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BEGIN:VEVENT
SUMMARY:David Stewart (University of Newcastle)
DTSTART;VALUE=DATE-TIME:20200515T150000Z
DTEND;VALUE=DATE-TIME:20200515T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/1
DESCRIPTION:Title: Irreducible modules for pseudo-reductive groups\nby David Stew
art (University of Newcastle) as part of Quadratic forms\, linear algebrai
c groups and beyond\n\n\nAbstract\n(Jt with Michael Bate) For any smooth c
onnected group G over an arbitrary field k\, its irreducible modules are i
n 1-1 correspondence with those of the pseudo-reductive quotient G/R_{u\,k
}(G) where R_{u\,k}(G) is the k-defined unipotent radical of G. If k is im
perfect\, a pseudo-reductive group may not be reductive. That means that o
ver the algebraic closure of k\, one sees some unipotent radical which is
not visible over k. If G has a split maximal torus\, much of the theory of
split reductive groups carries over and we give dimension formulae for ir
reducible G-modules which reduce the study to the split reductive case and
commutative pseudo-reductive case.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Duncan (University of South Carolina)
DTSTART;VALUE=DATE-TIME:20200522T150000Z
DTEND;VALUE=DATE-TIME:20200522T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/2
DESCRIPTION:Title: Cohomological invariants and separable algebras\nby Alexander
Duncan (University of South Carolina) as part of Quadratic forms\, linear
algebraic groups and beyond\n\n\nAbstract\nA separable algebra over a fiel
d k is a finite direct sum of central simple algebras over finite separabl
e extensions of k. It is natural to attach separable algebras to k-forms o
f algebraic objects. The fundamental example is the central simple algebra
corresponding to a Severi-Brauer variety. Blunk considered a pair of Azum
aya algebras attached to a del Pezzo surface of degree 6. More generally\,
one can consider endomorphism algebras of exceptional objects in derived
categories. Alternatively\, one can view these constructions as cohomologi
cal invariants of degree 2 with values in quasitrivial tori.\n\nIn the cas
e of Severi-Brauer varieties and Blunk's example of del Pezzo surfaces of
degree 6\, these invariants suffice to completely determine the isomorphis
m classes of the underlying objects. However\, in general they are not suf
ficient. We characterize which k-forms can be distinguished from one anoth
er using the theory of coflasque resolutions of reductive algebraic groups
. Moreover\, we discuss connections to rationality questions and to the Ta
te-Shafarevich group for number fields. \n\nThis is based on joint work wi
th Matthew Ballard\, Alicia Lamarche\, and Patrick McFaddin.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Federico Scavia (University of British Columbia)
DTSTART;VALUE=DATE-TIME:20200529T150000Z
DTEND;VALUE=DATE-TIME:20200529T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/3
DESCRIPTION:Title: Codimension two cycles on classifying stacks of algebraic tori
\nby Federico Scavia (University of British Columbia) as part of Quadratic
forms\, linear algebraic groups and beyond\n\n\nAbstract\nWe give an exam
ple of an algebraic torus $T$ such that the group ${\\rm CH}^2(BT)_{\\rm t
ors}$ is non-trivial. This answers a question of Blinstein and Merkurjev.\
n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Danny Krashen (Rutgers University)
DTSTART;VALUE=DATE-TIME:20200605T150000Z
DTEND;VALUE=DATE-TIME:20200605T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/4
DESCRIPTION:Title: Field patching\, local-global principles and rationality\nby D
anny Krashen (Rutgers University) as part of Quadratic forms\, linear alge
braic groups and beyond\n\n\nAbstract\nThis talk will describe local-globa
l principles for torsors for algebraic groups over a semiglobal field - th
at is\, a one variable function field over a complete discretely valued fi
eld.\nIn particular\, I will describe recent joint work with Colliot-Thél
ène\, Harbater\, Hartmann\, Parimala and Suresh in which we connect this
question in certain cases to questions of R-equivalence for the group\, an
d in some cases are able to give finiteness results and combinatorial desc
riptions for the obstruction to local-global principles.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roberto Pirisi (KTH Royal Institute of Technology)
DTSTART;VALUE=DATE-TIME:20200612T150000Z
DTEND;VALUE=DATE-TIME:20200612T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/5
DESCRIPTION:Title: Brauer groups of moduli of hyperelliptic curves\, via cohomologica
l invariants\nby Roberto Pirisi (KTH Royal Institute of Technology) as
part of Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstrac
t\nGiven an algebraic variety X\, the Brauer group of X is the group of Az
umaya algebras over X\, or equivalently the group of Severi-Brauer varieti
es over X\, i.e. fibrations over X which are étale locally isomorphic to
a projective space. It was first studied in the case where X is the spectr
um of a field by Noether and Brauer\, and has since became a central objec
t in algebraic and arithmetic geometry\, being for example one of the firs
t obstructions to rationality used to produce counterexamples to Noether's
problem of whether given a representation V of a finite group G the quoti
ent V/G is rational. While the Brauer group has been widely studied for sc
hemes\, computations at the level of moduli stacks are relatively recent\,
the most prominent of them being the computations by Antieau and Meier of
the Brauer group of the moduli stack of elliptic curves over a variety of
bases\, including Z\, Q\, and finite fields.\nIn a recent joint work with
A. Di Lorenzo\, we use the theory of cohomological invariants\, and its e
xtension to algebraic stacks\, to completely describe the Brauer group of
the moduli stacks of hyperelliptic curves over fields of characteristic ze
ro\, and the prime-to-char(k) part in positive characteristic. It turns ou
t that the (non-trivial part of the) group is generated by cyclic algebras
\, by an element coming from a map to the classifying stack of étale alge
bras of degree 2g+2\, and when g is odd by the Brauer-Severi fibration ind
uced by taking the quotient of the universal curve by the hyperelliptic in
volution. This paints a richer picture than in the case of elliptic curves
\, where all non-trivial elements come from cyclic algebras.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maike Gruchot (University of Aachen)
DTSTART;VALUE=DATE-TIME:20200619T150000Z
DTEND;VALUE=DATE-TIME:20200619T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/6
DESCRIPTION:Title: Variations of G-complete reducibility\nby Maike Gruchot (Unive
rsity of Aachen) as part of Quadratic forms\, linear algebraic groups and
beyond\n\n\nAbstract\nIn this talk we discuss variations of Serre’s noti
on of complete reducibility. Let $G$ be reductive algebraic group and $K$
be a reductive subgroup. First we consider a relative version in the case
of a subgroup of the $G$ which normalizes the identity component $K^0$ of
$K$. It turns that such a subgroup is relatively $G$-completely reducible
with respect to $K$ if and only if its image in the automorphism group of
$K^0$ is completely reducible. This allows us to generalize a number of fu
ndamental results from the absolute to the relative setting.\nBy results o
f Serre and Bate–Martin–Röhrle\, the usual notion of $G$-complete red
ucibility can be re-framed as a property of an action of a group on the sp
herical building of the identity component of $G$. We discuss that other v
ariations of this notion\, such as relative complete reducibility and σ-c
omplete reducibility which can also be viewed as special cases of this bui
lding-theoretic definition.\nThis is based on joint work with A. Litterick
and G. Röhrle.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Burt Totaro (UCLA)
DTSTART;VALUE=DATE-TIME:20200626T150000Z
DTEND;VALUE=DATE-TIME:20200626T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/7
DESCRIPTION:Title: Cohomological invariants in positive characteristic\nby Burt T
otaro (UCLA) as part of Quadratic forms\, linear algebraic groups and beyo
nd\n\n\nAbstract\nWe determine the mod p cohomological invariants for seve
ral affine group schemes G in chararacteristic p. These are invariants of
G-torsors with values in etale motivic cohomology\, or equivalently in Kat
o's version of Galois cohomology based on differential forms. In particula
r\, we find the mod 2 cohomological invariants for the symmetric groups an
d the orthogonal groups in characteristic 2\, which Serre computed in char
acteristic not 2.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benedict Williams (University of British Columbia)
DTSTART;VALUE=DATE-TIME:20200710T150000Z
DTEND;VALUE=DATE-TIME:20200710T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/8
DESCRIPTION:Title: Algebras requiring many generators\nby Benedict Williams (Univ
ersity of British Columbia) as part of Quadratic forms\, linear algebraic
groups and beyond\n\n\nAbstract\nA result of Forster says that if R is a n
oetherian ring of (Krull) dimension d\, then a rank-n projective module ov
er R can be generated by d+n elements\, and results of Chase and Swan impl
y that this bound is sharp—there exist examples that cannot be generated
by fewer than d+n elements. We view "projective modules" as forms of the
most trivial kind of non-unital R-algebra\, i.e.\, where the multiplicatio
n is identically 0. We take the results of Forster\, Chase and Swan as a s
tarting point for investigations into forms of other algebras.\n\nFix a fi
eld k and a k-algebra B\, not assumed unital or commutative. Let G denote
the automorphism group scheme of B as an algebra. Let U_r denote the varie
ty of r-tuples of elements that generate B as a k-algebra. In favourable c
ircumstances\, U_r/G is a k-variety\, generalizing the Grassmannian\, that
classifies forms of the algebra B equipped with r generators. In addition
\, as far as A1-invariant cohomology theories are concerned U_r/G approxim
ates the classifying stack BG. By measuring the non-injectivity of the map
of Chow rings CH(BG)->CH(U_r/G)\, we can produce examples of algebras (ov
er a ring R) requiring many generators\, generalizing the example of Chase
and Swan. I will tell a fuller version of this story\, with emphasis on t
he case where B is a matrix algebra\, so that U_r/G classifies Azumaya alg
ebras with r generators. The majority of the talk concerns joint work with
Uriya First and Zinovy Reichstein\, but I will mention some joint work wi
th Taeuk Nam & Cindy Tan and some independent work of Sebastian Gant.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maurice Chayet (ECAM-EPMI)
DTSTART;VALUE=DATE-TIME:20200703T150000Z
DTEND;VALUE=DATE-TIME:20200703T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/9
DESCRIPTION:Title: E8 and a new class of commutative non-associative algebras with a
continuous Pierce Spectrum\nby Maurice Chayet (ECAM-EPMI) as part of Q
uadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nT.A. Sp
ringer knew decades ago of the existence of a Group invariant commutative
algebra structure on the 3875 dimensional representation of $E_8$. It was
recently shown by S. Garibaldi and R. Guralnick that the automorphism grou
p of this unique commutative algebra coincides with $E_8$. However a desc
ription of this algebra has been a lingering question\, ever since it was
noticed by T.A. Springer.\n\nIn this talk\, based on joint work with Skip
Garibaldi\, we explain a correspondence which associates to each simple Li
e algebra\, a commutative non associative unital algebra\, and provide an
explicit closed form expression for the product. This correspondence encom
passes the 3875 invariant algebra for $E_8$ via the addition of a unit. Th
ese algebras turn out to be simple and are endowed with a non-degenerate
“associative” bilinear invariant form. Unlike their closet cousins\, t
he Jordan Algebras\, these algebras are not power associative and share th
e unusual property of having the unit interval as part of their Pierce Spe
ctrum.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raman Parimala (Emory University)
DTSTART;VALUE=DATE-TIME:20200914T150000Z
DTEND;VALUE=DATE-TIME:20200914T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/10
DESCRIPTION:Title: A Hasse principle for simply connected groups\nby Raman Parim
ala (Emory University) as part of Quadratic forms\, linear algebraic group
s and beyond\n\n\nAbstract\nKneser proposed a conjecture that if $G$ is a
semi-simple simply connected linear algebraic group defined over a number
field $k$ and $Y$ a principal homogeneous space under $G$\, then $Y$ satis
fies Hasse principle\, i.e.\, $Y$ has a rational point over $k$ if it does
over completions of $k$ at all its places. This is now a theorem due to
Kneser for classical groups\, Harder for exceptional groups of type other
than $E_8$ and Chernousov for groups of type $E_8$. There were questions
and conjectures on similar Hasse principles over function fields of $p$-a
dic curves and more generally\, semi global fields\, i.e.\,\nfunction fiel
ds of curves over complete discrete valued fields\, with respect to all th
eir discrete valuations. We shall discuss recent progress in this directi
on.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikhail Borovoi (Tel Aviv University)
DTSTART;VALUE=DATE-TIME:20200921T150000Z
DTEND;VALUE=DATE-TIME:20200921T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/11
DESCRIPTION:Title: Galois cohomology of real reductive groups\nby Mikhail Borovo
i (Tel Aviv University) as part of Quadratic forms\, linear algebraic grou
ps and beyond\n\n\nAbstract\nUsing ideas of Kac and Vinberg\, we give a si
mple combinatorial method of computing the Galois cohomology of semisimple
groups over the field $\\mathbb R$ of real numbers. I will explain the me
thod by the examples of simple groups of type $E_7$ (both adjoint and simp
ly connected).\n\nThis is a joint work with Dmitry A. Timashev\, Moscow\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandre Lourdeaux (University of Lyon)
DTSTART;VALUE=DATE-TIME:20200928T150000Z
DTEND;VALUE=DATE-TIME:20200928T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/12
DESCRIPTION:Title: Brauer invariants of linear algebraic groups\nby Alexandre Lo
urdeaux (University of Lyon) as part of Quadratic forms\, linear algebraic
groups and beyond\n\n\nAbstract\nOur talk deals with the cohomological in
variants of smooth and connected linear algebraic groups over an arbitrary
field. The notion of cohomological invariants was formalized by Serre in
the 90’s. It enables to study via Galois cohomology the geometry of line
ar algebraic groups or forms of algebraic stuctures (such as central simpl
e algebras with involution).\n\nWe intend to introduce the general ideas o
f the theory and to present a generalization of a result by Blinstein and
Merkurjev on degree 2 invariants with coefficients Q/Z(1)\, that is invari
ants taking values in the Brauer group. More precisely our result gives a
description of these invariants for every smooth and connected linear grou
ps\, in particular for non reductive groups over an imperfect field (as ps
eudo-reductive or unipotent groups for instance).\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Rapinchuk (Michigan State University)
DTSTART;VALUE=DATE-TIME:20201005T150000Z
DTEND;VALUE=DATE-TIME:20201005T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/13
DESCRIPTION:Title: Algebraic groups with good reduction\nby Igor Rapinchuk (Mich
igan State University) as part of Quadratic forms\, linear algebraic group
s and beyond\n\n\nAbstract\nTechniques involving reduction are very common
in number theory and arithmetic geometry. In particular\, elliptic curves
and general abelian varieties having good reduction have been the subject
of very intensive investigations over the years. The purpose of this talk
is to report on recent work that focuses on good reduction in the context
of reductive linear algebraic groups over finitely generated fields. In a
ddition\, we will highlight some applications to the study of local-global
principles and the analysis of algebraic groups having the same maximal t
ori. (Parts of this work are joint with V. Chernousov and A. Rapinchuk.)\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Merkurjev (UCLA)
DTSTART;VALUE=DATE-TIME:20201012T150000Z
DTEND;VALUE=DATE-TIME:20201012T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/14
DESCRIPTION:Title: Operations in connective K-theory\nby Alexander Merkurjev (UC
LA) as part of Quadratic forms\, linear algebraic groups and beyond\n\n\nA
bstract\nThis is a joint work with A.Vishik. A relation between Chow theor
y and algebraic K-theory of smooth algebraic varieties is given by a ring
homomorphism from the Chow ring to the graded Grothendieck ring of a varie
ty associated with the topological filtration. A much better relation can
be established via connective K-theory that maps to both Chow theory and K
-theory\, so the connective K-theory deserves detailed study.\n\nSteenrod
operations (mod p) and Adams operations are essentially all additive opera
tions in Chow theory and K-theory respectively. In the talk we describe th
e ring of additive operations in connective K-theory.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cyril Demarche (Institut de Mathématiques de Jussieu)
DTSTART;VALUE=DATE-TIME:20201019T150000Z
DTEND;VALUE=DATE-TIME:20201019T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/15
DESCRIPTION:Title: Splitting families in Galois cohomology\nby Cyril Demarche (I
nstitut de Mathématiques de Jussieu) as part of Quadratic forms\, linear
algebraic groups and beyond\n\n\nAbstract\nLet k be a field and A a finite
discrete Galois module. For any integer $n >1$\, let $x$ be a cohomology
class in $H^n(k\, A)$. We show that there exists a countable familiy of (s
mooth\, geometrically integral) $k$-varieties\, such that the following ho
lds: for any field extension $K/k$\, the restriction of $x$ vanishes in $H
^n(K\, A)$ if and only if one of the varieties has an $K$-point. In the ca
se $n= 2$\, we note that one variety (called a splitting variety for $x$)
is enough. The question of the existence of splitting varieties (or splitt
ing families) is insprired by the construction of norm varieties for symbo
ls by Rost. This is joint work with Mathieu Florence.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cameron Ruether (University of Ottawa)
DTSTART;VALUE=DATE-TIME:20201026T150000Z
DTEND;VALUE=DATE-TIME:20201026T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/16
DESCRIPTION:Title: Injections from Kronecker Products and the Cohomological Invarian
ts of Half-Spin\nby Cameron Ruether (University of Ottawa) as part of
Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nLet $G
$ be a linear algebraic group over a field $F$. As introduced by Serre\, d
egree $n$ cohomological invariants of $G$ with coefficients in a group $A$
\, where $A$ is equipped with an action of the absolute Galois group of $F
$\, are natural transformations of Galois cohomology functors $H^1(-\,G) \
\to H^n(-\,A)$. Commonly studied are the degree three invariants with coef
ficients in $\\mathbb{Q}/\\mathbb{Z} \\otimes \\mathbb{Q}/\\mathbb{Z}$. Th
ese invariants were recently described by Merkurjev for the semisimple adj
oint case\, and by Bermudez and Ruozzi for semisimple $G$ which are neithe
r simply connected nor adjoint. In particular\, they described the structu
re of the normalized degree three invariants (those which send the trivial
object to zero) of the half-spin group $\\operatorname{HSpin}_{16}$. By g
eneralizing a technique of Garibaldi we construct new injections into $\\o
peratorname{HSpin}$ induced by the Kronecker tensor product map. In partic
ular we construct an injection $\\operatorname{PSp}_{2n} \\times \\operato
rname{PSp}_2m \\to \\operatorname{HSpin}_{4nm}$ which we use to describe t
he normalized invariants of $\\operatorname{HSpin}_{4k}$ for any $k$\, gen
eralizing the result of Bermudez and Ruozzi.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anastasia Stavrova (St. Petersburg University)
DTSTART;VALUE=DATE-TIME:20201102T160000Z
DTEND;VALUE=DATE-TIME:20201102T170000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/17
DESCRIPTION:Title: Torsors of isotropic reductive groups over Laurent polynomials\nby Anastasia Stavrova (St. Petersburg University) as part of Quadratic
forms\, linear algebraic groups and beyond\n\n\nAbstract\nLet $k$ be a fie
ld of characteristic 0. Let $G$ be a reductive group over the ring of Laur
ent polynomials $R=k[x_1^{\\pm 1}\,\\ldots\,x_n^{\\pm 1}]$. We prove that
$G$ has isotropic rank $\\ge 1$ over $R$ iff it has isotropic rank $\\ge 1
$ over the field of fractions $k(x_1\,\\ldots\,x_n)$ of $R$\, and if this
is the case\, then the natural map $H^1_{et}(R\,G)\\to H^1_{et}(k(x_1\,\\l
dots\,x_n)\,G)$ has trivial kernel and $G$ is loop reductive\, i.e. $G$ co
ntains a maximal $R$-torus. We also deduce that if $G$ is a reductive grou
p over $R$ of isotropic rank $\\ge 2$\, then the natural map of non-stable
$K_1$-functors $K_1^G(R)\\to K_1^G\\bigl( k((x_1))\\ldots ((x_n)) \\bigr)
$ is injective\, and an isomorphism if $G$ is moreover semisimple.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lucy Moser-Jauslin (Université de Bourgogne)
DTSTART;VALUE=DATE-TIME:20201109T160000Z
DTEND;VALUE=DATE-TIME:20201109T170000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/18
DESCRIPTION:Title: Forms of almost homogeneous varieties\nby Lucy Moser-Jauslin
(Université de Bourgogne) as part of Quadratic forms\, linear algebraic g
roups and beyond\n\n\nAbstract\nIn this talk\, we will discuss almost homo
geneous varieties for reductive groups over a perfect field $k$. Let $K$ b
e an algebraic closure of $k$\, and let $G$ be a connected reductive $K$-g
roup with a fixed $k$-form $F$. A normal $G$-variety over $K$ is almost ho
mogeneous if it has an open dense orbit. Given an almost homogeneous $G$-
variety $X$\, the goal of this talk will be to determine $k$-forms of $X$
which are compatible with the $k$-form $F$ of $G$. In order to do this\, w
e describe an action of the Galois group on the combinatorics developed i
n Luna-Vust theory. This is joint work with Ronan Terpereau.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kęstutis Česnavičius (Université Paris-Sud)
DTSTART;VALUE=DATE-TIME:20201116T160000Z
DTEND;VALUE=DATE-TIME:20201116T170000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/19
DESCRIPTION:Title: Grothendieck–Serre in the split unramified case\nby Kęstut
is Česnavičius (Université Paris-Sud) as part of Quadratic forms\, line
ar algebraic groups and beyond\n\n\nAbstract\nThe Grothendieck–Serre con
jecture predicts that every generically trivial torsor under a reductive g
roup scheme G over a regular local ring R is trivial. We settle it in the
case when G is split and R is unramified. To overcome obstacles that have
so far kept the mixed characteristic case out of reach\, we rely on the re
cently-established Cohen–Macaulay version of the resolution of singulari
ties.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victor Petrov (St. Petersburg University)
DTSTART;VALUE=DATE-TIME:20201123T160000Z
DTEND;VALUE=DATE-TIME:20201123T170000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/20
DESCRIPTION:Title: Isotropy of Tits construction\nby Victor Petrov (St. Petersbu
rg University) as part of Quadratic forms\, linear algebraic groups and be
yond\n\n\nAbstract\nTits construction produces a Lie algebra out of a comp
osition 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 dim
ension 1\,2\,4 or 8 respectively. Garibaldi and Petersson noted that the T
its index ${}^2E_6^{35}$ cannot occur as a result of Tits construction. Re
cently Alex Henke proved that the Tits index $E_7^{66}$ is also not possib
le. 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 re
lies on basic facts about symmetric spaces and our joint result with Garib
aldi and Semenov about isotropy of groups of type $E_7$ in terms of the Ro
st invariant. This is a part of a work in progress joint with Simon Rigby.
\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Wertheim (UCLA)
DTSTART;VALUE=DATE-TIME:20201130T160000Z
DTEND;VALUE=DATE-TIME:20201130T170000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/21
DESCRIPTION:Title: Degree One Milnor K-Invariants of Groups of Multiplicative Type\nby Alex Wertheim (UCLA) as part of Quadratic forms\, linear algebraic
groups and beyond\n\n\nAbstract\nMany important algebraic objects can be v
iewed as $G$-torsors over a field $F$\, where $G$ is an algebraic group ov
er $F$. For example\, there is a natural bijection between $F$-isomorphism
classes of central simple $F$-algebras of degree n and $\\operatorname{PG
L}_n(F)$-torsors over $\\operatorname{Spec}(F)$. Much as one may study pri
ncipal bundles on a manifold via characteristic classes\, one may likewise
study G-torsors over a field via certain associated Galois cohomology cla
sses. This principle is made precise by the notion of a cohomological inva
riant\, which was first introduced by Serre. \n\nIn this talk\, we will de
termine the cohomological invariants for algebraic groups of multiplicativ
e type with values in $H^{1}(-\, Q/Z(1))$. Our main technical analysis wil
l center around a careful examination of $\\mu_n$-torsors over a smooth\,
connected\, reductive algebraic group. Along the way\, we will compute a r
elated group of invariants for smooth\, connected\, reductive groups.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pavel Sechin (University of Regensburg)
DTSTART;VALUE=DATE-TIME:20201207T160000Z
DTEND;VALUE=DATE-TIME:20201207T170000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/22
DESCRIPTION:Title: Morava K-theory pure motives with applications to quadrics\nb
y Pavel Sechin (University of Regensburg) as part of Quadratic forms\, lin
ear algebraic groups and beyond\n\n\nAbstract\nMorava K-theories $K(n)$ ar
e cohomology theories that have graded fields of positive characteristic a
s coefficient rings and that are obtained from algebraic cobordism of Levi
ne-Morel by change of coefficients. Pure motives with respect to $K(n)$ fi
t in-between Chow motives and $K_0$-motives (with $p$-localized or $p$-tor
sion coefficients)\, e.g. allowing to transfer $K(n)$-decompositions to $K
(m)$-decompositions whenever $m < n$. Thus\, it might be a reasonable appr
oach in the study of motivic decompositions to start with $K(1)$-motives (
i.e. more or less $K_0$-motives) and continue to $K(2)$-\, $K(3)$-motives
and so on\, eventually arriving to Chow-motives.\nOn the other hand we for
mulate a conjectural principle that connects the splitting of $K(n)$-motiv
e. \nwith the triviality of cohomological invariants of degrees less than
$n+1$.\nI plan to outline the proof of this principle for quadrics and exp
lain its consequences \nfor Chow groups of quadrics lying in powers of the
fundamental ideal in the Witt ring.\nThe talk is mostly based on the join
t work with Nikita Semenov.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Primosic (University of Alberta)
DTSTART;VALUE=DATE-TIME:20201214T160000Z
DTEND;VALUE=DATE-TIME:20201214T170000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/23
DESCRIPTION:Title: Motivic cohomology and infinitesimal group schemes\nby Eric P
rimosic (University of Alberta) as part of Quadratic forms\, linear algebr
aic groups and beyond\n\n\nAbstract\nFor $k$ a perfect field of characteri
stic $p > 0$ and $G$ a split reductive group over $k$ with $p$ a non-torsi
on prime for $G$\, we compute the mod $p$ motivic cohomology of the geomet
ric classifying space $BG_{(r)}$\, where $G_{(r)}$ is the $r$th Frobenius
kernel of $G$. Our main tool is a motivic version of the Eilenberg-Moore s
pectral sequence\, due to Krishna.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Uriya First (University of Haifa)
DTSTART;VALUE=DATE-TIME:20210120T163000Z
DTEND;VALUE=DATE-TIME:20210120T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/24
DESCRIPTION:Title: The Grothendieck--Serre conjecture for classical groups in low di
mensions\nby Uriya First (University of Haifa) as part of Quadratic fo
rms\, linear algebraic groups and beyond\n\n\nAbstract\nA famous conjectur
e of Grothendieck and Serre predicts that if $G$ is a reductive group sche
me over a semilocal regular domain $R$ and $X$ is a G-torsor\, then $X$ ha
s a point over the fraction field of $R$ if and only if it has an $R$-poin
t. I will discuss recent work with Eva Bayer-Fluckiger and Raman Parimala
in which we prove the conjecture for all forms of ${\\rm GL}_n$\, ${\\rm S
p}_n$ and ${\\rm SO}_n$ when $R$ is 2-dimensional\, and all forms of ${\\r
m GL}_{2n+1}$ when $R$ is 4-dimensional. (Here the ring $R$ is not require
d to contain a field.) We approach the problem using the hermitian Gersten
-Witt complex associated to an Azumaya algebra with involution $(A\,s)$ ov
er a semilocal regular ring $R$. Specifically\, we show that it is exact w
hen the Krull dimension of $R$ or the index of $A$ are sufficiently small.
\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Kuznetsov (Steklov Mathematics Institute)
DTSTART;VALUE=DATE-TIME:20210127T163000Z
DTEND;VALUE=DATE-TIME:20210127T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/25
DESCRIPTION:Title: Exceptional collection of vector bundles on F4/P4\nby Alexand
er Kuznetsov (Steklov Mathematics Institute) as part of Quadratic forms\,
linear algebraic groups and beyond\n\n\nAbstract\nIn the talk I will expla
in a construction of a full exceptional collection of vector bundles on th
e homogeneous variety of the simple algebraic group of Dynkin type $F_4$ c
orresponding to its maximal parabolic subgroup $P_4$. The construction is
based on the relation of this homogeneous variety to a homogeneous variety
of type $E_6 / P_1$ and uses an exceptional collection constructed by Fae
nzi and Manivel. This is joint work with Pieter Belmans and Maxim Smirnov.
\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giancarlo Lucchini-Arteche (University of Chile)
DTSTART;VALUE=DATE-TIME:20210203T163000Z
DTEND;VALUE=DATE-TIME:20210203T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/26
DESCRIPTION:Title: Local-global principles for homogeneous spaces over some two-dime
nsional geometric global fields\nby Giancarlo Lucchini-Arteche (Univer
sity of Chile) as part of Quadratic forms\, linear algebraic groups and be
yond\n\n\nAbstract\nOver number fields\, there is a classic obstruction to
the local-global principle for the existence of rational points\, known a
s the Brauer-Manin obstruction\, which is conjectured to explain all failu
res of this principle for homogeneous spaces of connected linear groups. I
n the last few years\, there has been an increasing interest in fields of
a more geometric nature that are amenable to local-global principles and B
rauer-Manin obstructions as well. These include\, for instance\, function
fields of curves over discretely valued fields\, by analogy with the case
of global fields of positive characteristic. It is in this context that I
will present recent work with Diego Izquierdo on local-global principles f
or homogeneous spaces with connected stabilizers. We will see that\, altho
ugh some of the known results for number fields have direct analogs (that
can be obtained in the same way)\, the particularities of these new fields
bring up new counterexamples that cannot be explained by the Brauer-Manin
obstruction\, contrary to the number field case.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ivan Panin (Steklov Institute at St.Petersburg)
DTSTART;VALUE=DATE-TIME:20210217T163000Z
DTEND;VALUE=DATE-TIME:20210217T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/27
DESCRIPTION:Title: Rationally isotropic quadratic spaces are locally isotropic (mixe
d characteristic case)\nby Ivan Panin (Steklov Institute at St.Petersb
urg) as part of Quadratic forms\, linear algebraic groups and beyond\n\n\n
Abstract\nA well-known conjecture of Colliot-Thélène asserts that a rati
onally isotropic quadratic space over a regular local ring is isotropic. I
f the ring contains a field\, then this conjecture was proved by the effor
ts of the speaker\, Pimenov and Scully. In the talk we will present new re
sults in the mixed characteristic case.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Patrick Brosnan (University of Maryland)
DTSTART;VALUE=DATE-TIME:20210210T163000Z
DTEND;VALUE=DATE-TIME:20210210T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/28
DESCRIPTION:Title: Fixed Points in Toroidal Compactifications and Essential Dimensio
n of Covers\nby Patrick Brosnan (University of Maryland) as part of Qu
adratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nEssentia
l dimension is a numerical measure of the complexity of algebraic\nobjects
invented by J. Buhler and Z. Reichstein in the 90s. Roughly speaking\,\n
the essential dimension of an algebraic object is the number of parameters
it\ntakes to define the object over a field. For example\, by Kummer theo
ry\, it\ntakes one parameter to define a mu_n torsor\, so the essential di
mension of the\nfunctor of mu_n torsors (or the essential dimension of mu_
n for short) is 1.\nIn a preprint from 2019\, Farb\, Kisin and Wolfson (FK
W) prove theorems about the\nessential dimension of congruence covers of S
himura varieties using arithmetic\nmethods. In many cases\, they are able
to prove that the congruence covers are\nincompressible\, that is\, they
are not obtainable by base change from varieties\nof strictly smaller dime
nsion. \n\nIn my talk\, I will discuss recent work with Najmuddin Fakhrudd
in\, where we recover many (but definitely not all) of the results of FKW\
, by geometric\narguments using a new fixed point theorem. This also allow
s us to extend the\nincompressibility results of FKW to Shimura varieties
of exceptional type to\nwhich the arithmetic methods of FKW do not apply.
I will also discuss a general\nconjecture we make on the essential dimensi
on of congruence covers arising from\nHodge theory. (With some caveats\, w
e conjecture that it is equal to the\ndimension of the image of the period
map.)\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julia Hartmann (University of Pennsylvania)
DTSTART;VALUE=DATE-TIME:20210224T163000Z
DTEND;VALUE=DATE-TIME:20210224T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/29
DESCRIPTION:Title: Local-global principles for constant reductive groups over arithm
etic function fields\nby Julia Hartmann (University of Pennsylvania) a
s part of Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstra
ct\nArithmetic function fields are one variable function fields over compl
ete discretely valued fields. They naturally admit several collections of
overfields with respect to which one can study local-global principles. We
will focus on studying local-global principles for torsors under reductiv
e groups that are defined over the underlying discrete valuation ring\, re
porting on joint work with J.L.-Colliot-Thélène\, D. Harbater\, D. Krash
en\, R. Parimala\, and V. Suresh.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthieu Romagny (Université Rennes 1)
DTSTART;VALUE=DATE-TIME:20210317T153000Z
DTEND;VALUE=DATE-TIME:20210317T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/30
DESCRIPTION:Title: Smooth affine group schemes over the dual numbers\nby Matthie
u Romagny (Université Rennes 1) as part of Quadratic forms\, linear algeb
raic groups and beyond\n\n\nAbstract\nWe provide a geometric construction
for the equivalence between the category of smooth affine group schemes ov
er the ring of dual numbers $k[ε]$ and the category of extensions \\[ 1
→ {\\rm Lie}(G) → E → G → 1\, \\] where G is a smooth affine group
scheme over k. The equivalence is given by Weil restriction\, and we prov
ide a quasi-inverse which we call Weil extension. As an application\, we e
stablish a Dieudonné classification for smooth\, commutative\, unipotent
group schemes over $k[ε]$ when k is a perfect field.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Popov (Steklov Institute\, Moscow)
DTSTART;VALUE=DATE-TIME:20210310T163000Z
DTEND;VALUE=DATE-TIME:20210310T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/31
DESCRIPTION:Title: Root systems and root lattices in number fields\nby Vladimir
Popov (Steklov Institute\, Moscow) as part of Quadratic forms\, linear alg
ebraic groups and beyond\n\n\nAbstract\nThe following construction of a ro
ot system of type G_2 is given in J.-P. Serre’s book “Complex Semisimp
le Lie algebras” (Chapter V\, Section 16): “It can be described as the
set of algebraic integers of a cyclotomic field generated by a cubic root
of unity\, with norm 1 and 3”. The talk\, based on joint work with Yu.
G. Zarhin\, concerns the problem of realization of root systems\, their We
yl groups and their root lattices in the form of groups and lattices natur
ally associated with number fields.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Benoist (École normale supérieure\, Paris)
DTSTART;VALUE=DATE-TIME:20210303T163000Z
DTEND;VALUE=DATE-TIME:20210303T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/32
DESCRIPTION:Title: Sums of squares in local fields\nby Olivier Benoist (École n
ormale supérieure\, Paris) as part of Quadratic forms\, linear algebraic
groups and beyond\n\n\nAbstract\nArtin and Pfister have shown that a nonne
gative real\npolynomial in n variables is a sum of $2^n$ squares of ration
al functions. In other words\, the Pythagoras number of the field $\\mathb
b R(x_1\,…\,x_n)$ is at most $2^n$. In this talk\, I will consider local
variants of this statement. In particular\, I will give a proof of a conj
ecture of Choi\, Dai\, Lam and Reznick: the Pythagoras number of the field
of Laurent series $\\mathbb R((x_1\,…\,x_n))$ is at most $2^{n-1}$.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michel Brion (Institut Fourier\, Université Grenoble Alpes)
DTSTART;VALUE=DATE-TIME:20210324T153000Z
DTEND;VALUE=DATE-TIME:20210324T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/33
DESCRIPTION:Title: Homomorphisms of algebraic groups: representability and rigidity<
/a>\nby Michel Brion (Institut Fourier\, Université Grenoble Alpes) as pa
rt of Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\n
The talk will address the following questions: given two algebraic groups
G\, H over a field\, is the functor of group homomorphisms from G to H rep
resentable by a scheme M\, locally of finite type? If so\, how to describe
the orbits of H acting on M via conjugation of homomorphisms? The represe
ntability question has a positive answer when G is reductive and H is smoo
th and affine\, by a result of Demazure in SGA3 (which holds over an arbit
ary base).The talk will present an extension of this result to the class o
f “semi-reductive” algebraic groups\, which includes reductive groups\
, finite groups and abelian varieties. In characteristic 0\, we will also
see that all the H-orbits in M are open. This rigidity property gives back
results of Vinberg and Margaux.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Lavrenov (St.Petersburg State University)
DTSTART;VALUE=DATE-TIME:20210331T153000Z
DTEND;VALUE=DATE-TIME:20210331T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/34
DESCRIPTION:Title: Morava motives of projective quadrics\nby Andrei Lavrenov (St
.Petersburg State University) as part of Quadratic forms\, linear algebrai
c groups and beyond\n\n\nAbstract\nThe category of Chow motives defined by
Grothendieck has plenty of various applications to quadratic forms\, and\
, more generally\, to projective homogeneous varieties. However\, there ar
e many open questions about the behaviour of Chow motives. In contrast\, i
f we change the Chow group by Grothendieck’s $K^0$ in the definition of
motives\, the resulting category behaves much more simply. One can define
the category of motives corresponding to any oriented cohomology theory A
and hopefully obtain invariants that are simpler than Chow motives but kee
p more information than $K^0$-motives.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kelly McKinnie (University of Montana)
DTSTART;VALUE=DATE-TIME:20210407T153000Z
DTEND;VALUE=DATE-TIME:20210407T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/35
DESCRIPTION:Title: Common Splitting Fields of Symbol Algebras\nby Kelly McKinnie
(University of Montana) as part of Quadratic forms\, linear algebraic gro
ups and beyond\n\n\nAbstract\nEvery central simple algebra of p-power degr
ee over a field of characteristic p is Brauer equivalent to a cyclic algeb
ra by a result of Albert. The proof of this and other similar p-algebra re
sults rely on the interplay between purely inseparable splitting fields an
d cyclic splitting fields of p-algebras. This talk on joint work with Adam
Chapman and Mathieu Florence looks at new results on common splitting fie
lds of symbol p-algebras with applications to symbol length.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Haution (LMU Munich)
DTSTART;VALUE=DATE-TIME:20210414T153000Z
DTEND;VALUE=DATE-TIME:20210414T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/36
DESCRIPTION:Title: The cobordism ring of algebraic involutions\nby Olivier Hauti
on (LMU Munich) as part of Quadratic forms\, linear algebraic groups and b
eyond\n\n\nAbstract\nI will provide an elementary definition of the cobord
ism ring of involutions of smooth projective varieties over a field (of ch
aracteristic not 2). I will describe its structure\, and give explicit “
stable” polynomial generators. I will draw some concrete consequences co
ncerning the geometry of fixed loci of involutions\, in terms of Chern num
bers. I will in particular mention an algebraic version of Boardman’s fi
ve halves theorem.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Suresh Venapally (Emory University)
DTSTART;VALUE=DATE-TIME:20210505T153000Z
DTEND;VALUE=DATE-TIME:20210505T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/37
DESCRIPTION:Title: Degree three cohomology groups of function fields of curves over
number fields\nby Suresh Venapally (Emory University) as part of Quadr
atic forms\, linear algebraic groups and beyond\n\n\nAbstract\nLet $F$ be
a field and $l$ a prime not equal to the characteristic of $F$. Given $a_1
\, \\ldots \, a_n \\in F^∗$\, the cup product gives an element $(a_1)\\c
dot \\ldots \\cdot (a_n)$ in \n$H^n(F\, µ_l^{\\otimes n})$ and such an el
ement is called a symbol. Class field\ntheory asserts that if $F$ is a glo
bal field or a local field\, then every element in $H^2(F\, µ_l^{\\otimes
2})$ is a symbol. Let $F$ be the function field of a curve over a totally
imaginary number field or a local field. If $F$ contains a primitive $l$t
h root of unity\, then we show that every element in $H^3(F\, µ_l^{\\otim
es 3})$ is a symbol. We describe an\napplication to the isotropy of quadra
tic forms over F. We also give an application to the finite generation of
the Chow group of zero-cycles on quadric fibrations of curves over totally
imaginary number fields.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Guralnick (University of Southern California)
DTSTART;VALUE=DATE-TIME:20210512T153000Z
DTEND;VALUE=DATE-TIME:20210512T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/38
DESCRIPTION:Title: Generic Stabilizers for Simple Algebraic Groups\nby Robert Gu
ralnick (University of Southern California) as part of Quadratic forms\, l
inear algebraic groups and beyond\n\n\nAbstract\nConsider an algebraic gro
up $G$ acting on an irreducible variety $X$. We say there exists a gener
ic stabilizer for this action if there exists a nonempty open subset $Y$ o
f $X$ such that the stabilizers of any $y$ in $Y$ are all conjugate in $G$
. In characteristic $0$\, there are general results of Richardson proving
the existence of a generic stabilizer in many cases. We especially consi
der the case that $G$ is a simple algebraic group in positive characteris
tic and $X$ is an irreducible $G$-module. We show that a generic stabili
zer always exists and determine the generic stabilizer in all cases. This
fails for semisimple groups. This is joint work with Skip Garibaldi and R
oss Lawther.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Susanna Zimmermann (Université d’Angers)
DTSTART;VALUE=DATE-TIME:20210519T153000Z
DTEND;VALUE=DATE-TIME:20210519T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/39
DESCRIPTION:Title: Algebraic groups acting birationally on surfaces over a perfect f
ield\nby Susanna Zimmermann (Université d’Angers) as part of Quadra
tic forms\, linear algebraic groups and beyond\n\n\nAbstract\nWhich linear
algebraic groups act birationally on a rational surface? And which are th
ese actions\, up to conjugacy by a birational map? The classification hist
ory is quite long over the field of complex numbers and cumulates in the w
orks of Blanc and Dolgachev-Iskovskikh. Over non-closed fields\, the class
ification is not complete yet\, but there are many partial results. In thi
s talk\, I would like to present the way to attack the classification in g
eneral\, as well as explain the complete list of actions (up to conjugacy)
when the linear algebraic group is infinite.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Vishik (University of Nottingham)
DTSTART;VALUE=DATE-TIME:20210526T153000Z
DTEND;VALUE=DATE-TIME:20210526T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/40
DESCRIPTION:Title: Torsion Motives\nby Alexander Vishik (University of Nottingha
m) as part of Quadratic forms\, linear algebraic groups and beyond\n\n\nAb
stract\nTorsion motives are Chow motives which disappear with rational coe
fficients. Surprisingly\, such objects exist - examples were constructed b
y Gorchinsky-Orlov. Hypothetically\, such motives should generate the kern
el of the family of "isotropic realization" functors. I will discuss some
invariants of torsion motives which\, in particular\, shed light on their
size.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jan Minac and Tung T. Nguyen (University of Western Ontario)
DTSTART;VALUE=DATE-TIME:20210602T153000Z
DTEND;VALUE=DATE-TIME:20210602T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/41
DESCRIPTION:Title: A panoramic view of absolute Galois groups\nby Jan Minac and
Tung T. Nguyen (University of Western Ontario) as part of Quadratic forms\
, linear algebraic groups and beyond\n\n\nAbstract\nFrom the very beginnin
g of its origin\, Galois theory has gained an air of depth\, beauty\, eleg
ance\, and interactions with deep arithmetic\, geometric\, and topological
considerations. Basic fundamental open questions in this area include the
characterization of absolute Galois groups among profinite groups\, a cha
racterization of maximal pro-p-quotients of absolute Galois groups\, the s
tudy of Massey products in Galois cohomology and the sharpening of Rost-Vo
evodsky’s remarkable work on the Bloch-Kato conjecture. These fundamenta
l questions are deeply intertwined with considerations of the special valu
es of L-functions and Galois representations. In this talk we plan to pres
ent some current research on some rather basic elementary aspects of Masse
y products\, Galois modules\, the nature of the values of zeta functions\,
and Fekete polynomials. These topics will include our joint work with Ngu
yen Duy Tˆan and Andrew Schultz. This talk will be in the form of a dialo
g where we shall try and reinact our researchefforts and bring to life our
actual research excitement pursuing questions and differentconnections wi
th a number of aspects of current Galois theory.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zev Rosengarten (The Hebrew University of Jerusalem)
DTSTART;VALUE=DATE-TIME:20210609T153000Z
DTEND;VALUE=DATE-TIME:20210609T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/42
DESCRIPTION:Title: Rigidity and Unirational groups\nby Zev Rosengarten (The Hebr
ew University of Jerusalem) as part of Quadratic forms\, linear algebraic
groups and beyond\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anand Sawant (Tata Institute of Fundamental Research)
DTSTART;VALUE=DATE-TIME:20210623T153000Z
DTEND;VALUE=DATE-TIME:20210623T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/43
DESCRIPTION:Title: Near-rationality properties of norm varieties\nby Anand Sawan
t (Tata Institute of Fundamental Research) as part of Quadratic forms\, li
near algebraic groups and beyond\n\n\nAbstract\nThe standard norm varietie
s played a crucial role in Voevodsky’s proof of the Bloch-Kato conjectur
e. I will discuss various near-rationality concepts for smooth projective
varieties and describe known near-rationality results for standard norm va
rieties. I will then outline an argument showing that a standard norm vari
ety over a field of characteristic 0 is universally R-trivial after passin
g to the algebraic closure of the base field. The talk is based on joint w
ork with Chetan Balwe and Amit Hogadi.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefan Schreieder (Leibniz Universität Hannover)
DTSTART;VALUE=DATE-TIME:20210707T153000Z
DTEND;VALUE=DATE-TIME:20210707T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/44
DESCRIPTION:Title: Refined unramified cohomology of schemes\nby Stefan Schreiede
r (Leibniz Universität Hannover) as part of Quadratic forms\, linear alge
braic groups and beyond\n\n\nAbstract\nWe introduce refined unramified coh
omology of algebraic schemes and show that it interpolates between Borel
–Moore homology and algebraic cycles. Over finitely generated fields\, l
-adic Chow groups of algebraic schemes are computed by refined unramified
cohomology. Over the complex numbers\, our approach simplifies and general
izes to cycles of arbitrary codimensions on possibly singular schemes\, pr
evious results of Bloch—Ogus\, Colliot-Thélène—Voisin\, Voisin\, and
Ma. Our approach has several applications. For instance\, it allows to pr
oduce the first example of a smooth complex projective variety whose Griff
iths group has infinite torsion.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Saltman (Center for Communications Research)
DTSTART;VALUE=DATE-TIME:20210721T153000Z
DTEND;VALUE=DATE-TIME:20210721T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/45
DESCRIPTION:Title: Lifting in Mixed Characteristics\nby David Saltman (Center fo
r Communications Research) as part of Quadratic forms\, linear algebraic g
roups and beyond\n\n\nAbstract\nIn Suresh's talk in this seminar\, he outl
ined the proof of his main theorem and in that process talked about choosi
ng elements with a "half dozen" and then a "dozen" properties. To a signif
icant extent\, this amounted to lifting degree $p$ cyclic Galois extension
s with particular properties at characteristic $p$ and characteristic not
$p$ points. This suggests there should be a "generic" polynomial that is t
he Artin-Schreier polynomial modulo $p$ and gives Kummer extensions in all
other characteristics. Starting from the change of variables noted by Sur
esh and others\, we present such a polynomial when the ground ring has a p
rimitive $p$ root of one\, $\\rho$. Then we define a generic extension wit
hout assuming the presence of that root of unity. If $x$ is a root of the
polynomial above\, it is useful to note that the Galois action is $\\sigma
(x) = \\rho x + 1$. This\nimplies a description\, in the obvious way\, of
a cyclic algebra. However in characteristic $p$ the differential degree p
crossed products are more useful\, and so that is the object we want to li
ft. We are led to consider algebras generated by $x$\, $y$ with $x^p$\, $y
^p$ central and $xy - \\rho yx = 1$. Finally we will describe some steps o
ne can make to generalize all of the above to degrees a power of $p$.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ning Guo (Institut de Mathématique d’Orsay)
DTSTART;VALUE=DATE-TIME:20210616T153000Z
DTEND;VALUE=DATE-TIME:20210616T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/46
DESCRIPTION:Title: The Grothendieck–Serre conjecture over valuation rings\nby
Ning Guo (Institut de Mathématique d’Orsay) as part of Quadratic forms\
, linear algebraic groups and beyond\n\n\nAbstract\nThe Grothendieck–Ser
re conjecture predicts that torsors under reductive group schemes over reg
ular local rings are trivial if they trivialize generically. In this talk\
, we consider the variant when the bases are valuation rings. This result
is predicted by the original Grothendieck–Serre conjecture and the resol
ution of singularities. The novelty of our proof lies in overcoming subtle
ties brought by general nondiscrete valuation rings. By using flasque reso
lutions and inducting with local cohomology\, we prove a non-Noetherian co
unterpart of Colliot-Thélène– Sansuc’s case of tori. Then\, taking a
dvantage of techniques in algebraization\, we obtain the passage to the He
nselian rank one case. Finally\, we induct on Levi subgroups and use the i
ntegrality of rational points of anisotropic groups to reduce to the semis
imple anisotropic case\, in which we appeal to properties of parahoric sub
groups in Bruhat–Tits theory to conclude. In the last section\, by using
properties of reflexive sheaves\, we also prove a variant of Nisnevich’
s purity conjecture.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seidon Alsaody (Uppsala University)
DTSTART;VALUE=DATE-TIME:20210714T153000Z
DTEND;VALUE=DATE-TIME:20210714T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/47
DESCRIPTION:by Seidon Alsaody (Uppsala University) as part of Quadratic fo
rms\, linear algebraic groups and beyond\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hanspeter Kraft (University of Basel)
DTSTART;VALUE=DATE-TIME:20210929T153000Z
DTEND;VALUE=DATE-TIME:20210929T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/48
DESCRIPTION:Title: Small $G$-varieties (joint work with Andriy Regeta and Susanna Zi
mmermann)\nby Hanspeter Kraft (University of Basel) as part of Quadrat
ic forms\, linear algebraic groups and beyond\n\n\nAbstract\nAbstract: An
affine variety with an action of a semisimple group $G$ is called ``small'
' if every non-trivial $G$-orbit in $X$ is isomorphic to the orbit of a hi
ghest weight vector. Such a variety $X$ carries a canonical action of the
multiplicative group $K^*$ commuting with the $G$-action. We show that $X$
is determined by the $K^*$-variety $X^U$ of fixed points under a maximal
unipotent subgroup $U \\subset G$. Moreover\, if $X$ is smooth\, then $X$
is a $G$-vector bundle over the algebraic quotient $X // G$.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Louis Colliot-Thelene (Université Paris-Saclay\, Orsay)
DTSTART;VALUE=DATE-TIME:20211020T153000Z
DTEND;VALUE=DATE-TIME:20211020T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/50
DESCRIPTION:Title: Quadratic forms and stable rationality I\nby Jean-Louis Colli
ot-Thelene (Université Paris-Saclay\, Orsay) as part of Quadratic forms\,
linear algebraic groups and beyond\n\n\nAbstract\nIn this survey split ov
er two talks\, I shall review how - over 50 years - quadratic forms\, an
d in particular Pfister forms\, have been used to produce more and more e
xamples of rationally connected varieties over the complex field which ar
e not stably rational. We shall start with the Artin-Mumford examples on c
onic bundles over the plane and end with the work of Schreieder on hypers
urfaces of small slope.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Louis Colliot-Thelene (Université Paris-Saclay\, Orsay)
DTSTART;VALUE=DATE-TIME:20211027T153000Z
DTEND;VALUE=DATE-TIME:20211027T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/51
DESCRIPTION:Title: Quadratic forms and stable rationality II\nby Jean-Louis Coll
iot-Thelene (Université Paris-Saclay\, Orsay) as part of Quadratic forms\
, linear algebraic groups and beyond\n\nAbstract: TBA\n\nIn this survey sp
lit over two talks\, I shall review how - over 50 years - quadratic form
s\, and in particular Pfister forms\, have been used to produce more and
more examples of rationally connected varieties over the complex field wh
ich are not stably rational. We shall start with the Artin-Mumford example
s on conic bundles over the plane and end with the work of Schreieder on
hypersurfaces of small slope.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Yakerson (ETH Zurich)
DTSTART;VALUE=DATE-TIME:20211110T163000Z
DTEND;VALUE=DATE-TIME:20211110T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/52
DESCRIPTION:Title: Twisted K-theory in motivic homotopy theory\nby Maria Yakerso
n (ETH Zurich) as part of Quadratic forms\, linear algebraic groups and be
yond\n\n\nAbstract\nIn this talk\, we will speak about algebraic K-theory
of vector bundles twisted by a Brauer class. In particular\, we will discu
ss a new approach to the motivic spectral sequence for twisted K-theory\,
constructed earlier by Bruno Kahn and Marc Levine. Time permitting\, we wi
ll mention potential applications for K3 surfaces. The talk is based on jo
int work with Elden Elmanto and Denis Nardin.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alena Pirutka (Courant Institute of Mathematical Sciences)
DTSTART;VALUE=DATE-TIME:20211117T163000Z
DTEND;VALUE=DATE-TIME:20211117T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/53
DESCRIPTION:Title: Quadrics and computation of the unramified Brauer group\nby A
lena Pirutka (Courant Institute of Mathematical Sciences) as part of Quadr
atic forms\, linear algebraic groups and beyond\n\n\nAbstract\nIn this tal
k we will discuss examples of computations of the unramified Brauer group
for fibrations in quadrics.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Erhard Neher (University of Ottawa)
DTSTART;VALUE=DATE-TIME:20211201T163000Z
DTEND;VALUE=DATE-TIME:20211201T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/54
DESCRIPTION:Title: Quadratic forms over semilocal rings\nby Erhard Neher (Univer
sity of Ottawa) as part of Quadratic forms\, linear algebraic groups and b
eyond\n\n\nAbstract\nWe discuss several results\, well-known for quadratic
forms over fields\, in the setting of quadratic forms over arbitrary semi
local rings. Among them are Springer's odd degree extension theorem and th
e norm principles of Scharlau and of Knebusch. The talk is based on joint
work with Philippe Gille.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikita Karpenko (University of Alberta)
DTSTART;VALUE=DATE-TIME:20210922T153000Z
DTEND;VALUE=DATE-TIME:20210922T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/55
DESCRIPTION:Title: Yagita’s counter-examples and beyond\nby Nikita Karpenko (U
niversity of Alberta) as part of Quadratic forms\, linear algebraic groups
and beyond\n\n\nAbstract\nA conjecture on a relationship between the Chow
and Grothendieck rings for the generically twisted variety of Borel subgr
oups in a split semisimple group $G$\, stated by myself\, has been disprov
ed by Nobuaki Yagita in characteristic $0$ for $G=\\operatorname{Spin}(2n+
1)$ with $n=8$ and $n=9$. For $n=8$\, I provided an alternative simpler pr
oof of Yagita’s result\, working in any characteristic\, but failed to d
o so for $n=9$. In a current joint work with Sanghoon Baek\, this gap is f
illed by involving a new ingredient – Pieri type K-theoretic formulas fo
r highest orthogonal Grassmannians. Besides\, a similar counter-example fo
r $n=10$ is produced. Note that the conjecture on $\\operatorname{Spin}(2n
+1)$ holds for $n$ up to $5$\; it remains open for $n=6$\, $n=7$\, and eve
ry $n>10$.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eva Bayer-Fluckiger (EPFL Lausanne)
DTSTART;VALUE=DATE-TIME:20211013T153000Z
DTEND;VALUE=DATE-TIME:20211013T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/56
DESCRIPTION:Title: Isometries of lattices and automorphisms of K3 surfaces\nby E
va Bayer-Fluckiger (EPFL Lausanne) as part of Quadratic forms\, linear alg
ebraic groups and beyond\n\n\nAbstract\nThe aim of this talk is to give ne
cessary and sufficient conditions for an integral polynomial to be the cha
racteristic polynomial of an isometry of some even\, unimodular lattice of
given signature\, a result with applications to automorphisms of K3 surfa
ces.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Pepin Lehalleur (Radboud University Nijmegen)
DTSTART;VALUE=DATE-TIME:20211006T153000Z
DTEND;VALUE=DATE-TIME:20211006T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/57
DESCRIPTION:Title: Quadratic enumerative geometry and the Deligne-Milnor formula
\nby Simon Pepin Lehalleur (Radboud University Nijmegen) as part of Quadra
tic forms\, linear algebraic groups and beyond\n\n\nAbstract\nClassical en
umerative geometry often involves identities between coherent and topologi
cal (or motivic) invariants. For instance\, the Deligne-Milnor formula exp
resses the Euler characteristic of the vanishing cycles at an isolated hyp
ersurface singularity in terms of the Jacobi algebra of the singularity. I
n quadratic enumerative geometry\, numerical invariants are refined into c
lasses in the Grothendieck-Witt ring of the base field. On the coherent si
de\, this refinement process involves Grothendieck duality\, while on the
motivic side\, it involves stable motivic homotopy theory. Both sides of t
he Deligne-Milnor formula admit such a natural quadratic refinement. In a
joint work with Marc Levine and Vasudevan Srinivas\, we compute both sides
in a class of simple examples of singularities and show that the two side
s do not match up: correction terms appear\, whose origin is still mysteri
ous.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joshua Ruiter (Michigan State University)
DTSTART;VALUE=DATE-TIME:20211124T163000Z
DTEND;VALUE=DATE-TIME:20211124T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/58
DESCRIPTION:Title: Abstract homomorphisms of some special unitary groups\nby Jos
hua Ruiter (Michigan State University) as part of Quadratic forms\, linear
algebraic groups and beyond\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Bate (University of York)
DTSTART;VALUE=DATE-TIME:20211208T163000Z
DTEND;VALUE=DATE-TIME:20211208T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/59
DESCRIPTION:Title: Overgroups of regular unipotent elements in algebraic groups\
nby Michael Bate (University of York) as part of Quadratic forms\, linear
algebraic groups and beyond\n\n\nAbstract\nI will talk about a recent pape
r with Ben Martin and Gerhard Roehrle on subgroups containing regular unip
otent elements in reductive algebraic groups. The main result which I will
describe is not itself new (it is due to Testerman and Zalesski)\, but th
e new proof we came up with is worth sharing\, since it is very short\, fr
ee from the case-checking of the original\, and only rests on quite well-k
nown basic properties of algebraic groups. I will also describe some gener
alisations and extensions.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Wittenberg (Université Sorbonne Paris Nord)
DTSTART;VALUE=DATE-TIME:20211215T163000Z
DTEND;VALUE=DATE-TIME:20211215T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/60
DESCRIPTION:Title: Massey products in the Galois cohomology of number fields\nby
Olivier Wittenberg (Université Sorbonne Paris Nord) as part of Quadratic
forms\, linear algebraic groups and beyond\n\n\nAbstract\n(Joint work wit
h Yonatan Harpaz.) Let k be a field and p be a prime.\nAccording to a con
jecture of Mináč and Tân\, Massey products of $n>2$ classes\nin $H^1(k\
,\\mathbb Z/p \\mathbb Z)$ should vanish whenever they are defined. We es
tablish this\nconjecture when $k$ is a number field\, for any $n$. This c
onstraint on the\nabsolute Galois group of k was previously known to hold
when $n=3$ and\nwhen $n=4$\, $p=2$.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Asher Auel (Dartmouth College)
DTSTART;VALUE=DATE-TIME:20220119T163000Z
DTEND;VALUE=DATE-TIME:20220119T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/61
DESCRIPTION:Title: Brauer classes split by genus one curves\nby Asher Auel (Dart
mouth College) as part of Quadratic forms\, linear algebraic groups and be
yond\n\n\nAbstract\nIt is an open problem\, even over the rational numbers
\, to decide whether every Brauer class is split by the function field of
a genus one curve. The problem has been solved for Brauer classes of index
at most 6 over any field. In this talk\, I’ll report on work with Ben A
ntieau relating the problem to the arithmetic of modular curves and method
s from explicit descent for elliptic curves\, which in particular allow us
to solve the case of index 7 over number fields.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alberto Elduque (Universidad de Zaragoza)
DTSTART;VALUE=DATE-TIME:20220126T163000Z
DTEND;VALUE=DATE-TIME:20220126T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/62
DESCRIPTION:Title: Gradings on simple Lie algebras\nby Alberto Elduque (Universi
dad de Zaragoza) as part of Quadratic forms\, linear algebraic groups and
beyond\n\n\nAbstract\nAfter reviewing the basic definitions about gradings
\, it will be shown how gradings by abelian groups on a (not necessarily a
ssociative) algebra correspond to morphisms from diagonalizable group sche
mes to the automorphism group scheme of the algebra. This is the clue to c
lassify gradings on simple Lie algebras. The known classification results
of such gradings will be surveyed.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Utiralova (Massachusetts Institute of Technology)
DTSTART;VALUE=DATE-TIME:20220216T163000Z
DTEND;VALUE=DATE-TIME:20220216T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/63
DESCRIPTION:Title: Harish-Chandra bimodules in complex rank\nby Alexandra Utiral
ova (Massachusetts Institute of Technology) as part of Quadratic forms\, l
inear algebraic groups and beyond\n\n\nAbstract\nDeligne tensor categories
are defined as an interpolation of the categories of representations of g
roups $\\operatorname{GL}_n$\, $\\operatorname{O}_n$\, $\\operatorname{Sp}
_{2n}$ or $\\operatorname{S}_n$ to the complex values of the parameter n.
One can extend many classical representation-theoretic notions and constru
ctions to this context. These complex rank analogs of classical objects pr
ovide insights into their stable behavior patterns as n goes to infinity.
I will talk about some of my results on Harish-Chandra bimodules in the De
ligne categories. It is known that in the classical case simple Harish-Cha
ndra bimodules admit a classification in terms of W-orbits of certain pair
s of weights. However\, the notion of weight is not well-defined in the se
tting of the Deligne categories. I will explain how in complex rank the ab
ove-mentioned classification translates to a condition on the correspondin
g (left and right) central characters. Another interesting phenomenon aris
ing in complex rank is that there are two ways to define Harish-Chandra bi
modules. That is\, one can either require that the center acts locally fin
itely on a bimodule M or that M has a finite K-type. The two conditions ar
e known to be equivalent for a semi-simple Lie algebra in the classical se
tting\, however\, in Deligne categories that is no longer the case. I will
talk about a way to construct examples of Harish-Chandra bimodules of fin
ite K-type using the ultraproduct realization of Deligne categories.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander S. Sivatski (Universidade Federal do Rio Grande do Norte
)
DTSTART;VALUE=DATE-TIME:20220202T163000Z
DTEND;VALUE=DATE-TIME:20220202T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/64
DESCRIPTION:Title: Nonstandard quadratic forms over rational function fields\nby
Alexander S. Sivatski (Universidade Federal do Rio Grande do Norte) as pa
rt of Quadratic forms\, linear algebraic groups and beyond\n\nAbstract: TB
A\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roman Fedorov (University of Pittsburg)
DTSTART;VALUE=DATE-TIME:20220209T163000Z
DTEND;VALUE=DATE-TIME:20220209T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/65
DESCRIPTION:Title: On the purity conjecture of Nisnevich for torsors under reductive
group schemes\nby Roman Fedorov (University of Pittsburg) as part of
Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nLet $R
$ be a regular semilocal integral domain containing an infinite field $k$.
Let $f$ be an element of $R$ that does not belong to the square of any ma
ximal ideal of $R$ (equivalently\, the hypersurface $\\{f=0 \\}$ is regula
r). Let $G$ be a reductive group scheme over $R$. Under an isotropy assump
tion on $G$ we show that a $G$-torsor over the localization $R_f$ is trivi
al\, provided it is rationally trivial.\n\nThe statement is derived from i
ts abstract version concerning Nisnevich sheaves satisfying some propertie
s. Note that if $f=1$\, then we recover the conjecture of Grothendieck and
Serre (already known for regular semilocal rings containing fields). The
proof of Nisnevich conjecture follows the same strategy except that one ne
eds an additional statement concerning G-torsors defined on the complement
of a subscheme of $A^1_R$ that is etale and finite over $R$.\n\nIf time p
ermits\, we will also explain that the aforementioned isotropy assumption
is necessary.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Susanne Pumpluen (University of Nottingham)
DTSTART;VALUE=DATE-TIME:20220302T163000Z
DTEND;VALUE=DATE-TIME:20220302T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/66
DESCRIPTION:Title: Nonassociative algebras obtained from skew polynomials and their
applications\nby Susanne Pumpluen (University of Nottingham) as part o
f Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nUsin
g skew polynomials\, we define a class of unital nonassociative algebras i
ntroduced by Petit in 1966 (but largely ignored so far). Some of these alg
ebras are canonical generalizations of (associative) central simple algebr
as\, and classical results from Albert\, Amitsur and Jacobson can be gener
alized to this nonassociative setting. We discuss their structure and time
permitting also their use in coding theory. Their most prominent feature
is that their right nucleus is the eigenspace of the skew polynomial used
in their construction.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mathieu Florence (Institut de Mathématiques de Jussieu Sorbonne U
niversité)
DTSTART;VALUE=DATE-TIME:20220223T163000Z
DTEND;VALUE=DATE-TIME:20220223T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/67
DESCRIPTION:Title: Equivariant Witt vector bundles\nby Mathieu Florence (Institu
t de Mathématiques de Jussieu Sorbonne Université) as part of Quadratic
forms\, linear algebraic groups and beyond\n\n\nAbstract\nEquivariant Witt
vector bundles\, over a scheme of characteristic p>0\, were introduced in
collaboration with Charles De Clercq and Giancarlo Lucchini-Arteche. Thei
r purpose is to serve as a geometric tool\, in proving mod p^2 liftability
of mod p representations of smooth profinite groups- comprising the case
of Galois representations. This is still work in progress.\n\nI will expla
in some liftability statements (positive or negative) achieved so far\, an
d related techniques. I will then discuss a "general lifting statement"\,
which I believe is within reach.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fabio Tanania (LMU Munich)
DTSTART;VALUE=DATE-TIME:20220309T163000Z
DTEND;VALUE=DATE-TIME:20220309T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/68
DESCRIPTION:Title: On the motivic cohomology of the Nisnevich classifying space of P
GL_n\nby Fabio Tanania (LMU Munich) as part of Quadratic forms\, linea
r algebraic groups and beyond\n\n\nAbstract\nn this talk I will present th
e construction of a (kind of) Serre spectral sequence for motivic cohomolo
gy associated to a map of simplicial schemes with motivically cellular fib
er. Then I will show how to apply it in order to obtain information about
the motivic cohomology of the Nisnevich classifying space of projective ge
neral linear groups. At the end I will also give a description of the moti
ve of a Severi-Brauer variety in terms of twisted motives of its Čech sim
plicial scheme.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maneesh Thakur (Indian Statistical Institute)
DTSTART;VALUE=DATE-TIME:20220316T153000Z
DTEND;VALUE=DATE-TIME:20220316T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/69
DESCRIPTION:Title: The cyclicity problem for Albert division algebras\nby Manees
h Thakur (Indian Statistical Institute) as part of Quadratic forms\, linea
r algebraic groups and beyond\n\n\nAbstract\nAn old question of Adrian Alb
ert\, raised more than fifty years ago\, asks the following\n\nQuestion: d
oes every Albert division algebra over a field of characteristic different
from 2 and 3 contain a cyclic cubic subfield?\n\nThis was answered in the
affirmative by Petersson and Racine in 1984\, assuming the ground field c
ontains cube roots of unity and in characteristic 3\, by Petersson in 1999
.\nIn this talk we will describe a proof of\n\nTheorem: Let A be an Albert
division algebra over a field k of arbitrary characteristic. Then there i
s an isotope of A that contains a cyclic cubic extension.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Antieau
DTSTART;VALUE=DATE-TIME:20220518T153000Z
DTEND;VALUE=DATE-TIME:20220518T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/70
DESCRIPTION:Title: Benjamin Antieu's talk is postponed to June 15\nby Benjamin A
ntieau as part of Quadratic forms\, linear algebraic groups and beyond\n\n
\nAbstract\nBenjamin Antieu's talk is postponed to June 15.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicole Lemire (University of Western Ontario)
DTSTART;VALUE=DATE-TIME:20220323T153000Z
DTEND;VALUE=DATE-TIME:20220323T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/71
DESCRIPTION:Title: Codimension 2 cycles of classifying spaces of low-dimensional alg
ebraic tori\nby Nicole Lemire (University of Western Ontario) as part
of Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nLet
T be an algebraic torus over a field F and let\n$CH^2(BT)$ be the Chow gr
oup of codimension 2 cycles in its classifying space. Following work of Bl
instein and Merkurjev on the structure of the torsion part of $CH^2(BT)$\,
Scavia\, in a recent preprint\, found an example of an algebraic torus wi
th non-trivial torsion in\n$CH^2(BT)$. In joint work with Alexander Neshit
ov\, we show computationally that the group $CH^2(BT)$ is torsion-free for
all algebraic tori of dimension at most 5 and determine the conjugacy cla
sses of\nfinite subgroups of $\\operatorname{GL}_6(\\mathbb Z)$ which corr
espond to 6-dimensional tori\nwith nontrivial torsion in $CH^2(BT)$. Some
interesting properties\nof the structure of low-dimensional algebraic tori
will be discussed.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudio Quadrelli (University of Milan)
DTSTART;VALUE=DATE-TIME:20220330T153000Z
DTEND;VALUE=DATE-TIME:20220330T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/72
DESCRIPTION:Title: The Bogomolov-Positselski conjecture for maximal pro-p Galois gro
ups\nby Claudio Quadrelli (University of Milan) as part of Quadratic f
orms\, linear algebraic groups and beyond\n\n\nAbstract\nLet $p$ be a prim
e. In the '90s\, F. Bogomolov asked the following question: if $K$ is a fi
eld containing an algebraically closed field\, then is the closure of the
commutator subgroup of the Sylow pro-$p$ subgroup of the absolute Galois g
roup $G_{K}$ of $K$\, a free pro-$p$ group (cf. [1])?\nLater on\, L. Posit
selski generalized Bogomolov's question to the following:\nif $K$ is a fie
ld containing a root of 1 of order $p$ - and $\\sqrt[p^\\infty]{K}$ denote
s the compositum of all radical extensions $\\KK(\\sqrt[p^n]{a})$\, with $
n\\geq1$ and $a\\in K$ - then the maximal pro-$p$ Galois group of $\\sqrt[
p^\\infty]{K}$ is a free pro-$p$ group (cf. [3]).\n\nIn a recent work [4]\
, Thomas Weigel and myself translated Positselski's version of Bogomolov's
conjecture into purely group theoretic language\, and verified it for tho
se fields whose maximal pro-$p$ Galois group is a pro-$p$ group of element
ary type\, as defined by I. Efrat (e.g.\, local fields\, PAC fields\, $p$-
rigid fields\, algebraic extensions of global fields with finite $K^\\time
s/(K^\\times)^p$...).\n\nThis group-theoretic formulation is tightly relat
ed to a cohomological property enjoyed by maximal pro-$p$ Galois groups of
fields\, called ``Kummerian property'' (introduced in [2]\, and related\,
in turn\, to smooth profinite groups introduced by M. Florence)\, which m
ay be used also to detect pro-$p$ groups that do not occur as absolute Gal
ois groups --- if time allows I will show some significant examples.\n\n\n
[1] F. Bogomolov\, On the structure of Galois groups of the fields of rati
onal functions\, $K$-theory and algebraic geometry: connections with quadr
atic forms and division algebra. In: Proceedings of Symposia on Pure Mathe
matics\, 1992\, Santa Barbara CA\, vol. 58 (1995).\n\n[2] I. Efrat and C.
Quadrelli\, Efrat\, The Kummerian property and maximal pro-$p$ Galois grou
ps. J. Algebra 525 (2019).\n\n[3] L. Positselski\, Koszul property and Bog
omolov’s conjecture. Int. Math. Res. Not. 31 (2005).\n\n[4] C. Quadrelli
and Th. Weigel\, Oriented pro-$\\ell$ groups with the Bogomolov-Positsels
ki property. Res. Number theory 8\, no. 2 (2022).\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Holger Petersson (Fernuniversität in Hagen)
DTSTART;VALUE=DATE-TIME:20220406T153000Z
DTEND;VALUE=DATE-TIME:20220406T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/73
DESCRIPTION:Title: Integral octonions: history and perspectives\nby Holger Peter
sson (Fernuniversität in Hagen) as part of Quadratic forms\, linear algeb
raic groups and beyond\n\n\nAbstract\nAfter setting the stage by recalling
the basic properties of composition algebras over commutative rings\, I s
ketch the history of intergral octonions\, from its infancy in the 1860s t
o Coxeter’s groundbreaking paper of 1946. Inspired by results due to Mah
ler (1942) and Allcock (1999)\, I proceed to describe the one-sided ideal
structure of octonion algebras over arbitrary commutative rings. The lectu
re concludes with a non-orthogonal version of the classical Cayley-Dickson
construction that allows for a description of integral octonions (more pr
ecisely\, of their multiplicative structure) in an intrinsic manner.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aravind Asok (University of Southern California)
DTSTART;VALUE=DATE-TIME:20220601T153000Z
DTEND;VALUE=DATE-TIME:20220601T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/74
DESCRIPTION:Title: On P^1-stabilization in unstable motivic homotopy theory\nby
Aravind Asok (University of Southern California) as part of Quadratic form
s\, linear algebraic groups and beyond\n\n\nAbstract\nI will discuss joint
work with Tom Bachmann and Mike Hopkins regarding an analog of the Freude
nthal suspension theorem in unstable motivic homotopy theory. To motivate
the result\, I will quickly introduce the unstable motivic homotopy categ
ory and discuss some concrete applications. Time permitting\, I will sket
ch the key idea behind the proof.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fatma Kader Bingöl (University of Antwerpen)
DTSTART;VALUE=DATE-TIME:20220525T153000Z
DTEND;VALUE=DATE-TIME:20220525T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/75
DESCRIPTION:Title: On the 4-torsion part of the Brauer group\nby Fatma Kader Bin
göl (University of Antwerpen) as part of Quadratic forms\, linear algebra
ic groups and beyond\n\n\nAbstract\nThe 4-torsion part of the Brauer group
of a field $F$ is generated by cyclic algebras of degree $2$ and $4$. Thi
s has been known for the case when $-1$ is a square in $F$. I will present
a proof for this statement without any assumption on $F$. In the same con
text\, one further obtains a bound on the index of exponent-$4$ algebras o
ver $F$ in terms of the $u$-invariant of $F$. This is joint work with K.J.
Becher.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Max Lieblich (University of Washington)
DTSTART;VALUE=DATE-TIME:20220622T153000Z
DTEND;VALUE=DATE-TIME:20220622T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/76
DESCRIPTION:Title: Murphy's Law for gerbes\nby Max Lieblich (University of Washi
ngton) as part of Quadratic forms\, linear algebraic groups and beyond\n\n
\nAbstract\nThis is a report on joint work with Daniel Bragg. We study wha
t gerbes are possible as residual gerbes in natural moduli stacks. Among o
ther things\, we show that every gerbe with finite structure group arises
as a residual gerbe in the stack of smooth projective curves. This gives e
xamples of "versal gerbes" that arise organically in algebraic geometry. T
he key to producing such things is the transport of various classical cons
tructions from equivariant geometry to corresponding constructions for spa
ces over non-trivial gerbes.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Danny Ofek (University of British Columbia)
DTSTART;VALUE=DATE-TIME:20220706T153000Z
DTEND;VALUE=DATE-TIME:20220706T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/77
DESCRIPTION:by Danny Ofek (University of British Columbia) as part of Quad
ratic forms\, linear algebraic groups and beyond\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ido Efrat (Ben-Gurion University)
DTSTART;VALUE=DATE-TIME:20220629T153000Z
DTEND;VALUE=DATE-TIME:20220629T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/78
DESCRIPTION:Title: Steinberg Relations for Massey Products\nby Ido Efrat (Ben-Gu
rion University) as part of Quadratic forms\, linear algebraic groups and
beyond\n\n\nAbstract\nLet $F$ be a field of characteristic prime to $m$ wh
ich contains the $m$th roots of unity\, and let $G_F$ be its absolute Galo
is group.\n\nAs shown by Tate\, for $a\\neq 0\,1$ in $F$\, the Kummer elem
ents $(a)_F$\, $(1-a)_F$ in $H^1(G_F\,\\mathbb{Z}/m)$ have trivial cup pro
duct.\n\nIn fact\, by the celebrated Voevodsky-Rost theorem\, this relatio
n completely determines the cohomology ring $H^{\\bullet}(G_F\,\\mathbb Z/
m)$ with the cup product.\n\nA natural generalization of the cup product i
s the $n$-fold Massey product\, where $n\\geq2$. Extending results by Kirs
ten Wickelgren\, we show how Tate's relation generalizes to the Massey pro
duct context.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Goncalo Tabuada (University of Warwick)
DTSTART;VALUE=DATE-TIME:20220713T153000Z
DTEND;VALUE=DATE-TIME:20220713T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/79
DESCRIPTION:by Goncalo Tabuada (University of Warwick) as part of Quadrati
c forms\, linear algebraic groups and beyond\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Antieau (Northwestern University)
DTSTART;VALUE=DATE-TIME:20220615T153000Z
DTEND;VALUE=DATE-TIME:20220615T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/80
DESCRIPTION:Title: The K-theory of Z/p^n\nby Benjamin Antieau (Northwestern Univ
ersity) as part of Quadratic forms\, linear algebraic groups and beyond\n\
n\nAbstract\nI will report on joint work with Achim Krause and Thomas Niko
laus where we give an algorithm to compute the K-groups of rings such as $
Z/p^n$.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eliahu Matzri (Bar Ilan University)
DTSTART;VALUE=DATE-TIME:20220608T153000Z
DTEND;VALUE=DATE-TIME:20220608T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/81
DESCRIPTION:Title: On the symbol length of symbols in Galois cohomology.\nby Eli
ahu Matzri (Bar Ilan University) as part of Quadratic forms\, linear algeb
raic groups and beyond\n\n\nAbstract\nFix a prime $p$ and let $F$ be a fie
ld with characteristic not $p$. Let $G_F$ be the absolute Galois group of
$F$ and let $\\mu_{p^s}$ be the $G_F$-module of roots of unity of order\n
dividing $p^s$ in a fixed algebraic closure of $F$. Let $\\alpha \\in H^n
(F\,\\mu_{p^s}^{\\otimes n})$ be a symbol (i.e $\\alpha=a_1\\cup \\dots \\
cup a_n$ where $a_i\\in H^1(F\, \\mu_{p^s})$) with effective exponent divi
ding $p^{s-1}$ (that is $p^{s-1} \\alpha=0 \\in H^n(G_F\,\\mu_p^{\\otimes
n}))$. In this talk I will explain how to write $\\alpha$ as a sum of symb
ols coming from $H^n(F\,\\mu_{p^{s-1}}^{\\otimes n})$ that is symbols of t
he form $p\\gamma$ for $\\gamma \\in H^n(F\,\\mu_{p^s}^{\\otimes n})$. If
$n>3$ and $p\\neq 2$ we assume $F$ is prime to $p$ closed and of character
istic zero.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karim Johannes Becher (University of Antwerp)
DTSTART;VALUE=DATE-TIME:20220928T153000Z
DTEND;VALUE=DATE-TIME:20220928T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/82
DESCRIPTION:Title: Fields with bounded Brauer 2-torsion index\nby Karim Johannes
Becher (University of Antwerp) as part of Quadratic forms\, linear algebr
aic groups and beyond\n\n\nAbstract\nAlexander Merkurjev’s groundbreakin
g results from the 1980ies showed that the $u$-invariant (maximal dimensio
n of an anisotropic quadratic form) of a field is related to the degrees o
f division algebras of exponent $2$. \n\nThis relation led Bruno Kahn to c
onjecture that the $u$-invariant is bounded in terms of the $2$-symbol len
gth (the number of quaternion algebras necessary to represent an arbitrary
element in the $2$-torsion of the Brauer group). He showed that if every
central simple algebra of exponent $2$ over a field $F$ of characteristic
not $2$ is equivalent to a tensor product of n quaternion algebras\, then
every $(2n+2)$-fold Pfister form over F is split after adjoining the squar
e-root of $-1$.\n\nIn my talk I want to present a variation and refinement
of this observation.\nAssuming that every central division algebra of exp
onent $2$ over $F$ has degree at most $2^n$\, I show that every $(2n+2)$-f
old Pfister form is hyperbolic if $-1$ is a sum of squares in $F$ and that
it is in any case equal to 8 times a $(2n-1)$-fold Pfister form.\nThe pro
of of this result is based on computations with trace forms of central sim
ple algebras.\nUsing this fact\, one can now remove in a result of Daniel
Krashen from 2016\, which characterises fields for which all symbol length
s in the Milnor $K$-groups modulo $2$ of a field are bounded\, the conditi
on that $-1$ be a square. In fact\, these fields are the same as those wit
h finite $u$-invariant (in the sense of Elman-Lam) and finite stability in
dex. This latter result is joint work in progress with Saurabh Gosavi.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stephen Scully (University of Victoria)
DTSTART;VALUE=DATE-TIME:20221123T163000Z
DTEND;VALUE=DATE-TIME:20221123T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/83
DESCRIPTION:Title: On the dimensions of quadratic forms isotropic over the function
field of a quadric\nby Stephen Scully (University of Victoria) as part
of Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nLe
t $p$ and $q$ be anisotropic quadratic forms over a field of any character
istic\, and $i$ the isotropy index of q over the function field of the qua
dric defined by p. In 2018\, we proposed a conjecture that constrains the
dimension of $q$ in terms of $i$ and the largest power of 2 strictly less
than the dimension of $p$. This can be viewed as a generalization of the "
separation theorem" originally proved by Hoffmann over fields of character
istic not 2. In this talk\, we'll explain that the conjecture is true when
$q$ is a quasilinear (i.e.\, diagonalizable) form over a field of charact
eristic 2. The proof\, which is elementary\, reveals a much stronger const
raint involving certain stable birational invariants of $p$ (which must al
so be quasilinear in order for the statement to be non-trivial in this cas
e). Examining the contribution from the "Izhboldin dimension" of $p$\, we
are led to formulate (for all forms in any characteristic) a strong versio
n of our original conjecture that incorporates other important results due
to Karpenko and Karpenko-Merkurjev. After discussing the quasilinear case
\, we'll explain what is known when $q$ is non-singular. Unlike the quasil
inear case\, the results here rely on algebraic-geometric methods.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Federico Scavia (UCLA)
DTSTART;VALUE=DATE-TIME:20221207T163000Z
DTEND;VALUE=DATE-TIME:20221207T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/84
DESCRIPTION:Title: Degenerate fourfold Massey products over arbitrary fields\nby
Federico Scavia (UCLA) as part of Quadratic forms\, linear algebraic grou
ps and beyond\n\n\nAbstract\nWe prove that\, for all fields $F$ of charact
eristic different from $2$ and all $a\,b\,c \\in F^*$\, the mod 2 Massey p
roduct $\\left< a\,b\,c\,a \\right>$ vanishes as soon as it is defined. Fo
r every field $E$ of characteristic different form $2$\, we construct a fi
eld $F$ containing $E$ and $a\,b\,c\,d \\in F^*$ such that $\\left< a\,b\,
c \\right>$ and $\\left< b\,c\,d \\right>$ vanish but $\\left< a\,b\,c\,d
\\right>$ is not defined. As a consequence\, we answer a question of Posit
selski by constructing the first examples of fields containing all roots o
f unity and such that the mod 2 cochain DGA of the absolute Galois group i
s not formal. This is joint work with Alexander Merkurjev.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Zhykhovich (LMU Munich)
DTSTART;VALUE=DATE-TIME:20221109T163000Z
DTEND;VALUE=DATE-TIME:20221109T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/85
DESCRIPTION:Title: The J-invariant of algebras with orthogonal involution.\nby M
axim Zhykhovich (LMU Munich) as part of Quadratic forms\, linear algebraic
groups and beyond\n\n\nAbstract\nThe $J$-invariant of a semi-simple algeb
raic group $G$ was introduced by Petrov\, Semenov and Zainoulline in 2008.
The $J$-invariant is a discrete invariant which encodes the motivic decom
position of the variety of Borel subgroups in $G$ (in this talk we conside
r Chow motives). Let $(A\, \\sigma)$ be a central simple algebra with orth
ogonal involution and trivial discriminant. The $J$-invariant of $(A\,\\si
gma)$ is defined as $J(\\mathrm{PGO}^+(A\,\\sigma))$. In this talk I will
discuss a conjecture of Quéguiner Mathieu\, Semenov and Zainoulline\, whi
ch allows to reduce the computation of $J(A\,\\sigma)$ to the case of quad
ratic forms.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Barry Demba (University of Bamako\, Mali)
DTSTART;VALUE=DATE-TIME:20221116T163000Z
DTEND;VALUE=DATE-TIME:20221116T173000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/86
DESCRIPTION:Title: Degree 3 relative invariant for unitary involutions\nby Barry
Demba (University of Bamako\, Mali) as part of Quadratic forms\, linear a
lgebraic groups and beyond\n\n\nAbstract\nThe Arason invariant in quadrati
c form theory is a degree 3 cohomological invariant attached to an even-di
mensional quadratic form with trivial discriminant and trivial Clifford in
variant. It is known that this invariant can also be described in terms of
the Rost invariant of a split Spin group. More generally\, using the Rost
invariant for non split Spin groups\, and for absolutely almost simple si
mply connected groups of other types\, one may try to define analogues of
the Arason invariant for the underlying algebraic objects\, namely hermiti
an forms and involutions. For involutions of the first kind\, degree 3 coh
omological invariants were investigated by several authors.\nIn this talk\
, following a suggestion of Tignol\, I will present the case of unitary in
volutions\, which correspond to groups of outer type A. Notice that the Ar
ason invariant for quadratic forms may be used to define an invariant for
unitary involutions with trivial discriminant algebra on split algebras. N
evertheless\, as for orthogonal involutions\, this invariant does not exte
nd in a functorial way to the non split case. Using the Rost invariant for
some torsors\, we define a relative Arason invariant for unitary involuti
ons. An important feature in this talk is that we do not restrict to invol
utions with trivial lower-degree invariants\, but also consider pairs of u
nitary involutions with isomorphic discriminant algebras. The end of the t
alk will be devoted to the properties of this relative invariant for algeb
ras of degree 8.\n\n(Joint work with A. Masquelein and A. Quéguiner-Mathi
eu)\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Martin (University of Aberdeen)
DTSTART;VALUE=DATE-TIME:20221012T153000Z
DTEND;VALUE=DATE-TIME:20221012T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/87
DESCRIPTION:Title: Geometric invariant theory for reductive groups over non-algebrai
cally closed fields\nby Benjamin Martin (University of Aberdeen) as pa
rt of Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\n
Let $G$ be a reductive linear algebraic group acting on an affine variety
$X$ over a field $k$. If $k$ is algebraically closed then the Hilbert-Mumf
ord Theorem gives a powerful tool for understanding the structure of $G(k)
$-orbits in $X$: an orbit is closed if and only if it is closed under taki
ng limits along cocharacters. Moreover\, if an orbit is not closed then on
e can reach a closed orbit by taking a limit along a cocharacter\, and by
work of Hesselink/Kempf/Rousseau one can choose this cocharacter in a cano
nical way.\n\nNow suppose $k$ is arbitrary\, and let $x\\in X(k)$. We say
the orbit $G(k)\\cdot x$ is cocharacter-closed over $k$ if it is closed un
der taking limits of $k$-defined cocharacters. There is a version of the H
ilbert-Mumford Theorem which holds in this more general setting. I will di
scuss the notion of cocharacter-closure and its interactions with the theo
ry of spherical buildings and the theory of $G$-complete reducibility.\n\n
This talk is based on joint work with Michael Bate\, Sebastian Herpel\, Ge
rhard R\\"ohrle and Rudolf Tange.\n\n\nReferences:\n\nBate M.E.\, Herpel S
.\, Martin B.\, Röhrle G. Cocharacter-closure and the rational Hilbert-Mu
mford Theorem. Math. Zeit. 287 (2017)\, 39–72.\nOpen access link: https:
//link.springer.com/article/10.1007/s00209-016-1816-5\n\nBate M.E.\, Marti
n B.\, Röhrle G.\, Tange R. Closed orbits and uniform S-instability in ge
ometric invariant theory. Trans. Amer. Math. Soc. 365 (2013)\, 3643–3673
.\nArxiv: https://arxiv.org/abs/0904.4853\n\nBate M.E.\, Martin B.\, Röhr
le G.\, Tange R. Complete reducibility and separable field extensions.C. R
. Math Acad. Sci. Paris 348 (2010)\, no. 9–10\, 495-497.\nArxiv: https:/
/arxiv.org/abs/1002.4319\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yashonidhi Pandey (Indian Institute of Science Education and Resea
rch\, Mohali)
DTSTART;VALUE=DATE-TIME:20221019T153000Z
DTEND;VALUE=DATE-TIME:20221019T163000Z
DTSTAMP;VALUE=DATE-TIME:20230925T230243Z
UID:Algebraicgroups/88
DESCRIPTION:Title: On Bruhat-Tits theory over a higher dimensional base\nby Yash
onidhi Pandey (Indian Institute of Science Education and Research\, Mohali
) as part of Quadratic forms\, linear algebraic groups and beyond\n\n\nAbs
tract\nThis is joint-work with Vikraman Balaji. The preprint is posted on
the arXiv.\n\nLet $\\mathcal O_{n} := k\\llbracket z_{1}\, \\ldots\, z_{n}
\\rrbracket$ over an algebraically closed residue field $k$ of characteris
tic zero. Set $K_{n}= {\\rm Fract}~\\mathcal{O}{n}$. Let $G$ be an almost-
simple\, simply-connected affine algebraic group over $k$ with a maximal t
orus $T$ and a Borel subgroup $B$. Given a $n$-tuple ${\\bf f} = (f_{1}\,
\\ldots\, f_{n})$ of concave functions on the root system of $G$ as in Bru
hat-Tits\, we define $n$-bounded subgroups $\\mathcal{P}_{\\bf f}(k)\\subs
et G(K_{n})$ as a direct generalization of Bruhat-Tits groups for the case
$n=1$. We show that these groups are schematic\, i.e. they are valued poi
nts of smooth quasi-affine group schemes with connected fibres and adapted
to the divisor with normal crossing $z_1 \\cdots z_n =0$ in the sense tha
t the restriction to the generic point of the divisor $z_i=0$ is given by
$f_i$. This provides a higher-dimensional analogue of the Bruhat-Tits grou
p schemes with natural specialization properties. Under suitable tameness
assumptions\, we extend all these results for a $n+1$-tuple ${\\bf f} = (f
_{0}\, \\ldots\, f_{n})$ of concave functions on the root system of $G$ re
placing $\\mathcal O_{n}$ by $\\mathcal{O} \\llbracket x_{1}\,\\cdots\,x_{
n} \\rrbracket$\, where $\\mathcal O$ is a complete discrete valuation rin
g with residue field of characteristic $p$. In particular\, if $x_0$ is th
e uniformizer of $\\mathcal{O}$\, then the group scheme is adapted to the
divisor $x_0 \\cdots x_n=0$. If time permits\, we may talk of applications
.\n
LOCATION:https://researchseminars.org/talk/Algebraicgroups/88/
END:VEVENT
END:VCALENDAR