. Then\, taking advantage of techniques in alge
braization\, we obtain the passage to the Henselian rank one case. Finally
\, we induct on Levi subgroups and use the integrality of rational points
of anisotropic groups to reduce to the semisimple anisotropic case\, in wh
ich we appeal to properties of parahoric subgroups in Bruhat–Tits theory
to conclude.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeroen Meulewaeter (Ghent University)
DTSTART:20201215T163000Z
DTEND:20201215T183000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/17
DESCRIPTION:Title: Structurable algebras and inner ideals: Moufang sets\,
triangles and hexagons\nby Jeroen Meulewaeter (Ghent University) as pa
rt of St. Petersburg algebraic groups seminar\n\n\nAbstract\nStructurable
algebras are a class of non-associative algebras introduced by Bruce Allis
on\, which includes the class of Jordan algebras.\n In earlier work of Lie
n Boelaert\, Tom De Medts and Anastasia Stavrova on low rank incidence geo
metries related to exceptional groups it became clear that structurable al
gebras play an important role in their description.\nThe natural question
arose to what extent it would be possible to recover those geometries dire
ctly from the structurable algebras and their associated Tits-Kantor-Koech
er Lie algebra (which are Lie algebras of algebraic groups). It turns out
that the notion of an inner ideal is essential. We have been able to recov
er many geometries of rank one and two directly from the algebras in a sur
prisingly direct fashion. More precisely\, we describe the so-called Moufa
ng sets\, triangles and hexagons.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrey Smolensky (St. Petersburg State University)
DTSTART:20210126T140000Z
DTEND:20210126T160000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/18
DESCRIPTION:Title: Root systems of type E_{k\,n}\nby Andrey Smolensky
(St. Petersburg State University) as part of St. Petersburg algebraic grou
ps seminar\n\n\nAbstract\nWe will discuss a construction of root systems o
f type $E_{k\,n}$ (including all finite simply-laced systems). This constr
uction provides a simpler description for many objects related to these ro
ot systems (fundamental weights\, affine roots\, etc.) and explains variou
s observed phenomena (monotonicity of root coefficients\, some of the bran
ching rules). We will discuss the ways one can arrive at this construction
\, in particular\, we will relate it to Manin`s "hyperbolic construction"
of $E_8$.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kuntal Chakraborty (IISER Pune)
DTSTART:20210205T140000Z
DTEND:20210205T160000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/19
DESCRIPTION:Title: NK1 of Bak’s unitary groups over graded rings\nb
y Kuntal Chakraborty (IISER Pune) as part of St. Petersburg algebraic grou
ps seminar\n\n\nAbstract\nIn 1966-67\, A Bak introduced the concept of “
form rings” and “form parameter”\nto give a uniform definition of cl
assical groups. This group is known as Bak’s Unitary\ngroup or general q
uadratic group. In this talk we recall the definition of Bak’s group\nan
d its elementary subgroups. After recalling the notion of Bak’s Unitary
group\, we\nhave deduced the graded Local-Global principle for this group.
The kernel 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 i
t as Bass’s nilpotent unitary $K_1$-group of $R$. We have proved that Ba
ss’s\nnil group has no k-torsion when $kR = R$. Using graded Local-Globa
l principle of\nUnitary group\, we also deduce the analog result for the g
raded rings.\nThis is a joint work with R. Basu.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Waldemar Holubowski (Silesian University of Technology)
DTSTART:20210216T141500Z
DTEND:20210216T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/20
DESCRIPTION:Title: On normal subgroups in infinite dimensional linear grou
ps\nby Waldemar Holubowski (Silesian University of Technology) as part
of St. Petersburg algebraic groups seminar\n\n\nAbstract\nI will give a s
urvey of old and new results on normal structure of subgroups in GL(V) wh
ere V is infinite dimensional vector space and some similar results on Li
e algebras.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Rapinchuk (Michigan State University)
DTSTART:20210223T141500Z
DTEND:20210223T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/21
DESCRIPTION:Title: Abstract homomorphisms of algebraic groups and applicat
ions\nby Igor Rapinchuk (Michigan State University) as part of St. Pet
ersburg algebraic groups seminar\n\n\nAbstract\nI will discuss several res
ults on abstract homomorphisms between the groups of rational points of al
gebraic groups. The main focus will be on a conjecture of Borel and Tits f
ormulated in their landmark 1973 paper. Our results settle this conjecture
in several cases\; the proofs make use of the notion of an algebraic ring
. I will mention several applications to character varieties of finitely g
enerated groups and representations of some non-arithmetic groups.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Lavrenov (St. Petersburg State University)
DTSTART:20210326T141500Z
DTEND:20210326T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/22
DESCRIPTION:Title: Morava motives of projective quadrics\nby Andrei La
vrenov (St. Petersburg State University) as part of St. Petersburg algebra
ic groups seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maneesh Thakur (Indian Statistical Institute\, Bangalore)
DTSTART:20210316T141500Z
DTEND:20210316T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/23
DESCRIPTION:Title: The Albert problem on Cyclicity of Albert division alge
bras\nby Maneesh Thakur (Indian Statistical Institute\, Bangalore) as
part of St. Petersburg algebraic groups seminar\n\n\nAbstract\nIn 1950's A
drian Albert\, inspired perhaps by Wedderburn's cyclicity theorem for degr
ee 3 central division algebras\, raised the question whether every Excepti
onal central simple Jordan algebra (now called an Albert division algebra)
always contains a cubic cyclic subfield. \nThe first progress on this pr
oblem is due to Holger Petersson and Michel Racine. They proved that the q
uestion has an affirmative answer when the base field contains a primitive
cube root of unity. \nRecently\, while attempting a proof of the Tits-
Weiss conjecture for Albert division algebras\, we proved that every Alber
t division algebra has an isotope that is cyclic\, i.e. contains a cubic c
yclic subfield\, with no assumptions on the base field. \nThis result has
interesting consequences for algebraic groups\, as well as Albert algebra
s. We will discuss a few of these in the seminar.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raimund Preusser (St. Petersburg State University)
DTSTART:20210402T141500Z
DTEND:20210402T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/24
DESCRIPTION:Title: Irreducible representations of Leavitt algebras\nby
Raimund Preusser (St. Petersburg State University) as part of St. Petersb
urg algebraic groups seminar\n\n\nAbstract\nI will talk about the paper "I
rreducible representations of Leavitt algebras"\, a joint work with Roozbe
h Hazrat and Alexander Shchegolev. In the paper we investigate representa
tions of weighted Leavitt path algebras L(E) defined by so-called represen
tation graphs F. We characterise the representation graphs F that yield ir
reducible representations of L(E). Specialising to weighted graphs E with
one vertex and m loops of weight n\, we obtain irreducible representations
for the celebrated Leavitt algebras L(m\,n).\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Evgeny Plotkin (Bar-Ilan University)
DTSTART:20210430T141500Z
DTEND:20210430T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/25
DESCRIPTION:Title: Logical equivalences of Chevalley and Kac-Moody groups<
/a>\nby Evgeny Plotkin (Bar-Ilan University) as part of St. Petersburg alg
ebraic groups seminar\n\n\nAbstract\nWe will survey a series of recent dev
elopments in the area of first-order descriptions of groups. The goal is t
o illuminate the known results and to pose new problems relevant to logica
l characterizations of Chevalley and Kac-Moody groups. We describe three t
ypes of logical equivalences: geometric similarity\, elementary equivalenc
e and isotipicity. We also dwell on the principal problem of isotipicity o
f finitely generated groups\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Rapinchuk (University of Virginia)
DTSTART:20210514T141500Z
DTEND:20210514T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/26
DESCRIPTION:Title: Groups with bounded generation: old and new\nby And
rei Rapinchuk (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 kno
wn examples of groups with (BG) and their properties. We will then report
on a recent result (joint with P. Corvaja\, J. Ren and U. Zannier) that no
n-virtually abelian anisotropic linear groups (i. e. those consisting enti
rely of semi-simple elements) are not boundedly generated. The proofs rely
on number-theoretic techniques.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Egor Voronetsky (St. Petersburg State University)
DTSTART:20210423T150000Z
DTEND:20210423T170000Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/27
DESCRIPTION:Title: Explicit presentation of relative Steinberg groups\
nby Egor Voronetsky (St. Petersburg State University) as part of St. Peter
sburg algebraic groups seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikolai Vavilov (St. Petersburg State University)
DTSTART:20210507T141500Z
DTEND:20210507T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/28
DESCRIPTION:Title: Exceptional uniform polytopes and conjugacy classes of
the Weyl groups (joint with V. Migrin)\nby Nikolai Vavilov (St. Peters
burg State University) as part of St. Petersburg algebraic groups seminar\
n\n\nAbstract\nThe present talk is mostly of expository nature\, but conta
ins\nalso some new results. We revisit the combinatorial\nstructure of the
semiregular and other uniform Gosset--Elte\npolytopes of exceptional symm
etry types 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\nfaces of these polytopes can be easily regained by using\nthe known de
scription of root subsystems and conjugacy\nclasses of the Weyl groups\, a
nd versions of the familiar\ngraphic means such as Schreier diagrams\, wei
ght diagrams\nor the like.\n In particular\, we calculate cycle indices
for [some of] these\npolytopes.\n As an interesting byproduct\, we noti
ced that the Carter\ndiagrams and Stekolshchik diagrams for cuspidal conju
gacy\nclasses of the Weyl groups are uniformly explained within the\nENHAN
CED Dynkin diagrams introduced by Dynkin and\nMinchenko.\n The present w
ork is part of the Diploma paper of the\nfirst-named author under the supe
rvision of the second-named\nauthor.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philippe Gille (Université Claude Bernard Lyon 1)
DTSTART:20210521T141500Z
DTEND:20210521T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/29
DESCRIPTION:Title: Local triviality for G-torsors\nby Philippe Gille (
Université Claude Bernard Lyon 1) as part of St. Petersburg algebraic gro
ups seminar\n\n\nAbstract\nThis is a report on joint work with Parimala an
d Suresh motivated by local-global principles \nfor function fields of p-a
dic curves. For a torsor E over a smooth projective curve X over the ring
of p-adic integers under a reductive X-group scheme G\, we provide a crit
erion for the local triviality of E with respect to the Zariski topology.\
n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Trost (Bochum University)
DTSTART:20210528T141500Z
DTEND:20210528T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/30
DESCRIPTION:Title: Quantitative aspects of normal generation of ${\\rm SL}
_2(R)$\nby Alexander Trost (Bochum University) as part of St. Petersbu
rg algebraic groups seminar\n\n\nAbstract\nIt has been known by work of Ca
rter-Keller and Tavgen since the 90s that split Chevalley groups $G(\\Phi\
,R)=:G$ defined using rings $R$ of S-algebraic integers and irreducible ro
ot systems $\\Phi$ of rank two are boundedly generated by root elements. W
ork by Kedra-Gal has further shown that if a finite collection of conjugac
y classes generates $G(\\Phi\,R)$\, then it boundedly generates $G(\\Phi\,
R)$. Also\, it was shown in the case of $G={\\rm SL}_n(R)$ for $n\\geq 3$
by Morris that there is a bound (for bounded generation) only depending on
the number of finitely many conjugacy classes (rather than the classes th
emselves) that are taken as a generating set and by Kedra-Libman-Martin th
at 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 \\text
it{strongly bounded.}\n In this talk\, I will explain a method to generali
ze strong boundedness results to other $G(\\Phi\,R)$ for arbitrary rings o
f algebraic integers and all split Chevalley groups groups by using G\\"od
els Compactness theorem together with classical Sandwich Classification Th
eorems of split Chevalley groups. I will demonstrate this method in the ca
se of ${\\rm SL}_2(R)$ for $R$ a ring of S-algebraic integers with infinit
ely many units. I will also\, if time allows\, talk about the existence of
small normally generating subsets of $G(\\Phi\,R)$ and explain how the ex
istence or non-existence of small normally generating sets distinguish ${\
\rm Sp}_4(R)\, G_2(R)$ and ${\\rm SL}_2(R)$ from the other $G(\\Phi\,R)$ i
n regards to strong boundedness.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rostislav Devyatov (Max-Planck Institute for Mathematics)
DTSTART:20210611T141500Z
DTEND:20210611T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/31
DESCRIPTION:Title: Multiplicity-free products of Schubert divisors and an
application to canonical dimension of torsors\nby Rostislav Devyatov (
Max-Planck 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 s
peak about Schubert calculus. Let G/B be a flag variety\, where G is a lin
ear simple algebraic group\, and B is a Borel subgroup. Schubert calculus
studies (in classical terms) multiplication in the cohomology ring of a fl
ag variety over the complex numbers\, or (in more algebraic terms) the Cho
w ring of the flag variety. This ring is generated as a group by the class
es of so-called Schubert varieties (or their Poincare duals\, if we speak
about the 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
generated by the classes of Schubert varieties of codimension 1\, called S
chubert divisors. More precisely\, the subring generated by Schubert divis
ors is a subgroup of finite index. These two facts lead to the following g
eneral question: how to decompose a product of Schubert divisors into a li
near combination of Schubert varieties. In my talk\, I am going to address
(and answer 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
combination equal 0? When does it equal 1?\n\nIn the second part of my tal
k I am going to say how to apply these results to theory of torsors and th
eir canonical dimensions. A torsor of an algebraic group G (over an arbitr
ary field\, here this is important) is a scheme E with an action of G such
that over 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 ca
nonical dimension of a scheme X understood as a scheme is the minimal dime
nsion of a subscheme Y of X such that there exists a rational map from X t
o Y. And the canonical dimension of an algebraic group G understood as a g
roup is the maximum over all field extensions L of the base field of G of
the canonical dimensions of all G_L-torsors. In my talk I am going to expl
ain how to get estimates on canonical dimension of certain groups understo
od as groups using the result from the first part.\n\nAttention! This talk
will not be in Zoom! To attend the talk\, go to https://bbb.mpim-bonn.mp
g.de/b/ros-z2x-mm6 and enter the password communicated by the organizers.\
n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anastasia Stavrova (St. Petersburg State University)
DTSTART:20210709T141500Z
DTEND:20210709T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/32
DESCRIPTION:Title: Non-stable K1-functors\, R-equivalence\, and A1-equival
ence\nby Anastasia Stavrova (St. Petersburg State University) as part
of St. Petersburg algebraic groups seminar\n\n\nAbstract\nI will speak abo
ut the relations between the non-stable $K_1$-functors (also called Whiteh
ead groups)\, the (generalized) R-equivalence class groups\, and $A^1$-equ
ivalence class groups for isotropic reductive groups over regular rings. T
his is part of the joint work with Philippe Gille https://arxiv.org/abs/21
07.01950.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anastasia Stavrova (St. Petersburg State University)
DTSTART:20210913T141500Z
DTEND:20210913T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/33
DESCRIPTION:Title: Non-stable K1-functors\, R-equivalence\, and A1-equival
ence: the case of henselian pairs\nby Anastasia Stavrova (St. Petersbu
rg State University) as part of St. Petersburg algebraic groups seminar\n\
n\nAbstract\nI will speak about the non-stable $K_1$-functors and R-equiva
lence class groups for reductive groups over henselian pairs. In case of a
n equicharacteristic henselian regular local ring $A$\, we show the existe
nce of a specialization isomorphism $G(K)/R\\cong G(k)/R$ between the R-eq
uivalence 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/abs/2107.01950.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Tsvetkov (St. Petersburg State University)
DTSTART:20210920T141500Z
DTEND:20210920T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/34
DESCRIPTION:Title: Unipotent elements in microweight representations of ex
ceptional groups\nby Konstantin Tsvetkov (St. Petersburg State Univers
ity) as part of St. Petersburg algebraic groups seminar\n\n\nAbstract\nI w
ill speak about unipotent elements in Chevalley groups of types E_6 and E_
7 which are analogues of linear transvections. These elements are also cal
led transvections.\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 e
very transvection T(u\, v) belongs to the elementary subgroup E(E_7\, R) a
nd if u is unimodular\, then T(u\, v) belongs to E(E_6\, R). These stateme
nts are analogues of Whitehead lemma and Suslin's normality theorem.\n\nIn
the second part we consider the group of type E_7. Namely\, we define tra
nsvections and prove relations between them.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joshua Ruiter (Michigan State University)
DTSTART:20210927T141500Z
DTEND:20210927T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/35
DESCRIPTION:Title: Abstract homomorphisms of some special unitary groups\nby Joshua Ruiter (Michigan State University) as part of St. Petersburg
algebraic groups seminar\n\n\nAbstract\nA conjecture of Borel and Tits sa
ys that an abstract group homomorphism between the groups of rational poin
ts of algebraic groups (over infinite fields) should admit a "standard des
cription\," which is a factorization as a composition of maps in which one
of those 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 algebraic ring structure on the Zariski-closure of the image of a on
e-dimensional (unipotent) root group. We prove a case of the conjecture fo
r a class of quasi-split special unitary groups by extending this method t
o treat two-dimensional root groups and their associated algebraic rings.\
n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Semenov (St. Petersburg State University)
DTSTART:20211004T141500Z
DTEND:20211004T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/36
DESCRIPTION:Title: Twisted forms of commutative monoid structures on affin
e spaces\nby Andrei Semenov (St. Petersburg State University) as part
of St. Petersburg algebraic groups seminar\n\n\nAbstract\nThe talk is base
d on a joint work with Pavel Gvozdevsky.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roman Isaev (Moscow State University)
DTSTART:20211011T141500Z
DTEND:20211011T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/37
DESCRIPTION:Title: Полная система инвариантов мн
огомерного Кубика Рубика\nby Roman Isaev (Moscow
State University) as part of St. Petersburg algebraic groups seminar\n\n\
nAbstract\nСистемой инвариантов Кубика Рубик
а называют набор величин\, зависящих от п
оложений и ориентаций его кубиков и сохр
аняющихся при вращении слоёв головоломк
и. Инварианты естественным образом возн
икают при исследовании связанных состоя
ний Кубика Рубика. Совокупность всех воз
можных инвариантов образует так называе
мую полную систему. Каждая из орбит\, на к
оторые разбиваются состояния головолом
ки при действии группы поворотов\, харак
теризуется уникальным значением полной
системы. Таким образом\, описание всех ин
вариантов позволяет сформулировать кри
терий связанности состояний. На семинар
е мы рассмотрим естественное обобщение
головоломки на многомерный случай\, опиш
ем полную систему и найдём количество ор
бит\, возникающих при действии группы Ку
бика Рубика.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roman Fedorov (University of Pittsburgh)
DTSTART:20211025T161500Z
DTEND:20211025T181500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/38
DESCRIPTION:Title: On the purity conjecture of Nisnevich for torsors under
reductive group schemes\nby Roman Fedorov (University of Pittsburgh)
as part of St. Petersburg algebraic groups seminar\n\n\nAbstract\nLet R be
a regular 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 id
eal of R (equivalently\, the hypersurface {f=0} is\nregular). Let G be a r
eductive group scheme over R. Under an isotropy\nassumption on G we show t
hat a G-torsor over the localization R_f is\ntrivial\, provided it is rati
onally trivial. (In fact\, the isotropy\nassumption is necessary.)\n\nThe
statement is derived from its abstract version concerning\nNisnevich sheav
es satisfying some properties. Note that if f=1\, then\nwe recover the con
jecture of Grothendieck and Serre (already known for\nregular semilocal ri
ngs containing fields). The proof of Nisnevich\nconjecture follows the sam
e strategy except that one needs an\nadditional statement concerning G-tor
sors defined on the complement of\na subscheme of A^1_R that is etale and
finite over R.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victor Petrov (St. Petersburg State University)
DTSTART:20211018T141500Z
DTEND:20211018T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/39
DESCRIPTION:Title: Geometry of symmetric spaces of types EIII and EVI\
nby Victor Petrov (St. Petersburg State University) as part of St. Petersb
urg algebraic groups seminar\n\n\nAbstract\nWe generalize results of Atsuy
ama and Vinberg on Rozenfeld planes to the case of arbitrary fields of cha
racteristic 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 classificatio
n of Freudenthal triple systems (or\, equivalently\, structurable algebras
of skew dimension 1) and use a recent result of Garibaldi and Gross on mi
nuscule embeddings.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Erhard Neher (University of Ottawa)
DTSTART:20211115T141500Z
DTEND:20211115T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/40
DESCRIPTION:Title: Steinberg groups and Jordan pairs\nby Erhard Neher
(University of Ottawa) as part of St. Petersburg algebraic groups seminar\
n\n\nAbstract\nLinear and unitary Steinberg groups are special cases of St
einberg groups associated with Jordan pairs. I will give an introduction t
o these types of Steinberg groups\, following my recent work with Ottmar L
oos. As an example\, I will describe the universal central extension of th
e projective elementary group of the Jordan pair of 1 x 2 matrices over a
division octonion algebra\, i.e.\, the automorphism group of an octonion p
lane\, a group of absolute type E_6.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Meirav Topol (Shamoon College of Engineering)
DTSTART:20211122T141500Z
DTEND:20211122T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/41
DESCRIPTION:Title: Algebra and geometry meet on the way - fundamental grou
ps\nby Meirav Topol (Shamoon College of Engineering) as part of St. Pe
tersburg algebraic groups seminar\n\n\nAbstract\nI consider an algebraic s
urface X embedded in some projective space and project it onto the project
ive plane CP^2\, using a generic projection\, and get the branch curve S.
The curve S is cuspidal with nodes and branch points\, and it can tell a l
ot about X. I calculate the fundamental group G of the complement of S in
CP^2. Group G does not change when the complex structure of X changes cont
inuously\, and this is the motivation for me to try and classify algebraic
surfaces in the moduli space. Because it is not easy to determine G\, I c
an calculate the fundamental group of the Galois cover of X\, it is a quot
ient of G and is considered also as an invariant of classification. The cu
rve S is usually hard to describe so I use the algorithm of degeneration o
f X to ease calculations for G. At the end of the talk I will present an o
utput of a new computer algorithm\, developed\njointly with Uriel Sinichki
n (TAU\, Israel)\, which gives a presentation of G.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pavel Gvozdevsky (St. Petersburg State University)
DTSTART:20211129T141500Z
DTEND:20211129T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/42
DESCRIPTION:Title: Bounded reduction of orthogonal matrices over polynomia
l 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 coeffi
cients in a small-dimensional ring can be reduced to a smaller matrix by a
bounded number of elementary orthogonal transformations.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tariq Syed (Euler Institute in Saint Petersburg)
DTSTART:20220117T141500Z
DTEND:20220117T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/43
DESCRIPTION:Title: Spin-orbits of unimodular rows\nby Tariq Syed (Eule
r Institute in Saint Petersburg) as part of St. Petersburg algebraic group
s seminar\n\n\nAbstract\nMotivated by cancellation questions for projectiv
e modules\, I will discuss the degree maps for unimodular rows as defined
by Suslin. I will explain some connections to Spin-orbits of unimodular ro
ws and to Hermitian K-theory. Throughout the talk\, I will pose some relat
ed questions which allow for open discussion.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cameron Ruether (University of Ottawa)
DTSTART:20220125T141500Z
DTEND:20220125T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/44
DESCRIPTION:Title: Cohomological Invariants of Half-Spin Groups\nby Ca
meron Ruether (University of Ottawa) as part of St. Petersburg algebraic g
roups seminar\n\n\nAbstract\nCohomological invariants of a linear algebrai
c group are a tool introduced by Serre aimed at studying the first Galois
cohomology set of the group. So called degree three invariants form a grou
p\, and the structure of this group is known in many\, but not all\, cases
. In particular\, linear algebraic groups which are neither simply connect
ed nor adjoint have received less attention. We will discuss a recent comp
utation of these invariant for one such group\, the split half-spin group.
The computation exploits the functoriality of cohomological invariants by
using newly constructed homomorphisms into half-spin. Furthermore\, one c
an ask the same question about non-split half-spin groups. We will discuss
how many of the ingredients for the split computation can be adapted to t
he non-split setting using Galois descent. In particular\, we show how Gal
ois descent is compatible with the new morphisms used in the split case.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikolai Vavilov (St. Petersburg State University)
DTSTART:20220315T141500Z
DTEND:20220315T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/45
DESCRIPTION:Title: Bounded generation of Chevalley groups\, and around (jo
int with Inna Capdeboscq\, Boris Kunyavskii\, Eugene Plotkin)\nby Niko
lai Vavilov (St. Petersburg State University) as part of St. Petersburg al
gebraic groups seminar\n\n\nAbstract\nWe discuss several results on bounde
d elementary generation and\nbounded commutator width for Chevalley groups
over Dedekind\nrings of arithmetic type in positive characteristic. In pa
rticular\,\nChevalley groups of rank \\ge 2 over polynomial rings F_q[t] a
nd\nChevalley 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\neleme
ntarily generated. We sketch several proofs\, and establish\nrather plausi
ble explicit bounds\, which are better than the known\nones even in the nu
mber case. Using these bounds we can also\nproduce sharp bounds of the com
mutator width of these groups.\nWe also mention several applications and p
ossible generalisations.\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Sivatski (St. Petersburg State University)
DTSTART:20220517T141500Z
DTEND:20220517T161500Z
DTSTAMP:20260404T110826Z
UID:SPbAlgebraicGroups/46
DESCRIPTION:Title: Applications of conics in the theory of quadratic forms
and central simple algebras\nby Alexander Sivatski (St. Petersburg St
ate University) as part of St. Petersburg algebraic groups seminar\n\n\nAb
stract\nWe discuss several interrelated problems from the theory of quadra
tic forms and central simple algebras over fields. Their proofs have a com
mon\nfeature\, namely\, one of the main tools in the argument is a conic a
nd its function field.\n\\smallskip\n$\\bullet$ We construct an indecompo
sable division 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 suitabl
e field extension. This problem is related to nonexcellence of $2$-primary
field extensions and the homology\ngroups of the Brauer complex for a tri
quadratic extension\, which are of their own interest\, and will be also d
iscussed.\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 $Q_i$\nov
er a field $k$.\nWe prove that this extension becomes nonexcellent after r
eplacement of the ground field $k$ by a suitable extension $F$\, provided
that $ind (Q_1\\otimes_k Q_2)=4$. This result permits to show that\nthe t
orsion of the Chow group $CH^2(X_1\\times X_2\\times X_3 )$ over $F$ equa
ls $\\Bbb Z/2\\Bbb Z$ 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 extension $L/ F$ 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 construct biquaternion algebras $D_1$ and $D_2$ over a field $k$\nwitho
ut a common 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 quate
rnion algebras $Q_i$ over a field $F$. Assume that $ind(\\alpha)\\le 2$ fo
r any\n$\\alpha$ in the subgroup of ${}_2 Br(F)$ generated by all $Q_i$. W
e give a necessary condition on the level of quadratic forms for trivialit
y of the torsion of\n$CH^2(C_1\\times C_2\\times C_3 )$\, which is equival
ent to existence of a common slot for algebras $Q_i$.\n\n$\\bullet$ Applyi
ng the excellence property of conics\, we prove that for any quadratic for
ms $\\varphi_1$ and $\\varphi_2$ over a field $F$\, and\n$d\\in F^*$\, the
anisotropic part of the form $\\varphi_1\\perp (t^2-d)\\varphi_2$ over $F
(t)$ has a similar type\, i.e. there are forms $\\tau_1$ and\n$\\tau_2$ ov
er $F$ 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 for
m 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 defined over $F$.\n\nThe talk will take place in room 201 of the M&CS
building\n
LOCATION:https://stable.researchseminars.org/talk/SPbAlgebraicGroups/46/
END:VEVENT
END:VCALENDAR