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BEGIN:VEVENT
SUMMARY:David Stewart (University of Newcastle)
DTSTART:20200515T150000Z
DTEND:20200515T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/1
DESCRIPTION:Title: Irreducible modules for pseudo-reductive groups\nby Dav
id Stewart (University of Newcastle) as part of Quadratic forms\, linear a
lgebraic groups and beyond\n\n\nAbstract\n(Jt with Michael Bate) For any s
mooth connected group G over an arbitrary field k\, its irreducible module
s are in 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 imperfect\, a pseudo-reductive group may not be reductive. That means
that over the algebraic closure of k\, one sees some unipotent radical wh
ich is not visible over k. If G has a split maximal torus\, much of the th
eory of split reductive groups carries over and we give dimension formulae
for irreducible G-modules which reduce the study to the split reductive c
ase and commutative pseudo-reductive case.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Duncan (University of South Carolina)
DTSTART:20200522T150000Z
DTEND:20200522T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/2
DESCRIPTION:Title: Cohomological invariants and separable algebras\nby Ale
xander Duncan (University of South Carolina) as part of Quadratic forms\,
linear algebraic groups and beyond\n\n\nAbstract\nA separable algebra over
a field k is a finite direct sum of central simple algebras over finite s
eparable extensions of k. It is natural to attach separable algebras to k-
forms of algebraic objects. The fundamental example is the central simple
algebra corresponding to a Severi-Brauer variety. Blunk considered a pair
of Azumaya algebras attached to a del Pezzo surface of degree 6. More gene
rally\, one can consider endomorphism algebras of exceptional objects in d
erived categories. Alternatively\, one can view these constructions as coh
omological invariants of degree 2 with values in quasitrivial tori.\n\nIn
the case of Severi-Brauer varieties and Blunk's example of del Pezzo surfa
ces of degree 6\, these invariants suffice to completely determine the iso
morphism classes of the underlying objects. However\, in general they are
not sufficient. We characterize which k-forms can be distinguished from on
e another using the theory of coflasque resolutions of reductive algebraic
groups. Moreover\, we discuss connections to rationality questions and to
the Tate-Shafarevich group for number fields. \n\nThis is based on joint
work with Matthew Ballard\, Alicia Lamarche\, and Patrick McFaddin.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Federico Scavia (University of British Columbia)
DTSTART:20200529T150000Z
DTEND:20200529T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/3
DESCRIPTION:Title: Codimension two cycles on classifying stacks of algebraic t
ori\nby Federico Scavia (University of British Columbia) as part of Qu
adratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nWe give
an example of an algebraic torus $T$ such that the group ${\\rm CH}^2(BT)_
{\\rm tors}$ is non-trivial. This answers a question of Blinstein and Merk
urjev.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Danny Krashen (Rutgers University)
DTSTART:20200605T150000Z
DTEND:20200605T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/4
DESCRIPTION:Title: Field patching\, local-global principles and rationality\nby Danny Krashen (Rutgers University) as part of Quadratic forms\, line
ar algebraic groups and beyond\n\n\nAbstract\nThis talk will describe loca
l-global principles for torsors for algebraic groups over a semiglobal fie
ld - that is\, a one variable function field over a complete discretely va
lued field.\nIn particular\, I will describe recent joint work with Collio
t-Thélène\, Harbater\, Hartmann\, Parimala and Suresh in which we connec
t this question in certain cases to questions of R-equivalence for the gro
up\, and in some cases are able to give finiteness results and combinatori
al descriptions for the obstruction to local-global principles.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roberto Pirisi (KTH Royal Institute of Technology)
DTSTART:20200612T150000Z
DTEND:20200612T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/5
DESCRIPTION:Title: Brauer groups of moduli of hyperelliptic curves\, via cohom
ological invariants\nby Roberto Pirisi (KTH Royal Institute of Technol
ogy) as part of Quadratic forms\, linear algebraic groups and beyond\n\n\n
Abstract\nGiven an algebraic variety X\, the Brauer group of X is the grou
p of Azumaya algebras over X\, or equivalently the group of Severi-Brauer
varieties over X\, i.e. fibrations over X which are étale locally isomorp
hic to a projective space. It was first studied in the case where X is the
spectrum of a field by Noether and Brauer\, and has since became a centra
l object in algebraic and arithmetic geometry\, being for example one of t
he first obstructions to rationality used to produce counterexamples to No
ether's problem of whether given a representation V of a finite group G th
e quotient V/G is rational. While the Brauer group has been widely studied
for schemes\, computations at the level of moduli stacks are relatively r
ecent\, the most prominent of them being the computations by Antieau and M
eier of the Brauer group of the moduli stack of elliptic curves over a var
iety of bases\, including Z\, Q\, and finite fields.\nIn a recent joint wo
rk with A. Di Lorenzo\, we use the theory of cohomological invariants\, an
d its extension to algebraic stacks\, to completely describe the Brauer gr
oup of the moduli stacks of hyperelliptic curves over fields of characteri
stic zero\, and the prime-to-char(k) part in positive characteristic. It t
urns out that the (non-trivial part of the) group is generated by cyclic a
lgebras\, by an element coming from a map to the classifying stack of éta
le algebras of degree 2g+2\, and when g is odd by the Brauer-Severi fibrat
ion induced by taking the quotient of the universal curve by the hyperelli
ptic involution. This paints a richer picture than in the case of elliptic
curves\, where all non-trivial elements come from cyclic algebras.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maike Gruchot (University of Aachen)
DTSTART:20200619T150000Z
DTEND:20200619T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/6
DESCRIPTION:Title: Variations of G-complete reducibility\nby Maike Gruchot
(University of Aachen) as part of Quadratic forms\, linear algebraic grou
ps and beyond\n\n\nAbstract\nIn this talk we discuss variations of Serre
’s notion 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 r
educible 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 num
ber of fundamental results from the absolute to the relative setting.\nBy
results of Serre and Bate–Martin–Röhrle\, the usual notion of $G$-com
plete reducibility can be re-framed as a property of an action of a group
on the spherical building of the identity component of $G$. We discuss tha
t other variations of this notion\, such as relative complete reducibility
and σ-complete reducibility which can also be viewed as special cases of
this building-theoretic definition.\nThis is based on joint work with A.
Litterick and G. Röhrle.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Burt Totaro (UCLA)
DTSTART:20200626T150000Z
DTEND:20200626T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/7
DESCRIPTION:Title: Cohomological invariants in positive characteristic\nby
Burt Totaro (UCLA) as part of Quadratic forms\, linear algebraic groups a
nd beyond\n\n\nAbstract\nWe determine the mod p cohomological invariants f
or several affine group schemes G in chararacteristic p. These are invaria
nts of G-torsors with values in etale motivic cohomology\, or equivalently
in Kato's version of Galois cohomology based on differential forms. In pa
rticular\, we find the mod 2 cohomological invariants for the symmetric gr
oups and the orthogonal groups in characteristic 2\, which Serre computed
in characteristic not 2.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benedict Williams (University of British Columbia)
DTSTART:20200710T150000Z
DTEND:20200710T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/8
DESCRIPTION:Title: Algebras requiring many generators\nby Benedict William
s (University of British Columbia) as part of Quadratic forms\, linear alg
ebraic groups and beyond\n\n\nAbstract\nA result of Forster says that if R
is a noetherian ring of (Krull) dimension d\, then a rank-n projective mo
dule over R can be generated by d+n elements\, and results of Chase and Sw
an imply that this bound is sharp—there exist examples that cannot be ge
nerated 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 multip
lication is identically 0. We take the results of Forster\, Chase and Swan
as a starting point for investigations into forms of other algebras.\n\nF
ix a field 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 th
e variety of r-tuples of elements that generate B as a k-algebra. In favou
rable circumstances\, U_r/G is a k-variety\, generalizing the Grassmannian
\, that classifies forms of the algebra B equipped with r generators. In a
ddition\, as far as A1-invariant cohomology theories are concerned U_r/G a
pproximates 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 algeb
ras (over a ring R) requiring many generators\, generalizing the example o
f Chase and Swan. I will tell a fuller version of this story\, with emphas
is on the case where B is a matrix algebra\, so that U_r/G classifies Azum
aya algebras with r generators. The majority of the talk concerns joint wo
rk with Uriya First and Zinovy Reichstein\, but I will mention some joint
work with Taeuk Nam & Cindy Tan and some independent work of Sebastian Gan
t.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maurice Chayet (ECAM-EPMI)
DTSTART:20200703T150000Z
DTEND:20200703T160000Z
DTSTAMP:20260404T110655Z
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 pa
rt of Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\n
T.A. Springer knew decades ago of the existence of a Group invariant commu
tative algebra structure on the 3875 dimensional representation of $E_8$.
It was recently shown by S. Garibaldi and R. Guralnick that the automorphi
sm group of this unique commutative algebra coincides with $E_8$. However
a description 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 wit
h Skip Garibaldi\, we explain a correspondence which associates to each si
mple Lie algebra\, a commutative non associative unital algebra\, and prov
ide an explicit closed form expression for the product. This correspondenc
e encompasses the 3875 invariant algebra for $E_8$ via the addition of a u
nit. These algebras turn out to be simple and are endowed with a non-degen
erate “associative” bilinear invariant form. Unlike their closet cousi
ns\, the Jordan Algebras\, these algebras are not power associative and sh
are the unusual property of having the unit interval as part of their Pier
ce Spectrum.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raman Parimala (Emory University)
DTSTART:20200914T150000Z
DTEND:20200914T160000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/10
DESCRIPTION:Title: A Hasse principle for simply connected groups\nby Rama
n Parimala (Emory University) as part of Quadratic forms\, linear algebrai
c groups 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
$ satisfies 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 typ
e other than $E_8$ and Chernousov for groups of type $E_8$. There were qu
estions and conjectures on similar Hasse principles over function fields o
f $p$-adic curves and more generally\, semi global fields\, i.e.\,\nfuncti
on fields of curves over complete discrete valued fields\, with respect to
all their discrete valuations. We shall discuss recent progress in this
direction.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikhail Borovoi (Tel Aviv University)
DTSTART:20200921T150000Z
DTEND:20200921T160000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/11
DESCRIPTION:Title: Galois cohomology of real reductive groups\nby Mikhail
Borovoi (Tel Aviv University) as part of Quadratic forms\, linear algebra
ic groups and beyond\n\n\nAbstract\nUsing ideas of Kac and Vinberg\, we gi
ve a simple combinatorial method of computing the Galois cohomology of sem
isimple groups over the field $\\mathbb R$ of real numbers. I will explain
the method by the examples of simple groups of type $E_7$ (both adjoint a
nd simply connected).\n\nThis is a joint work with Dmitry A. Timashev\, Mo
scow\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandre Lourdeaux (University of Lyon)
DTSTART:20200928T150000Z
DTEND:20200928T160000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/12
DESCRIPTION:Title: Brauer invariants of linear algebraic groups\nby Alexa
ndre Lourdeaux (University of Lyon) as part of Quadratic forms\, linear al
gebraic groups and beyond\n\n\nAbstract\nOur talk deals with the cohomolog
ical invariants of smooth and connected linear algebraic groups over an ar
bitrary field. The notion of cohomological invariants was formalized by Se
rre in the 90’s. It enables to study via Galois cohomology the geometry
of linear algebraic groups or forms of algebraic stuctures (such as centra
l simple algebras with involution).\n\nWe intend to introduce the general
ideas of the theory and to present a generalization of a result by Blinste
in and Merkurjev on degree 2 invariants with coefficients Q/Z(1)\, that is
invariants taking values in the Brauer group. More precisely our result g
ives a description of these invariants for every smooth and connected line
ar groups\, in particular for non reductive groups over an imperfect field
(as pseudo-reductive or unipotent groups for instance).\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Rapinchuk (Michigan State University)
DTSTART:20201005T150000Z
DTEND:20201005T160000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/13
DESCRIPTION:Title: Algebraic groups with good reduction\nby Igor Rapinchu
k (Michigan State University) as part of Quadratic forms\, linear algebrai
c groups 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 th
is talk is to report on recent work that focuses on good reduction in the
context of reductive linear algebraic groups over finitely generated field
s. In addition\, we will highlight some applications to the study of local
-global principles and the analysis of algebraic groups having the same ma
ximal tori. (Parts of this work are joint with V. Chernousov and A. Rapinc
huk.)\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Merkurjev (UCLA)
DTSTART:20201012T150000Z
DTEND:20201012T160000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/14
DESCRIPTION:Title: Operations in connective K-theory\nby Alexander Merkur
jev (UCLA) as part of Quadratic forms\, linear algebraic groups and beyond
\n\n\nAbstract\nThis is a joint work with A.Vishik. A relation between Cho
w theory 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 variety associated with the topological filtration. A much better relati
on can be established via connective K-theory that maps to both Chow theor
y and K-theory\, so the connective K-theory deserves detailed study.\n\nSt
eenrod operations (mod p) and Adams operations are essentially all additiv
e operations in Chow theory and K-theory respectively. In the talk we desc
ribe the ring of additive operations in connective K-theory.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cyril Demarche (Institut de Mathématiques de Jussieu)
DTSTART:20201019T150000Z
DTEND:20201019T160000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/15
DESCRIPTION:Title: Splitting families in Galois cohomology\nby Cyril Dema
rche (Institut 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 coho
mology class in $H^n(k\, A)$. We show that there exists a countable famili
y of (smooth\, geometrically integral) $k$-varieties\, such that the follo
wing holds: for any field extension $K/k$\, the restriction of $x$ vanishe
s in $H^n(K\, A)$ if and only if one of the varieties has an $K$-point. In
the case $n= 2$\, we note that one variety (called a splitting variety fo
r $x$) is enough. The question of the existence of splitting varieties (or
splitting families) is insprired by the construction of norm varieties fo
r symbols by Rost. This is joint work with Mathieu Florence.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cameron Ruether (University of Ottawa)
DTSTART:20201026T150000Z
DTEND:20201026T160000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/16
DESCRIPTION:Title: Injections from Kronecker Products and the Cohomological I
nvariants of Half-Spin\nby Cameron Ruether (University of Ottawa) as p
art 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 Se
rre\, degree $n$ cohomological invariants of $G$ with coefficients in a gr
oup $A$\, where $A$ is equipped with an action of the absolute Galois grou
p of $F$\, are natural transformations of Galois cohomology functors $H^1(
-\,G) \\to H^n(-\,A)$. Commonly studied are the degree three invariants wi
th coefficients in $\\mathbb{Q}/\\mathbb{Z} \\otimes \\mathbb{Q}/\\mathbb{
Z}$. These invariants were recently described by Merkurjev for the semisim
ple adjoint case\, and by Bermudez and Ruozzi for semisimple $G$ which are
neither simply connected nor adjoint. In particular\, they described the
structure of the normalized degree three invariants (those which send the
trivial object to zero) of the half-spin group $\\operatorname{HSpin}_{16}
$. By generalizing a technique of Garibaldi we construct new injections in
to $\\operatorname{HSpin}$ induced by the Kronecker tensor product map. In
particular we construct an injection $\\operatorname{PSp}_{2n} \\times \\
operatorname{PSp}_2m \\to \\operatorname{HSpin}_{4nm}$ which we use to des
cribe the normalized invariants of $\\operatorname{HSpin}_{4k}$ for any $k
$\, generalizing the result of Bermudez and Ruozzi.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anastasia Stavrova (St. Petersburg University)
DTSTART:20201102T160000Z
DTEND:20201102T170000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/17
DESCRIPTION:Title: Torsors of isotropic reductive groups over Laurent polynom
ials\nby Anastasia Stavrova (St. Petersburg University) as part of Qua
dratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nLet $k$ b
e a field of characteristic 0. Let $G$ be a reductive group over the ring
of Laurent polynomials $R=k[x_1^{\\pm 1}\,\\ldots\,x_n^{\\pm 1}]$. We prov
e 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 i
f this is the case\, then the natural map $H^1_{et}(R\,G)\\to H^1_{et}(k(x
_1\,\\ldots\,x_n)\,G)$ has trivial kernel and $G$ is loop reductive\, i.e.
$G$ contains a maximal $R$-torus. We also deduce that if $G$ is a reducti
ve group 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://stable.researchseminars.org/talk/Algebraicgroups/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lucy Moser-Jauslin (Université de Bourgogne)
DTSTART:20201109T160000Z
DTEND:20201109T170000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/18
DESCRIPTION:Title: Forms of almost homogeneous varieties\nby Lucy Moser-J
auslin (Université de Bourgogne) as part of Quadratic forms\, linear alge
braic groups and beyond\n\n\nAbstract\nIn this talk\, we will discuss almo
st homogeneous varieties for reductive groups over a perfect field $k$. Le
t $K$ be an algebraic closure of $k$\, and let $G$ be a connected reductiv
e $K$-group with a fixed $k$-form $F$. A normal $G$-variety over $K$ is al
most homogeneous if it has an open dense orbit. Given an almost homogeneo
us $G$-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 t
his\, we describe an action of the Galois group on the combinatorics devel
oped in Luna-Vust theory. This is joint work with Ronan Terpereau.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kęstutis Česnavičius (Université Paris-Sud)
DTSTART:20201116T160000Z
DTEND:20201116T170000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/19
DESCRIPTION:Title: Grothendieck–Serre in the split unramified case\nby
Kęstutis Česnavičius (Université Paris-Sud) as part of Quadratic forms
\, linear algebraic groups and beyond\n\n\nAbstract\nThe Grothendieck–Se
rre conjecture predicts that every generically trivial torsor under a redu
ctive group 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 tha
t have so far kept the mixed characteristic case out of reach\, we rely on
the recently-established Cohen–Macaulay version of the resolution of si
ngularities.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victor Petrov (St. Petersburg University)
DTSTART:20201123T160000Z
DTEND:20201123T170000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/20
DESCRIPTION:Title: Isotropy of Tits construction\nby Victor Petrov (St. P
etersburg University) as part of Quadratic forms\, linear algebraic groups
and beyond\n\n\nAbstract\nTits construction produces a Lie algebra out of
a composition algebra and an exceptional Jordan algebra. The type of the
result is $F_4$\, ${}^2E_6$\, $E_7$ or $E_8$ when the composition algebra
has dimension 1\,2\,4 or 8 respectively. Garibaldi and Petersson noted tha
t the Tits index ${}^2E_6^{35}$ cannot occur as a result of Tits construct
ion. Recently Alex Henke proved that the Tits index $E_7^{66}$ is also not
possible. We push the analogy further and show that Lie algebras of Tits
index $E_8^{133}$ don’t lie in the image of the Tits construction. The p
roof relies on basic facts about symmetric spaces and our joint result wit
h Garibaldi and Semenov about isotropy of groups of type $E_7$ in terms of
the Rost invariant. This is a part of a work in progress joint with Simon
Rigby.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Wertheim (UCLA)
DTSTART:20201130T160000Z
DTEND:20201130T170000Z
DTSTAMP:20260404T110655Z
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 alg
ebraic groups and beyond\n\n\nAbstract\nMany important algebraic objects c
an be viewed as $G$-torsors over a field $F$\, where $G$ is an algebraic g
roup over $F$. For example\, there is a natural bijection between $F$-isom
orphism classes of central simple $F$-algebras of degree n and $\\operator
name{PGL}_n(F)$-torsors over $\\operatorname{Spec}(F)$. Much as one may st
udy principal bundles on a manifold via characteristic classes\, one may l
ikewise study G-torsors over a field via certain associated Galois cohomol
ogy classes. This principle is made precise by the notion of a cohomologic
al invariant\, which was first introduced by Serre. \n\nIn this talk\, we
will determine the cohomological invariants for algebraic groups of multip
licative type with values in $H^{1}(-\, Q/Z(1))$. Our main technical analy
sis will center around a careful examination of $\\mu_n$-torsors over a sm
ooth\, connected\, reductive algebraic group. Along the way\, we will comp
ute a related group of invariants for smooth\, connected\, reductive group
s.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pavel Sechin (University of Regensburg)
DTSTART:20201207T160000Z
DTEND:20201207T170000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/22
DESCRIPTION:Title: Morava K-theory pure motives with applications to quadrics
\nby Pavel Sechin (University of Regensburg) as part of Quadratic form
s\, linear algebraic groups and beyond\n\n\nAbstract\nMorava K-theories $K
(n)$ are cohomology theories that have graded fields of positive character
istic as coefficient rings and that are obtained from algebraic cobordism
of Levine-Morel by change of coefficients. Pure motives with respect to $K
(n)$ fit in-between Chow motives and $K_0$-motives (with $p$-localized or
$p$-torsion coefficients)\, e.g. allowing to transfer $K(n)$-decomposition
s to $K(m)$-decompositions whenever $m < n$. Thus\, it might be a reasonab
le approach in the study of motivic decompositions to start with $K(1)$-mo
tives (i.e. more or less $K_0$-motives) and continue to $K(2)$-\, $K(3)$-m
otives and so on\, eventually arriving to Chow-motives.\nOn the other hand
we formulate a conjectural principle that connects the splitting of $K(n)
$-motive. \nwith the triviality of cohomological invariants of degrees les
s than $n+1$.\nI plan to outline the proof of this principle for quadrics
and explain its consequences \nfor Chow groups of quadrics lying in powers
of the fundamental ideal in the Witt ring.\nThe talk is mostly based on t
he joint work with Nikita Semenov.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Primosic (University of Alberta)
DTSTART:20201214T160000Z
DTEND:20201214T170000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/23
DESCRIPTION:Title: Motivic cohomology and infinitesimal group schemes\nby
Eric Primosic (University of Alberta) as part of Quadratic forms\, linear
algebraic groups and beyond\n\n\nAbstract\nFor $k$ a perfect field of cha
racteristic $p > 0$ and $G$ a split reductive group over $k$ with $p$ a no
n-torsion prime for $G$\, we compute the mod $p$ motivic cohomology of the
geometric classifying space $BG_{(r)}$\, where $G_{(r)}$ is the $r$th Fro
benius kernel of $G$. Our main tool is a motivic version of the Eilenberg-
Moore spectral sequence\, due to Krishna.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Uriya First (University of Haifa)
DTSTART:20210120T163000Z
DTEND:20210120T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/24
DESCRIPTION:Title: The Grothendieck--Serre conjecture for classical groups in
low dimensions\nby Uriya First (University of Haifa) as part of Quadr
atic forms\, linear algebraic groups and beyond\n\n\nAbstract\nA famous co
njecture of Grothendieck and Serre predicts that if $G$ is a reductive gro
up scheme over a semilocal regular domain $R$ and $X$ is a G-torsor\, then
$X$ has a point over the fraction field of $R$ if and only if it has an $
R$-point. I will discuss recent work with Eva Bayer-Fluckiger and Raman Pa
rimala in which we prove the conjecture for all forms of ${\\rm GL}_n$\, $
{\\rm Sp}_n$ and ${\\rm SO}_n$ when $R$ is 2-dimensional\, and all forms o
f ${\\rm GL}_{2n+1}$ when $R$ is 4-dimensional. (Here the ring $R$ is not
required to contain a field.) We approach the problem using the hermitian
Gersten-Witt complex associated to an Azumaya algebra with involution $(A\
,s)$ over a semilocal regular ring $R$. Specifically\, we show that it is
exact when the Krull dimension of $R$ or the index of $A$ are sufficiently
small.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Kuznetsov (Steklov Mathematics Institute)
DTSTART:20210127T163000Z
DTEND:20210127T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/25
DESCRIPTION:Title: Exceptional collection of vector bundles on F4/P4\nby
Alexander Kuznetsov (Steklov Mathematics Institute) as part of Quadratic f
orms\, linear algebraic groups and beyond\n\n\nAbstract\nIn the talk I wil
l explain a construction of a full exceptional collection of vector bundle
s on the homogeneous variety of the simple algebraic group of Dynkin type
$F_4$ corresponding to its maximal parabolic subgroup $P_4$. The construct
ion 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 Faenzi and Manivel. This is joint work with Pieter Belmans and Maxim S
mirnov.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giancarlo Lucchini-Arteche (University of Chile)
DTSTART:20210203T163000Z
DTEND:20210203T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/26
DESCRIPTION:Title: Local-global principles for homogeneous spaces over some t
wo-dimensional geometric global fields\nby Giancarlo Lucchini-Arteche
(University of Chile) as part of Quadratic forms\, linear algebraic groups
and beyond\n\n\nAbstract\nOver number fields\, there is a classic obstruc
tion to the local-global principle for the existence of rational points\,
known as the Brauer-Manin obstruction\, which is conjectured to explain al
l failures of this principle for homogeneous spaces of connected linear gr
oups. In the last few years\, there has been an increasing interest in fie
lds of a more geometric nature that are amenable to local-global principle
s and Brauer-Manin obstructions as well. These include\, for instance\, fu
nction fields of curves over discretely valued fields\, by analogy with th
e case of global fields of positive characteristic. It is in this context
that I will present recent work with Diego Izquierdo on local-global princ
iples for homogeneous spaces with connected stabilizers. We will see that\
, although 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 Braue
r-Manin obstruction\, contrary to the number field case.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ivan Panin (Steklov Institute at St.Petersburg)
DTSTART:20210217T163000Z
DTEND:20210217T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/27
DESCRIPTION:Title: Rationally isotropic quadratic spaces are locally isotropi
c (mixed characteristic case)\nby Ivan Panin (Steklov Institute at St.
Petersburg) as part of Quadratic forms\, linear algebraic groups and beyon
d\n\n\nAbstract\nA well-known conjecture of Colliot-Thélène asserts that
a rationally isotropic quadratic space over a regular local ring is isotr
opic. If the ring contains a field\, then this conjecture was proved by th
e efforts of the speaker\, Pimenov and Scully. In the talk we will present
new results in the mixed characteristic case.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Patrick Brosnan (University of Maryland)
DTSTART:20210210T163000Z
DTEND:20210210T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/28
DESCRIPTION:Title: Fixed Points in Toroidal Compactifications and Essential D
imension of Covers\nby Patrick Brosnan (University of Maryland) as par
t of Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nE
ssential dimension is a numerical measure of the complexity of algebraic\n
objects invented by J. Buhler and Z. Reichstein in the 90s. Roughly speak
ing\,\nthe essential dimension of an algebraic object is the number of par
ameters it\ntakes to define the object over a field. For example\, by Kumm
er theory\, it\ntakes one parameter to define a mu_n torsor\, so the essen
tial dimension 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 Wolf
son (FKW) prove theorems about the\nessential dimension of congruence cove
rs of Shimura varieties using arithmetic\nmethods. In many cases\, they a
re able to prove that the congruence covers are\nincompressible\, that is\
, they are not obtainable by base change from varieties\nof strictly small
er dimension. \n\nIn my talk\, I will discuss recent work with Najmuddin F
akhruddin\, where we recover many (but definitely not all) of the results
of FKW\, by geometric\narguments using a new fixed point theorem. This als
o allows us to extend the\nincompressibility results of FKW to Shimura var
ieties 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
dimension of congruence covers arising from\nHodge theory. (With some cave
ats\, we conjecture that it is equal to the\ndimension of the image of the
period map.)\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julia Hartmann (University of Pennsylvania)
DTSTART:20210224T163000Z
DTEND:20210224T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/29
DESCRIPTION:Title: Local-global principles for constant reductive groups over
arithmetic function fields\nby Julia Hartmann (University of Pennsylv
ania) as part of Quadratic forms\, linear algebraic groups and beyond\n\n\
nAbstract\nArithmetic function fields are one variable function fields ove
r complete discretely valued fields. They naturally admit several collecti
ons of overfields with respect to which one can study local-global princip
les. We will focus on studying local-global principles for torsors under r
eductive groups that are defined over the underlying discrete valuation ri
ng\, reporting on joint work with J.L.-Colliot-Thélène\, D. Harbater\, D
. Krashen\, R. Parimala\, and V. Suresh.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthieu Romagny (Université Rennes 1)
DTSTART:20210317T153000Z
DTEND:20210317T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/30
DESCRIPTION:Title: Smooth affine group schemes over the dual numbers\nby
Matthieu Romagny (Université Rennes 1) as part of Quadratic forms\, linea
r algebraic groups and beyond\n\n\nAbstract\nWe provide a geometric constr
uction for the equivalence between the category of smooth affine group sch
emes over 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 w
e provide a quasi-inverse which we call Weil extension. As an application\
, we establish a Dieudonné classification for smooth\, commutative\, unip
otent group schemes over $k[ε]$ when k is a perfect field.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Popov (Steklov Institute\, Moscow)
DTSTART:20210310T163000Z
DTEND:20210310T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/31
DESCRIPTION:Title: Root systems and root lattices in number fields\nby Vl
adimir Popov (Steklov Institute\, Moscow) as part of Quadratic forms\, lin
ear algebraic groups and beyond\n\n\nAbstract\nThe following construction
of a root system of type G_2 is given in J.-P. Serre’s book “Complex S
emisimple Lie algebras” (Chapter V\, Section 16): “It can be described
as the set of algebraic integers of a cyclotomic field generated by a cub
ic root of unity\, with norm 1 and 3”. The talk\, based on joint work wi
th Yu. G. Zarhin\, concerns the problem of realization of root systems\, t
heir Weyl groups and their root lattices in the form of groups and lattice
s naturally associated with number fields.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Benoist (École normale supérieure\, Paris)
DTSTART:20210303T163000Z
DTEND:20210303T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/32
DESCRIPTION:Title: Sums of squares in local fields\nby Olivier Benoist (
École normale supérieure\, Paris) as part of Quadratic forms\, linear al
gebraic groups and beyond\n\n\nAbstract\nArtin and Pfister have shown that
a nonnegative real\npolynomial in n variables is a sum of $2^n$ squares o
f rational functions. In other words\, the Pythagoras number of the field
$\\mathbb R(x_1\,…\,x_n)$ is at most $2^n$. In this talk\, I will consid
er local variants of this statement. In particular\, I will give a proof o
f a conjecture of Choi\, Dai\, Lam and Reznick: the Pythagoras number of t
he field of Laurent series $\\mathbb R((x_1\,…\,x_n))$ is at most $2^{n-
1}$.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michel Brion (Institut Fourier\, Université Grenoble Alpes)
DTSTART:20210324T153000Z
DTEND:20210324T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/33
DESCRIPTION:Title: Homomorphisms of algebraic groups: representability and ri
gidity\nby Michel Brion (Institut Fourier\, Université Grenoble Alpes
) as part of Quadratic forms\, linear algebraic groups and beyond\n\n\nAbs
tract\nThe talk will address the following questions: given two algebraic
groups G\, H over a field\, is the functor of group homomorphisms from G t
o H representable by a scheme M\, locally of finite type? If so\, how to d
escribe the orbits of H acting on M via conjugation of homomorphisms? The
representability question has a positive answer when G is reductive and H
is smooth and affine\, by a result of Demazure in SGA3 (which holds over a
n arbitary base).The talk will present an extension of this result to the
class of “semi-reductive” algebraic groups\, which includes reductive
groups\, finite groups and abelian varieties. In characteristic 0\, we wil
l also see that all the H-orbits in M are open. This rigidity property giv
es back results of Vinberg and Margaux.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Lavrenov (St.Petersburg State University)
DTSTART:20210331T153000Z
DTEND:20210331T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/34
DESCRIPTION:Title: Morava motives of projective quadrics\nby Andrei Lavre
nov (St.Petersburg State University) as part of Quadratic forms\, linear a
lgebraic groups and beyond\n\n\nAbstract\nThe category of Chow motives def
ined by Grothendieck has plenty of various applications to quadratic forms
\, and\, more generally\, to projective homogeneous varieties. However\, t
here are many open questions about the behaviour of Chow motives. In contr
ast\, if we change the Chow group by Grothendieck’s $K^0$ in the definit
ion of motives\, the resulting category behaves much more simply. One can
define the category of motives corresponding to any oriented cohomology th
eory A and hopefully obtain invariants that are simpler than Chow motives
but keep more information than $K^0$-motives.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kelly McKinnie (University of Montana)
DTSTART:20210407T153000Z
DTEND:20210407T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/35
DESCRIPTION:Title: Common Splitting Fields of Symbol Algebras\nby Kelly M
cKinnie (University of Montana) as part of Quadratic forms\, linear algebr
aic groups and beyond\n\n\nAbstract\nEvery central simple algebra of p-pow
er degree over a field of characteristic p is Brauer equivalent to a cycli
c algebra by a result of Albert. The proof of this and other similar p-alg
ebra results rely on the interplay between purely inseparable splitting fi
elds and cyclic splitting fields of p-algebras. This talk on joint work wi
th Adam Chapman and Mathieu Florence looks at new results on common splitt
ing fields of symbol p-algebras with applications to symbol length.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Haution (LMU Munich)
DTSTART:20210414T153000Z
DTEND:20210414T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/36
DESCRIPTION:Title: The cobordism ring of algebraic involutions\nby Olivie
r Haution (LMU Munich) as part of Quadratic forms\, linear algebraic group
s and beyond\n\n\nAbstract\nI will provide an elementary definition of the
cobordism ring of involutions of smooth projective varieties over a field
(of characteristic not 2). I will describe its structure\, and give expli
cit “stable” polynomial generators. I will draw some concrete conseque
nces concerning the geometry of fixed loci of involutions\, in terms of Ch
ern numbers. I will in particular mention an algebraic version of Boardman
’s five halves theorem.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Suresh Venapally (Emory University)
DTSTART:20210505T153000Z
DTEND:20210505T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/37
DESCRIPTION:Title: Degree three cohomology groups of function fields of curve
s over number fields\nby Suresh Venapally (Emory University) as part o
f Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nLet
$F$ be a field and $l$ a prime not equal to the characteristic of $F$. Giv
en $a_1\, \\ldots \, a_n \\in F^∗$\, the cup product gives an element $(
a_1)\\cdot \\ldots \\cdot (a_n)$ in \n$H^n(F\, µ_l^{\\otimes n})$ and suc
h an element is called a symbol. Class field\ntheory asserts that if $F$ i
s a global 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 primiti
ve $l$th root of unity\, then we show that every element in $H^3(F\, µ_l^
{\\otimes 3})$ is a symbol. We describe an\napplication to the isotropy of
quadratic forms over F. We also give an application to the finite generat
ion of the Chow group of zero-cycles on quadric fibrations of curves over
totally imaginary number fields.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Guralnick (University of Southern California)
DTSTART:20210512T153000Z
DTEND:20210512T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/38
DESCRIPTION:Title: Generic Stabilizers for Simple Algebraic Groups\nby Ro
bert Guralnick (University of Southern California) as part of Quadratic fo
rms\, linear algebraic groups and beyond\n\n\nAbstract\nConsider an algebr
aic group $G$ acting on an irreducible variety $X$. We say there exists
a generic stabilizer for this action if there exists a nonempty open subse
t $Y$ of $X$ 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 especiall
y consider the case that $G$ is a simple algebraic group in positive char
acteristic and $X$ is an irreducible $G$-module. We show that a generic
stabilizer always exists and determine the generic stabilizer in all cases
. This fails for semisimple groups. This is joint work with Skip Garibald
i and Ross Lawther.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Susanna Zimmermann (Université d’Angers)
DTSTART:20210519T153000Z
DTEND:20210519T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/39
DESCRIPTION:Title: Algebraic groups acting birationally on surfaces over a pe
rfect field\nby Susanna Zimmermann (Université d’Angers) as part of
Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nWhich
linear algebraic groups act birationally on a rational surface? And which
are these actions\, up to conjugacy by a birational map? The classificati
on history is quite long over the field of complex numbers and cumulates i
n the works of Blanc and Dolgachev-Iskovskikh. Over non-closed fields\, th
e classification is not complete yet\, but there are many partial results.
In this talk\, I would like to present the way to attack the classificati
on in general\, as well as explain the complete list of actions (up to con
jugacy) when the linear algebraic group is infinite.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Vishik (University of Nottingham)
DTSTART:20210526T153000Z
DTEND:20210526T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/40
DESCRIPTION:Title: Torsion Motives\nby Alexander Vishik (University of No
ttingham) as part of Quadratic forms\, linear algebraic groups and beyond\
n\n\nAbstract\nTorsion motives are Chow motives which disappear with ratio
nal coefficients. Surprisingly\, such objects exist - examples were constr
ucted by Gorchinsky-Orlov. Hypothetically\, such motives should generate t
he kernel of the family of "isotropic realization" functors. I will discus
s some invariants of torsion motives which\, in particular\, shed light on
their size.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jan Minac and Tung T. Nguyen (University of Western Ontario)
DTSTART:20210602T153000Z
DTEND:20210602T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/41
DESCRIPTION:Title: A panoramic view of absolute Galois groups\nby Jan Min
ac and Tung T. Nguyen (University of Western Ontario) as part of Quadratic
forms\, linear algebraic groups and beyond\n\n\nAbstract\nFrom the very b
eginning of its origin\, Galois theory has gained an air of depth\, beauty
\, elegance\, and interactions with deep arithmetic\, geometric\, and topo
logical considerations. Basic fundamental open questions in this area incl
ude the characterization of absolute Galois groups among profinite groups\
, a characterization of maximal pro-p-quotients of absolute Galois groups\
, the study of Massey products in Galois cohomology and the sharpening of
Rost-Voevodsky’s remarkable work on the Bloch-Kato conjecture. These fun
damental questions are deeply intertwined with considerations of the speci
al values of L-functions and Galois representations. In this talk we plan
to present some current research on some rather basic elementary aspects o
f Massey products\, Galois modules\, the nature of the values of zeta func
tions\, and Fekete polynomials. These topics will include our joint work w
ith Nguyen Duy Tˆan and Andrew Schultz. This talk will be in the form of
a dialog where we shall try and reinact our researchefforts and bring to l
ife our actual research excitement pursuing questions and differentconnect
ions with a number of aspects of current Galois theory.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zev Rosengarten (The Hebrew University of Jerusalem)
DTSTART:20210609T153000Z
DTEND:20210609T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/42
DESCRIPTION:Title: Rigidity and Unirational groups\nby Zev Rosengarten (T
he Hebrew University of Jerusalem) as part of Quadratic forms\, linear alg
ebraic groups and beyond\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anand Sawant (Tata Institute of Fundamental Research)
DTSTART:20210623T153000Z
DTEND:20210623T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/43
DESCRIPTION:Title: Near-rationality properties of norm varieties\nby Anan
d Sawant (Tata Institute of Fundamental Research) as part of Quadratic for
ms\, linear algebraic groups and beyond\n\n\nAbstract\nThe standard norm v
arieties played a crucial role in Voevodsky’s proof of the Bloch-Kato co
njecture. I will discuss various near-rationality concepts for smooth proj
ective varieties and describe known near-rationality results for standard
norm varieties. I will then outline an argument showing that a standard no
rm variety over a field of characteristic 0 is universally R-trivial after
passing to the algebraic closure of the base field. The talk is based on
joint work with Chetan Balwe and Amit Hogadi.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefan Schreieder (Leibniz Universität Hannover)
DTSTART:20210707T153000Z
DTEND:20210707T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/44
DESCRIPTION:Title: Refined unramified cohomology of schemes\nby Stefan Sc
hreieder (Leibniz Universität Hannover) as part of Quadratic forms\, line
ar algebraic groups and beyond\n\n\nAbstract\nWe introduce refined unramif
ied cohomology of algebraic schemes and show that it interpolates between
Borel–Moore homology and algebraic cycles. Over finitely generated field
s\, l-adic Chow groups of algebraic schemes are computed by refined unrami
fied cohomology. Over the complex numbers\, our approach simplifies and ge
neralizes to cycles of arbitrary codimensions on possibly singular schemes
\, previous results of Bloch—Ogus\, Colliot-Thélène—Voisin\, Voisin\
, and Ma. Our approach has several applications. For instance\, it allows
to produce the first example of a smooth complex projective variety whose
Griffiths group has infinite torsion.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Saltman (Center for Communications Research)
DTSTART:20210721T153000Z
DTEND:20210721T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/45
DESCRIPTION:Title: Lifting in Mixed Characteristics\nby David Saltman (Ce
nter for Communications Research) as part of Quadratic forms\, linear alge
braic groups and beyond\n\n\nAbstract\nIn Suresh's talk in this seminar\,
he outlined the proof of his main theorem and in that process talked about
choosing elements with a "half dozen" and then a "dozen" properties. To a
significant extent\, this amounted to lifting degree $p$ cyclic Galois ex
tensions with particular properties at characteristic $p$ and characterist
ic not $p$ points. This suggests there should be a "generic" polynomial th
at is the Artin-Schreier polynomial modulo $p$ and gives Kummer extensions
in all other characteristics. Starting from the change of variables noted
by Suresh and others\, we present such a polynomial when the ground ring
has a primitive $p$ root of one\, $\\rho$. Then we define a generic extens
ion without 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 wa
y\, of a cyclic algebra. However in characteristic $p$ the differential de
gree p crossed products are more useful\, and so that is the object we wan
t to lift. 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 one can make to generalize all of the above to degrees a power of $p
$.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ning Guo (Institut de Mathématique d’Orsay)
DTSTART:20210616T153000Z
DTEND:20210616T163000Z
DTSTAMP:20260404T110655Z
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 Grothendiec
k–Serre conjecture predicts that torsors under reductive group schemes o
ver regular local rings are trivial if they trivialize generically. In thi
s talk\, we consider the variant when the bases are valuation rings. This
result is predicted by the original Grothendieck–Serre conjecture and th
e resolution of singularities. The novelty of our proof lies in overcoming
subtleties brought by general nondiscrete valuation rings. By using flasq
ue resolutions and inducting with local cohomology\, we prove a non-Noethe
rian counterpart of Colliot-Thélène– Sansuc’s case of tori. Then\, t
aking advantage of techniques in algebraization\, we obtain the passage to
the Henselian rank one case. Finally\, we induct on Levi subgroups and us
e the integrality of rational points of anisotropic groups to reduce to th
e semisimple anisotropic case\, in which we appeal to properties of paraho
ric subgroups in Bruhat–Tits theory to conclude. In the last section\, b
y using properties of reflexive sheaves\, we also prove a variant of Nisne
vich’s purity conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seidon Alsaody (Uppsala University)
DTSTART:20210714T153000Z
DTEND:20210714T163000Z
DTSTAMP:20260404T110655Z
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://stable.researchseminars.org/talk/Algebraicgroups/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hanspeter Kraft (University of Basel)
DTSTART:20210929T153000Z
DTEND:20210929T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/48
DESCRIPTION:Title: Small $G$-varieties (joint work with Andriy Regeta and Sus
anna Zimmermann)\nby Hanspeter Kraft (University of Basel) as part of
Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nAbstra
ct: 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 highest weight vector. Such a variety $X$ carries a canonical action
of the multiplicative group $K^*$ commuting with the $G$-action. We show t
hat $X$ is determined by the $K^*$-variety $X^U$ of fixed points under a m
aximal unipotent subgroup $U \\subset G$. Moreover\, if $X$ is smooth\, th
en $X$ is a $G$-vector bundle over the algebraic quotient $X // G$.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Louis Colliot-Thelene (Université Paris-Saclay\, Orsay)
DTSTART:20211020T153000Z
DTEND:20211020T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/50
DESCRIPTION:Title: Quadratic forms and stable rationality I\nby Jean-Loui
s Colliot-Thelene (Université Paris-Saclay\, Orsay) as part of Quadratic
forms\, linear algebraic groups and beyond\n\n\nAbstract\nIn this survey s
plit over two talks\, I shall review how - over 50 years - quadratic for
ms\, and in particular Pfister forms\, have been used to produce more and
more examples of rationally connected varieties over the complex field w
hich are not stably rational. We shall start with the Artin-Mumford exampl
es on conic bundles over the plane and end with the work of Schreieder on
hypersurfaces of small slope.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Louis Colliot-Thelene (Université Paris-Saclay\, Orsay)
DTSTART:20211027T153000Z
DTEND:20211027T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/51
DESCRIPTION:Title: Quadratic forms and stable rationality II\nby Jean-Lou
is Colliot-Thelene (Université Paris-Saclay\, Orsay) as part of Quadratic
forms\, linear algebraic groups and beyond\n\nAbstract: TBA\n\nIn this su
rvey split over two talks\, I shall review how - over 50 years - quadrat
ic forms\, and in particular Pfister forms\, have been used to produce mo
re and more examples of rationally connected varieties over the complex fi
eld which are not stably rational. We shall start with the Artin-Mumford
examples on conic bundles over the plane and end with the work of Schreie
der on hypersurfaces of small slope.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Yakerson (ETH Zurich)
DTSTART:20211110T163000Z
DTEND:20211110T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/52
DESCRIPTION:Title: Twisted K-theory in motivic homotopy theory\nby Maria
Yakerson (ETH Zurich) as part of Quadratic forms\, linear algebraic groups
and beyond\n\n\nAbstract\nIn this talk\, we will speak about algebraic K-
theory of vector bundles twisted by a Brauer class. In particular\, we wil
l discuss a new approach to the motivic spectral sequence for twisted K-th
eory\, constructed earlier by Bruno Kahn and Marc Levine. Time permitting\
, we will mention potential applications for K3 surfaces. The talk is base
d on joint work with Elden Elmanto and Denis Nardin.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alena Pirutka (Courant Institute of Mathematical Sciences)
DTSTART:20211117T163000Z
DTEND:20211117T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/53
DESCRIPTION:Title: Quadrics and computation of the unramified Brauer group\nby Alena Pirutka (Courant Institute of Mathematical Sciences) as part o
f Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nIn t
his talk we will discuss examples of computations of the unramified Brauer
group for fibrations in quadrics.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Erhard Neher (University of Ottawa)
DTSTART:20211201T163000Z
DTEND:20211201T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/54
DESCRIPTION:Title: Quadratic forms over semilocal rings\nby Erhard Neher
(University of Ottawa) as part of Quadratic forms\, linear algebraic group
s and beyond\n\n\nAbstract\nWe discuss several results\, well-known for qu
adratic forms over fields\, in the setting of quadratic forms over arbitra
ry semilocal rings. Among them are Springer's odd degree extension theorem
and the norm principles of Scharlau and of Knebusch. The talk is based on
joint work with Philippe Gille.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikita Karpenko (University of Alberta)
DTSTART:20210922T153000Z
DTEND:20210922T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/55
DESCRIPTION:Title: Yagita’s counter-examples and beyond\nby Nikita Karp
enko (University of Alberta) as part of Quadratic forms\, linear algebraic
groups and beyond\n\n\nAbstract\nA conjecture on a relationship between t
he Chow and Grothendieck rings for the generically twisted variety of Bore
l subgroups in a split semisimple group $G$\, stated by myself\, has been
disproved by Nobuaki Yagita in characteristic $0$ for $G=\\operatorname{Sp
in}(2n+1)$ with $n=8$ and $n=9$. For $n=8$\, I provided an alternative sim
pler proof of Yagita’s result\, working in any characteristic\, but fail
ed to do so for $n=9$. In a current joint work with Sanghoon Baek\, this g
ap is filled by involving a new ingredient – Pieri type K-theoretic form
ulas for highest orthogonal Grassmannians. Besides\, a similar counter-exa
mple for $n=10$ is produced. Note that the conjecture on $\\operatorname{S
pin}(2n+1)$ holds for $n$ up to $5$\; it remains open for $n=6$\, $n=7$\,
and every $n>10$.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eva Bayer-Fluckiger (EPFL Lausanne)
DTSTART:20211013T153000Z
DTEND:20211013T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/56
DESCRIPTION:Title: Isometries of lattices and automorphisms of K3 surfaces\nby Eva Bayer-Fluckiger (EPFL Lausanne) as part of Quadratic forms\, lin
ear algebraic groups and beyond\n\n\nAbstract\nThe aim of this talk is to
give necessary and sufficient conditions for an integral polynomial to be
the characteristic polynomial of an isometry of some even\, unimodular lat
tice of given signature\, a result with applications to automorphisms of K
3 surfaces.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Pepin Lehalleur (Radboud University Nijmegen)
DTSTART:20211006T153000Z
DTEND:20211006T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/57
DESCRIPTION:Title: Quadratic enumerative geometry and the Deligne-Milnor form
ula\nby Simon Pepin Lehalleur (Radboud University Nijmegen) as part of
Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nClass
ical enumerative geometry often involves identities between coherent and t
opological (or motivic) invariants. For instance\, the Deligne-Milnor form
ula expresses the Euler characteristic of the vanishing cycles at an isola
ted hypersurface singularity in terms of the Jacobi algebra of the singula
rity. In quadratic enumerative geometry\, numerical invariants are refined
into classes in the Grothendieck-Witt ring of the base field. On the cohe
rent side\, this refinement process involves Grothendieck duality\, while
on the motivic side\, it involves stable motivic homotopy theory. Both sid
es of the Deligne-Milnor formula admit such a natural quadratic refinement
. In a joint work with Marc Levine and Vasudevan Srinivas\, we compute bot
h sides in a class of simple examples of singularities and show that the t
wo sides do not match up: correction terms appear\, whose origin is still
mysterious.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joshua Ruiter (Michigan State University)
DTSTART:20211124T163000Z
DTEND:20211124T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/58
DESCRIPTION:Title: Abstract homomorphisms of some special unitary groups\
nby Joshua Ruiter (Michigan State University) as part of Quadratic forms\,
linear algebraic groups and beyond\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Bate (University of York)
DTSTART:20211208T163000Z
DTEND:20211208T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/59
DESCRIPTION:Title: Overgroups of regular unipotent elements in algebraic grou
ps\nby Michael Bate (University of York) as part of Quadratic forms\,
linear algebraic groups and beyond\n\n\nAbstract\nI will talk about a rece
nt paper with Ben Martin and Gerhard Roehrle on subgroups containing regul
ar unipotent elements in reductive algebraic groups. The main result which
I will describe is not itself new (it is due to Testerman and Zalesski)\,
but the new proof we came up with is worth sharing\, since it is very sho
rt\, free from the case-checking of the original\, and only rests on quite
well-known basic properties of algebraic groups. I will also describe som
e generalisations and extensions.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Wittenberg (Université Sorbonne Paris Nord)
DTSTART:20211215T163000Z
DTEND:20211215T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/60
DESCRIPTION:Title: Massey products in the Galois cohomology of number fields<
/a>\nby Olivier Wittenberg (Université Sorbonne Paris Nord) as part of Qu
adratic forms\, linear algebraic groups and beyond\n\n\nAbstract\n(Joint w
ork with Yonatan Harpaz.) Let k be a field and p be a prime.\nAccording t
o a conjecture 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 establish this\nconjecture when $k$ is a number field\, for any $n$.
This constraint on the\nabsolute Galois group of k was previously known t
o hold when $n=3$ and\nwhen $n=4$\, $p=2$.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Asher Auel (Dartmouth College)
DTSTART:20220119T163000Z
DTEND:20220119T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/61
DESCRIPTION:Title: Brauer classes split by genus one curves\nby Asher Aue
l (Dartmouth College) as part of Quadratic forms\, linear algebraic groups
and beyond\n\n\nAbstract\nIt is an open problem\, even over the rational
numbers\, to decide whether every Brauer class is split by the function fi
eld of a genus one curve. The problem has been solved for Brauer classes o
f index at most 6 over any field. In this talk\, I’ll report on work wit
h Ben Antieau relating the problem to the arithmetic of modular curves and
methods from explicit descent for elliptic curves\, which in particular a
llow us to solve the case of index 7 over number fields.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alberto Elduque (Universidad de Zaragoza)
DTSTART:20220126T163000Z
DTEND:20220126T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/62
DESCRIPTION:Title: Gradings on simple Lie algebras\nby Alberto Elduque (U
niversidad de Zaragoza) as part of Quadratic forms\, linear algebraic grou
ps and beyond\n\n\nAbstract\nAfter reviewing the basic definitions about g
radings\, it will be shown how gradings by abelian groups on a (not necess
arily associative) algebra correspond to morphisms from diagonalizable gro
up schemes to the automorphism group scheme of the algebra. This is the cl
ue to classify gradings on simple Lie algebras. The known classification r
esults of such gradings will be surveyed.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Utiralova (Massachusetts Institute of Technology)
DTSTART:20220216T163000Z
DTEND:20220216T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/63
DESCRIPTION:Title: Harish-Chandra bimodules in complex rank\nby Alexandra
Utiralova (Massachusetts Institute of Technology) as part of Quadratic fo
rms\, linear algebraic groups and beyond\n\n\nAbstract\nDeligne tensor cat
egories are defined as an interpolation of the categories of representatio
ns of groups $\\operatorname{GL}_n$\, $\\operatorname{O}_n$\, $\\operatorn
ame{Sp}_{2n}$ or $\\operatorname{S}_n$ to the complex values of the parame
ter n. One can extend many classical representation-theoretic notions and
constructions to this context. These complex rank analogs of classical obj
ects provide insights into their stable behavior patterns as n goes to inf
inity. I will talk about some of my results on Harish-Chandra bimodules in
the Deligne categories. It is known that in the classical case simple Har
ish-Chandra bimodules admit a classification in terms of W-orbits of certa
in pairs of weights. However\, the notion of weight is not well-defined in
the setting of the Deligne categories. I will explain how in complex rank
the above-mentioned classification translates to a condition on the corre
sponding (left and right) central characters. Another interesting phenomen
on arising in complex rank is that there are two ways to define Harish-Cha
ndra bimodules. That is\, one can either require that the center acts loca
lly finitely on a bimodule M or that M has a finite K-type. The two condit
ions are known to be equivalent for a semi-simple Lie algebra in the class
ical setting\, however\, in Deligne categories that is no longer the case.
I will talk about a way to construct examples of Harish-Chandra bimodules
of finite K-type using the ultraproduct realization of Deligne categories
.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander S. Sivatski (Universidade Federal do Rio Grande do Norte
)
DTSTART:20220202T163000Z
DTEND:20220202T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/64
DESCRIPTION:Title: Nonstandard quadratic forms over rational function fields<
/a>\nby Alexander S. Sivatski (Universidade Federal do Rio Grande do Norte
) as part of Quadratic forms\, linear algebraic groups and beyond\n\nAbstr
act: TBA\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roman Fedorov (University of Pittsburg)
DTSTART:20220209T163000Z
DTEND:20220209T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/65
DESCRIPTION:Title: On the purity conjecture of Nisnevich for torsors under re
ductive group schemes\nby Roman Fedorov (University of Pittsburg) as p
art of Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstract\
nLet $R$ be a regular semilocal integral domain containing an infinite fie
ld $k$. Let $f$ be an element of $R$ that does not belong to the square of
any maximal ideal of $R$ (equivalently\, the hypersurface $\\{f=0 \\}$ is
regular). Let $G$ be a reductive group scheme over $R$. Under an isotropy
assumption on $G$ we show that a $G$-torsor over the localization $R_f$ i
s trivial\, provided it is rationally trivial.\n\nThe statement is derived
from its abstract version concerning Nisnevich sheaves satisfying some pr
operties. Note that if $f=1$\, then we recover the conjecture of Grothendi
eck and Serre (already known for regular semilocal rings containing fields
). The proof of Nisnevich conjecture follows the same strategy except that
one needs an additional statement concerning G-torsors defined on the com
plement of a subscheme of $A^1_R$ that is etale and finite over $R$.\n\nIf
time permits\, we will also explain that the aforementioned isotropy assu
mption is necessary.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Susanne Pumpluen (University of Nottingham)
DTSTART:20220302T163000Z
DTEND:20220302T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/66
DESCRIPTION:Title: Nonassociative algebras obtained from skew polynomials and
their applications\nby Susanne Pumpluen (University of Nottingham) as
part of Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstrac
t\nUsing skew polynomials\, we define a class of unital nonassociative alg
ebras introduced by Petit in 1966 (but largely ignored so far). Some of th
ese algebras are canonical generalizations of (associative) central simple
algebras\, and classical results from Albert\, Amitsur and Jacobson can b
e generalized to this nonassociative setting. We discuss their structure a
nd time permitting also their use in coding theory. Their most prominent f
eature is that their right nucleus is the eigenspace of the skew polynomia
l used in their construction.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mathieu Florence (Institut de Mathématiques de Jussieu Sorbonne U
niversité)
DTSTART:20220223T163000Z
DTEND:20220223T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/67
DESCRIPTION:Title: Equivariant Witt vector bundles\nby Mathieu Florence (
Institut de Mathématiques de Jussieu Sorbonne Université) as part of Qua
dratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nEquivaria
nt Witt vector bundles\, over a scheme of characteristic p>0\, were introd
uced in collaboration with Charles De Clercq and Giancarlo Lucchini-Artech
e. Their purpose is to serve as a geometric tool\, in proving mod p^2 lift
ability of mod p representations of smooth profinite groups- comprising th
e case of Galois representations. This is still work in progress.\n\nI wil
l explain some liftability statements (positive or negative) achieved so f
ar\, and related techniques. I will then discuss a "general lifting statem
ent"\, which I believe is within reach.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fabio Tanania (LMU Munich)
DTSTART:20220309T163000Z
DTEND:20220309T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/68
DESCRIPTION:Title: On the motivic cohomology of the Nisnevich classifying spa
ce of PGL_n\nby Fabio Tanania (LMU Munich) as part of Quadratic forms\
, linear algebraic groups and beyond\n\n\nAbstract\nn this talk I will pre
sent the construction of a (kind of) Serre spectral sequence for motivic c
ohomology associated to a map of simplicial schemes with motivically cellu
lar fiber. Then I will show how to apply it in order to obtain information
about the motivic cohomology of the Nisnevich classifying space of projec
tive general linear groups. At the end I will also give a description of t
he motive of a Severi-Brauer variety in terms of twisted motives of its Č
ech simplicial scheme.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maneesh Thakur (Indian Statistical Institute)
DTSTART:20220316T153000Z
DTEND:20220316T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/69
DESCRIPTION:Title: The cyclicity problem for Albert division algebras\nby
Maneesh Thakur (Indian Statistical Institute) as part of Quadratic forms\
, linear algebraic groups and beyond\n\n\nAbstract\nAn old question of Adr
ian Albert\, raised more than fifty years ago\, asks the following\n\nQues
tion: does every Albert division algebra over a field of characteristic di
fferent 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 contains 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 is an isotope of A that contains a cyclic cubic extension.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Antieau
DTSTART:20220518T153000Z
DTEND:20220518T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/70
DESCRIPTION:Title: Benjamin Antieu's talk is postponed to June 15\nby Ben
jamin Antieau as part of Quadratic forms\, linear algebraic groups and bey
ond\n\n\nAbstract\nBenjamin Antieu's talk is postponed to June 15.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicole Lemire (University of Western Ontario)
DTSTART:20220323T153000Z
DTEND:20220323T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/71
DESCRIPTION:Title: Codimension 2 cycles of classifying spaces of low-dimensio
nal algebraic tori\nby Nicole Lemire (University of Western Ontario) a
s part of Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstra
ct\nLet T be an algebraic torus over a field F and let\n$CH^2(BT)$ be the
Chow group of codimension 2 cycles in its classifying space. Following wor
k of Blinstein and Merkurjev on the structure of the torsion part of $CH^2
(BT)$\, Scavia\, in a recent preprint\, found an example of an algebraic t
orus with non-trivial torsion in\n$CH^2(BT)$. In joint work with Alexander
Neshitov\, we show computationally that the group $CH^2(BT)$ is torsion-f
ree for all algebraic tori of dimension at most 5 and determine the conjug
acy classes of\nfinite subgroups of $\\operatorname{GL}_6(\\mathbb Z)$ whi
ch correspond to 6-dimensional tori\nwith nontrivial torsion in $CH^2(BT)$
. Some interesting properties\nof the structure of low-dimensional algebra
ic tori will be discussed.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudio Quadrelli (University of Milan)
DTSTART:20220330T153000Z
DTEND:20220330T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/72
DESCRIPTION:Title: The Bogomolov-Positselski conjecture for maximal pro-p Gal
ois groups\nby Claudio Quadrelli (University of Milan) as part of Quad
ratic forms\, linear algebraic groups and beyond\n\n\nAbstract\nLet $p$ be
a prime. In the '90s\, F. Bogomolov asked the following question: if $K$
is a field containing an algebraically closed field\, then is the closure
of the commutator subgroup of the Sylow pro-$p$ subgroup of the absolute G
alois group $G_{K}$ of $K$\, a free pro-$p$ group (cf. [1])?\nLater on\, L
. Positselski generalized Bogomolov's question to the following:\nif $K$ i
s a field containing a root of 1 of order $p$ - and $\\sqrt[p^\\infty]{K}$
denotes 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 wo
rk [4]\, Thomas Weigel and myself translated Positselski's version of Bogo
molov's conjecture into purely group theoretic language\, and verified it
for those fields whose maximal pro-$p$ Galois group is a pro-$p$ group of
elementary type\, as defined by I. Efrat (e.g.\, local fields\, PAC fields
\, $p$-rigid fields\, algebraic extensions of global fields with finite $K
^\\times/(K^\\times)^p$...).\n\nThis group-theoretic formulation is tightl
y related to a cohomological property enjoyed by maximal pro-$p$ Galois gr
oups of fields\, called ``Kummerian property'' (introduced in [2]\, and re
lated\, in turn\, to smooth profinite groups introduced by M. Florence)\,
which may be used also to detect pro-$p$ groups that do not occur as absol
ute Galois 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 rational functions\, $K$-theory and algebraic geometry: connections wit
h quadratic forms and division algebra. In: Proceedings of Symposia on Pur
e Mathematics\, 1992\, Santa Barbara CA\, vol. 58 (1995).\n\n[2] I. Efrat
and C. Quadrelli\, Efrat\, The Kummerian property and maximal pro-$p$ Galo
is groups. J. Algebra 525 (2019).\n\n[3] L. Positselski\, Koszul property
and Bogomolov’s conjecture. Int. Math. Res. Not. 31 (2005).\n\n[4] C. Qu
adrelli and Th. Weigel\, Oriented pro-$\\ell$ groups with the Bogomolov-Po
sitselski property. Res. Number theory 8\, no. 2 (2022).\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Holger Petersson (Fernuniversität in Hagen)
DTSTART:20220406T153000Z
DTEND:20220406T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/73
DESCRIPTION:Title: Integral octonions: history and perspectives\nby Holge
r Petersson (Fernuniversität in Hagen) as part of Quadratic forms\, linea
r algebraic groups and beyond\n\n\nAbstract\nAfter setting the stage by re
calling the basic properties of composition algebras over commutative ring
s\, I sketch the history of intergral octonions\, from its infancy in the
1860s to Coxeter’s groundbreaking paper of 1946. Inspired by results due
to Mahler (1942) and Allcock (1999)\, I proceed to describe the one-sided
ideal structure of octonion algebras over arbitrary commutative rings. Th
e lecture concludes with a non-orthogonal version of the classical Cayley-
Dickson construction that allows for a description of integral octonions (
more precisely\, of their multiplicative structure) in an intrinsic manner
.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aravind Asok (University of Southern California)
DTSTART:20220601T153000Z
DTEND:20220601T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/74
DESCRIPTION:Title: On P^1-stabilization in unstable motivic homotopy theory\nby Aravind Asok (University of Southern California) as part of Quadrat
ic forms\, linear algebraic groups and beyond\n\n\nAbstract\nI will discus
s joint work with Tom Bachmann and Mike Hopkins regarding an analog of the
Freudenthal suspension theorem in unstable motivic homotopy theory. To m
otivate the result\, I will quickly introduce the unstable motivic homotop
y category and discuss some concrete applications. Time permitting\, I wi
ll sketch the key idea behind the proof.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fatma Kader Bingöl (University of Antwerpen)
DTSTART:20220525T153000Z
DTEND:20220525T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/75
DESCRIPTION:Title: On the 4-torsion part of the Brauer group\nby Fatma Ka
der Bingöl (University of Antwerpen) as part of Quadratic forms\, linear
algebraic groups and beyond\n\n\nAbstract\nThe 4-torsion part of the Braue
r group of a field $F$ is generated by cyclic algebras of degree $2$ and $
4$. This 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 s
ame context\, one further obtains a bound on the index of exponent-$4$ alg
ebras over $F$ in terms of the $u$-invariant of $F$. This is joint work wi
th K.J. Becher.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Max Lieblich (University of Washington)
DTSTART:20220622T153000Z
DTEND:20220622T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/76
DESCRIPTION:Title: Murphy's Law for gerbes\nby Max Lieblich (University o
f Washington) as part of Quadratic forms\, linear algebraic groups and bey
ond\n\n\nAbstract\nThis is a report on joint work with Daniel Bragg. We st
udy what gerbes are possible as residual gerbes in natural moduli stacks.
Among other things\, we show that every gerbe with finite structure group
arises as a residual gerbe in the stack of smooth projective curves. This
gives examples of "versal gerbes" that arise organically in algebraic geom
etry. The key to producing such things is the transport of various classic
al constructions from equivariant geometry to corresponding constructions
for spaces over non-trivial gerbes.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Danny Ofek (University of British Columbia)
DTSTART:20220706T153000Z
DTEND:20220706T163000Z
DTSTAMP:20260404T110655Z
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://stable.researchseminars.org/talk/Algebraicgroups/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ido Efrat (Ben-Gurion University)
DTSTART:20220629T153000Z
DTEND:20220629T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/78
DESCRIPTION:Title: Steinberg Relations for Massey Products\nby Ido Efrat
(Ben-Gurion University) as part of Quadratic forms\, linear algebraic grou
ps and beyond\n\n\nAbstract\nLet $F$ be a field of characteristic prime to
$m$ which contains the $m$th roots of unity\, and let $G_F$ be its absolu
te Galois group.\n\nAs shown by Tate\, for $a\\neq 0\,1$ in $F$\, the Kumm
er elements $(a)_F$\, $(1-a)_F$ in $H^1(G_F\,\\mathbb{Z}/m)$ have trivial
cup product.\n\nIn fact\, by the celebrated Voevodsky-Rost theorem\, this
relation completely determines the cohomology ring $H^{\\bullet}(G_F\,\\ma
thbb Z/m)$ with the cup product.\n\nA natural generalization of the cup pr
oduct is the $n$-fold Massey product\, where $n\\geq2$. Extending results
by Kirsten Wickelgren\, we show how Tate's relation generalizes to the Mas
sey product context.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Goncalo Tabuada (University of Warwick)
DTSTART:20220713T153000Z
DTEND:20220713T163000Z
DTSTAMP:20260404T110655Z
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://stable.researchseminars.org/talk/Algebraicgroups/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Antieau (Northwestern University)
DTSTART:20220615T153000Z
DTEND:20220615T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/80
DESCRIPTION:Title: The K-theory of Z/p^n\nby Benjamin Antieau (Northweste
rn University) as part of Quadratic forms\, linear algebraic groups and be
yond\n\n\nAbstract\nI will report on joint work with Achim Krause and Thom
as Nikolaus where we give an algorithm to compute the K-groups of rings su
ch as $Z/p^n$.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eliahu Matzri (Bar Ilan University)
DTSTART:20220608T153000Z
DTEND:20220608T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/81
DESCRIPTION:Title: On the symbol length of symbols in Galois cohomology.\
nby Eliahu Matzri (Bar Ilan University) as part of Quadratic forms\, linea
r algebraic groups and beyond\n\n\nAbstract\nFix a prime $p$ and let $F$ b
e a field with characteristic not $p$. Let $G_F$ be the absolute Galois gr
oup of $F$ and let $\\mu_{p^s}$ be the $G_F$-module of roots of unity of o
rder\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 expone
nt dividing $p^{s-1}$ (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 symbols coming from $H^n(F\,\\mu_{p^{s-1}}^{\\otimes n})$ that is symbo
ls of the 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 ch
aracteristic zero.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karim Johannes Becher (University of Antwerp)
DTSTART:20220928T153000Z
DTEND:20220928T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/82
DESCRIPTION:Title: Fields with bounded Brauer 2-torsion index\nby Karim J
ohannes Becher (University of Antwerp) as part of Quadratic forms\, linear
algebraic groups and beyond\n\n\nAbstract\nAlexander Merkurjev’s ground
breaking results from the 1980ies showed that the $u$-invariant (maximal d
imension of an anisotropic quadratic form) of a field is related to the de
grees of division algebras of exponent $2$. \n\nThis relation led Bruno Ka
hn to conjecture that the $u$-invariant is bounded in terms of the $2$-sym
bol length (the number of quaternion algebras necessary to represent an ar
bitrary 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 characte
ristic 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 th
e square-root of $-1$.\n\nIn my talk I want to present a variation and ref
inement of this observation.\nAssuming that every central division algebra
of exponent $2$ over $F$ has degree at most $2^n$\, I show that every $(2
n+2)$-fold Pfister form is hyperbolic if $-1$ is a sum of squares in $F$ a
nd that it is in any case equal to 8 times a $(2n-1)$-fold Pfister form.\n
The proof of this result is based on computations with trace forms of cent
ral simple algebras.\nUsing this fact\, one can now remove in a result of
Daniel Krashen from 2016\, which characterises fields for which all symbol
lengths in the Milnor $K$-groups modulo $2$ of a field are bounded\, the
condition that $-1$ be a square. In fact\, these fields are the same as th
ose with finite $u$-invariant (in the sense of Elman-Lam) and finite stabi
lity index. This latter result is joint work in progress with Saurabh Gosa
vi.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stephen Scully (University of Victoria)
DTSTART:20221123T163000Z
DTEND:20221123T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/83
DESCRIPTION:Title: On the dimensions of quadratic forms isotropic over the fu
nction field of a quadric\nby Stephen Scully (University of Victoria)
as part of Quadratic forms\, linear algebraic groups and beyond\n\n\nAbstr
act\nLet $p$ and $q$ be anisotropic quadratic forms over a field of any ch
aracteristic\, and $i$ the isotropy index of q over the function field of
the quadric defined by p. In 2018\, we proposed a conjecture that constrai
ns the dimension of $q$ in terms of $i$ and the largest power of 2 strictl
y less than the dimension of $p$. This can be viewed as a generalization o
f the "separation theorem" originally proved by Hoffmann over fields of ch
aracteristic not 2. In this talk\, we'll explain that the conjecture is tr
ue when $q$ is a quasilinear (i.e.\, diagonalizable) form over a field of
characteristic 2. The proof\, which is elementary\, reveals a much stronge
r constraint involving certain stable birational invariants of $p$ (which
must also be quasilinear in order for the statement to be non-trivial in t
his case). Examining the contribution from the "Izhboldin dimension" of $p
$\, we are led to formulate (for all forms in any characteristic) a strong
version of our original conjecture that incorporates other important resu
lts due to Karpenko and Karpenko-Merkurjev. After discussing the quasiline
ar case\, we'll explain what is known when $q$ is non-singular. Unlike the
quasilinear case\, the results here rely on algebraic-geometric methods.\
n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Federico Scavia (UCLA)
DTSTART:20221207T163000Z
DTEND:20221207T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/84
DESCRIPTION:Title: Degenerate fourfold Massey products over arbitrary fields<
/a>\nby Federico Scavia (UCLA) as part of Quadratic forms\, linear algebra
ic groups and beyond\n\n\nAbstract\nWe prove that\, for all fields $F$ of
characteristic different from $2$ and all $a\,b\,c \\in F^*$\, the mod 2 M
assey product $\\left< a\,b\,c\,a \\right>$ vanishes as soon as it is defi
ned. For every field $E$ of characteristic different form $2$\, we constru
ct a field $F$ 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 o
f Positselski by constructing the first examples of fields containing all
roots of unity and such that the mod 2 cochain DGA of the absolute Galois
group is not formal. This is joint work with Alexander Merkurjev.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Zhykhovich (LMU Munich)
DTSTART:20221109T163000Z
DTEND:20221109T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/85
DESCRIPTION:Title: The J-invariant of algebras with orthogonal involution.\nby Maxim Zhykhovich (LMU Munich) as part of Quadratic forms\, linear al
gebraic groups and beyond\n\n\nAbstract\nThe $J$-invariant of a semi-simpl
e algebraic group $G$ was introduced by Petrov\, Semenov and Zainoulline i
n 2008. The $J$-invariant is a discrete invariant which encodes the motivi
c decomposition of the variety of Borel subgroups in $G$ (in this talk we
consider Chow motives). Let $(A\, \\sigma)$ be a central simple algebra wi
th orthogonal involution and trivial discriminant. The $J$-invariant of $(
A\,\\sigma)$ is defined as $J(\\mathrm{PGO}^+(A\,\\sigma))$. In this talk
I will discuss a conjecture of Quéguiner Mathieu\, Semenov and Zainoullin
e\, which allows to reduce the computation of $J(A\,\\sigma)$ to the case
of quadratic forms.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Barry Demba (University of Bamako\, Mali)
DTSTART:20221116T163000Z
DTEND:20221116T173000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/86
DESCRIPTION:Title: Degree 3 relative invariant for unitary involutions\nb
y Barry Demba (University of Bamako\, Mali) as part of Quadratic forms\, l
inear algebraic groups and beyond\n\n\nAbstract\nThe Arason invariant in q
uadratic form theory is a degree 3 cohomological invariant attached to an
even-dimensional quadratic form with trivial discriminant and trivial Clif
ford invariant. It is known that this invariant can also be described in t
erms of the Rost invariant of a split Spin group. More generally\, using t
he Rost invariant for non split Spin groups\, and for absolutely almost si
mple simply connected groups of other types\, one may try to define analog
ues of the Arason invariant for the underlying algebraic objects\, namely
hermitian forms and involutions. For involutions of the first kind\, degre
e 3 cohomological invariants were investigated by several authors.\nIn thi
s talk\, following a suggestion of Tignol\, I will present the case of uni
tary involutions\, which correspond to groups of outer type A. Notice that
the Arason invariant for quadratic forms may be used to define an invaria
nt for unitary involutions with trivial discriminant algebra on split alge
bras. Nevertheless\, as for orthogonal involutions\, this invariant does n
ot extend in a functorial way to the non split case. Using the Rost invari
ant for some torsors\, we define a relative Arason invariant for unitary i
nvolutions. An important feature in this talk is that we do not restrict t
o involutions with trivial lower-degree invariants\, but also consider pai
rs of unitary involutions with isomorphic discriminant algebras. The end o
f the talk will be devoted to the properties of this relative invariant fo
r algebras of degree 8.\n\n(Joint work with A. Masquelein and A. Quéguine
r-Mathieu)\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Martin (University of Aberdeen)
DTSTART:20221012T153000Z
DTEND:20221012T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/87
DESCRIPTION:Title: Geometric invariant theory for reductive groups over non-a
lgebraically closed fields\nby Benjamin Martin (University of Aberdeen
) as part of Quadratic forms\, linear algebraic groups and beyond\n\n\nAbs
tract\nLet $G$ be a reductive linear algebraic group acting on an affine v
ariety $X$ over a field $k$. If $k$ is algebraically closed then the Hilbe
rt-Mumford Theorem gives a powerful tool for understanding the structure o
f $G(k)$-orbits in $X$: an orbit is closed if and only if it is closed und
er taking limits along cocharacters. Moreover\, if an orbit is not closed
then one 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 canonical 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 cl
osed under taking limits of $k$-defined cocharacters. There is a version o
f the Hilbert-Mumford Theorem which holds in this more general setting. I
will discuss the notion of cocharacter-closure and its interactions with t
he theory of spherical buildings and the theory of $G$-complete reducibili
ty.\n\nThis talk is based on joint work with Michael Bate\, Sebastian Herp
el\, Gerhard R\\"ohrle and Rudolf Tange.\n\n\nReferences:\n\nBate M.E.\, H
erpel S.\, Martin B.\, Röhrle G. Cocharacter-closure and the rational Hil
bert-Mumford 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.\
, Martin B.\, Röhrle G.\, Tange R. Closed orbits and uniform S-instabilit
y in geometric invariant theory. Trans. Amer. Math. Soc. 365 (2013)\, 3643
–3673.\nArxiv: https://arxiv.org/abs/0904.4853\n\nBate M.E.\, Martin B.\
, Röhrle G.\, Tange R. Complete reducibility and separable field extensio
ns.C. R. Math Acad. Sci. Paris 348 (2010)\, no. 9–10\, 495-497.\nArxiv:
https://arxiv.org/abs/1002.4319\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yashonidhi Pandey (Indian Institute of Science Education and Resea
rch\, Mohali)
DTSTART:20221019T153000Z
DTEND:20221019T163000Z
DTSTAMP:20260404T110655Z
UID:Algebraicgroups/88
DESCRIPTION:Title: On Bruhat-Tits theory over a higher dimensional base\n
by Yashonidhi Pandey (Indian Institute of Science Education and Research\,
Mohali) as part of Quadratic forms\, linear algebraic groups and beyond\n
\n\nAbstract\nThis is joint-work with Vikraman Balaji. The preprint is pos
ted on the arXiv.\n\nLet $\\mathcal O_{n} := k\\llbracket z_{1}\, \\ldots\
, z_{n}\\rrbracket$ over an algebraically closed residue field $k$ of char
acteristic zero. Set $K_{n}= {\\rm Fract}~\\mathcal{O}{n}$. Let $G$ be an
almost-simple\, simply-connected affine algebraic group over $k$ with a ma
ximal torus $T$ 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 Bruhat-Tits\, we define $n$-bounded subgroups $\\mathcal{P}_{\\bf f}(k
)\\subset G(K_{n})$ as a direct generalization of Bruhat-Tits groups for t
he case $n=1$. We show that these groups are schematic\, i.e. they are val
ued points 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 se
nse that the restriction to the generic point of the divisor $z_i=0$ is gi
ven by $f_i$. This provides a higher-dimensional analogue of the Bruhat-Ti
ts group schemes with natural specialization properties. Under suitable ta
meness 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
$G$ replacing $\\mathcal O_{n}$ by $\\mathcal{O} \\llbracket x_{1}\,\\cdo
ts\,x_{n} \\rrbracket$\, where $\\mathcal O$ is a complete discrete valuat
ion ring with residue field of characteristic $p$. In particular\, if $x_0
$ is the 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 appli
cations.\n
LOCATION:https://stable.researchseminars.org/talk/Algebraicgroups/88/
END:VEVENT
END:VCALENDAR