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BEGIN:VEVENT
SUMMARY:Agatha Atkarskaya (Bar Ilan University)
DTSTART;VALUE=DATE-TIME:20200428T100000Z
DTEND;VALUE=DATE-TIME:20200428T120000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20200505T100000Z
DTEND;VALUE=DATE-TIME:20200505T120000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20200512T100000Z
DTEND;VALUE=DATE-TIME:20200512T120000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20200519T100000Z
DTEND;VALUE=DATE-TIME:20200519T120000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20200623T100000Z
DTEND;VALUE=DATE-TIME:20200623T120000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20200914T100000Z
DTEND;VALUE=DATE-TIME:20200914T120000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20200921T100000Z
DTEND;VALUE=DATE-TIME:20200921T120000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20200928T100000Z
DTEND;VALUE=DATE-TIME:20200928T120000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20201005T100000Z
DTEND;VALUE=DATE-TIME:20201005T120000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20201019T100000Z
DTEND;VALUE=DATE-TIME:20201019T120000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20201102T100000Z
DTEND;VALUE=DATE-TIME:20201102T120000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20201110T163000Z
DTEND;VALUE=DATE-TIME:20201110T183000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20201124T163000Z
DTEND;VALUE=DATE-TIME:20201124T183000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20201130T100000Z
DTEND;VALUE=DATE-TIME:20201130T120000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20201208T163000Z
DTEND;VALUE=DATE-TIME:20201208T183000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20201215T163000Z
DTEND;VALUE=DATE-TIME:20201215T183000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20210126T140000Z
DTEND;VALUE=DATE-TIME:20210126T160000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20210205T140000Z
DTEND;VALUE=DATE-TIME:20210205T160000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20210216T141500Z
DTEND;VALUE=DATE-TIME:20210216T161500Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20210223T141500Z
DTEND;VALUE=DATE-TIME:20210223T161500Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20210326T141500Z
DTEND;VALUE=DATE-TIME:20210326T161500Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20210316T141500Z
DTEND;VALUE=DATE-TIME:20210316T161500Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20210402T141500Z
DTEND;VALUE=DATE-TIME:20210402T161500Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20210430T141500Z
DTEND;VALUE=DATE-TIME:20210430T161500Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20210514T141500Z
DTEND;VALUE=DATE-TIME:20210514T161500Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20210423T150000Z
DTEND;VALUE=DATE-TIME:20210423T170000Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20210507T141500Z
DTEND;VALUE=DATE-TIME:20210507T161500Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20210521T141500Z
DTEND;VALUE=DATE-TIME:20210521T161500Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20210528T141500Z
DTEND;VALUE=DATE-TIME:20210528T161500Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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;VALUE=DATE-TIME:20210611T141500Z
DTEND;VALUE=DATE-TIME:20210611T161500Z
DTSTAMP;VALUE=DATE-TIME:20210612T231250Z
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
END:VCALENDAR