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BEGIN:VEVENT
SUMMARY:Eli Matzri (Bar-Ilan University)
DTSTART:20200908T151000Z
DTEND:20200908T153500Z
DTSTAMP:20260422T185859Z
UID:BIRS_20w5133/1
DESCRIPTION:Title: Polynomials over central division algebras (joint with Shira Gilat)\nby Eli Matzri (Bar-Ilan University) as part of BIRS workshop: Arithmet
ic Aspects of Algebraic Groups\n\n\nAbstract\nLet $F$ be a field which is
prime to $p$ closed. We show that any twisted polynomial in $D[y\,\\sigma]
$ of degree at most $p-1$ ($K/F$ a cyclic Galois extension of degree $p$)
splits into linear factors. As an application we show that a standard Kum
mer space is a degree $p$ symbol algebra $D=(a\,b)_{p\,F}$ generates the m
ultiplicative group $D^{\\times}$. We also show that $GL_p(F)$ is also gen
erated by any standard Kummer space.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5133/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raman Parimala (Emory University)
DTSTART:20200908T155000Z
DTEND:20200908T163500Z
DTSTAMP:20260422T185859Z
UID:BIRS_20w5133/2
DESCRIPTION:Title: The unramified Brauer group\nby Raman Parimala (Emory University)
as part of BIRS workshop: Arithmetic Aspects of Algebraic Groups\n\n\nAbs
tract\nIn this talk we shall explain a method to translate arithmetic info
rmation to algebraic data in the context of the study of the unramified Br
auer group of tori.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5133/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nivedita Bhaskhar (University of Southern California)
DTSTART:20200908T165500Z
DTEND:20200908T172000Z
DTSTAMP:20260422T185859Z
UID:BIRS_20w5133/3
DESCRIPTION:Title: SK1 triviality for l-torsion algebras over p-adic curves - a proof sk
etch\nby Nivedita Bhaskhar (University of Southern California) as part
of BIRS workshop: Arithmetic Aspects of Algebraic Groups\n\n\nAbstract\nA
ny central simple algebra $A$ over a field $K$ is a form of a matrix algeb
ra. Further $A/K$ comes equipped with a reduced norm map which is obtained
by twisting the determinant function. Every element in the commutator su
bgroup $[A^*\, A^*]$ has reduced norm 1 and hence lies in $SL_1(A)$\, the
group of reduced norm one elements of A. Whether the reverse inclusion hol
ds was formulated as a question in 1943 by Tannaka and Artin in terms of t
he triviality of the reduced Whitehead group $SK_1(A) := SL_1(A)/[A^*\, A
^*]$. \n\n$$ $$\n\nPlatonov negatively settled the Tannaka-Artin question
by giving a counter example over a cohomological dimension (cd) 4 base fie
ld. In the same paper however\, the triviality of $SK_1(A)$ was shown for
all algebras over cd at most 2 fields. In this talk\, we investigate the s
ituation for $l$-torsion algebras over a class of cd 3 fields of some arit
hmetic flavour\, namely function fields of $p$-adic curves where l is any
prime not equal to p. We partially answer a question of Suslin by proving
the triviality of the reduced Whitehead group for these algebras. The proo
f relies on the techniques of patching as developed by Harbater-Hartmann-K
rashen and exploits the arithmetic of these fields.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5133/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jinbo Ren (University of Virginia)
DTSTART:20200908T173500Z
DTEND:20200908T180000Z
DTSTAMP:20260422T185859Z
UID:BIRS_20w5133/4
DESCRIPTION:Title: Mathematical logic and its applications in arithmetics of algebraic g
roups and beyond\nby Jinbo Ren (University of Virginia) as part of BIR
S workshop: Arithmetic Aspects of Algebraic Groups\n\n\nAbstract\nA large
family of classical arithmetic problems (in algebraic groups) including\n\
n$$ $$\n\n(a) Finding rational solutions of the so-called trigonometric Di
ophantine equation $F(\\cos 2\\pi x_i\, \\sin 2\\pi x_i)=0$\, where $F$ is
an irreducible multivariate polynomial with rational coefficients\;\n\n$$
$$\n\n(b) Determining all $\\lambda \\in \\mathbb{C}$ such that $(2\,\\sq
rt{2(2-\\lambda)})$ and $(3\, \\sqrt{6(3-\\lambda)})$ are both torsion poi
nts of the elliptic curve $y^2=x(x-1)(x-\\lambda)$\;\n\n$$ $$\n\ncan be re
garded as special cases of the Zilber-Pink conjecture in Diophantine geome
try. In this short talk\, I will explain how we use tools from mathematica
l logic to attack this conjecture. In particular\, I will present a series
partial results toward the Zilber-Pink conjecture\, including those prove
d by Christopher Daw and myself.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5133/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Srimathy Srinivasan (University of Colorado)
DTSTART:20200908T181500Z
DTEND:20200908T184000Z
DTSTAMP:20260422T185859Z
UID:BIRS_20w5133/5
DESCRIPTION:Title: A finiteness theorem for special unitary groups of quaternionic skew-
hermitian forms with good reduction\nby Srimathy Srinivasan (Universit
y of Colorado) as part of BIRS workshop: Arithmetic Aspects of Algebraic G
roups\n\n\nAbstract\nI will give a brief sketch of why the number of spec
ial unitary groups of quaternionic skew-hermitian forms with good reductio
n at a set of discrete valuations is finite for certain fields. This answe
rs a conjecture of Chernousov\, Rapinchuk and Rapinchuk for groups of this
type.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5133/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew Stover (Temple University)
DTSTART:20200909T160000Z
DTEND:20200909T162500Z
DTSTAMP:20260422T185859Z
UID:BIRS_20w5133/6
DESCRIPTION:Title: Superrigidity in rank one\nby Matthew Stover (Temple University)
as part of BIRS workshop: Arithmetic Aspects of Algebraic Groups\n\n\nAbst
ract\nI will overview work with Uri Bader\, David Fisher\, and Nick Miller
on superrigidity of certain representations of lattices in $SO(n\,1)$ and
$SU(n\,1)$. Our primary application of this superrigidity theorem is to p
rove arithmeticitiy of finite volume real or complex hyperbolic manifold c
ontaining infinitely many maximal totally geodesic submanifolds\, answerin
g a question due independently to Alan Reid and Curtis McMullen.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5133/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Chernousov (University of Alberta)
DTSTART:20200909T164000Z
DTEND:20200909T172500Z
DTSTAMP:20260422T185859Z
UID:BIRS_20w5133/7
DESCRIPTION:Title: On the Tits-Weiss conjecture on U-operators and the Kneser-Tits conje
cture for some groups of type E_7 and E_8.\nby Vladimir Chernousov (U
niversity of Alberta) as part of BIRS workshop: Arithmetic Aspects of Alge
braic Groups\n\n\nAbstract\nJoint work with S. Alsaody and A. Pianzola. In
the first part of the talk we remind the definition of an $R$-equivalence
(introduced by Manin)\, state the Tits-Weiss conjecture on generation of
structure groups of Albert algebras by $U$-operators and the Kneser-Tits c
onjecture for isotropic groups. In the second part of the talk we focus on
computation of $R$-equivalence classes for groups of type $E_6$. As appli
cations of our result we prove the Tits-Weiss conjecture and the Kneser-Ti
ts conjecture for some isotropic groups of type $E_7$ and $E_8$.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5133/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zev Rosengarten (Hebrew University of Jerusalem)
DTSTART:20200909T174500Z
DTEND:20200909T181000Z
DTSTAMP:20260422T185859Z
UID:BIRS_20w5133/8
DESCRIPTION:Title: Rigidity for Unirational Groups\nby Zev Rosengarten (Hebrew Unive
rsity of Jerusalem) as part of BIRS workshop: Arithmetic Aspects of Algebr
aic Groups\n\n\nAbstract\nOne of the most fundamental results underlying t
he theory of abelian varieties is "rigidity" -- that is\, that any k-schem
e morphism of abelian varieties which preserves identities is actually a k
-group homomorphism. This result depends crucially upon the properness of
such varieties. For affine groups in general\, there is no analogous rigid
ity statement. We will nevertheless show that such a rigidity result holds
for unirational groups (which are always affine) satisfying certain condi
tions\, and discuss several implications.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5133/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Charlotte Ure (University of Virginia)
DTSTART:20200909T182500Z
DTEND:20200909T185000Z
DTSTAMP:20260422T185859Z
UID:BIRS_20w5133/9
DESCRIPTION:Title: The Generic Clifford Algebra and its Brauer Class\nby Charlotte U
re (University of Virginia) as part of BIRS workshop: Arithmetic Aspects o
f Algebraic Groups\n\n\nAbstract\nThe Clifford algebra is an object intima
tely connected with the theory of quadratic forms and orthogonal groups. T
his classical notion of Clifford algebras associated to quadratic forms ca
n be generalized to higher degree. In this talk\, I will discuss a generic
version of the Clifford algebra associated to a binary cubic form. This a
lgebra defines a nontrivial Brauer class in the Brauer group of a relative
elliptic curve – the Jacobian of the universal genus one curve obtained
from the Clifford algebra. This is joint work in progress with Rajesh Kul
karni.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5133/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David El-Chai Ben-Ezra (The Hebrew University)
DTSTART:20200909T190500Z
DTEND:20200909T193000Z
DTSTAMP:20260422T185859Z
UID:BIRS_20w5133/10
DESCRIPTION:Title: The Congruence Subgroup Problem for Automorphism Groups\nby Davi
d El-Chai Ben-Ezra (The Hebrew University) as part of BIRS workshop: Arith
metic Aspects of Algebraic Groups\n\n\nAbstract\nIn its classical setting\
, the Congruence Subgroup Problem (CSP) asks whether every finite index su
bgroup of $GL_{n}(\\mathbb{Z})$ contains a principal congruence subgroup o
f the form \n\\[\n\\ker(GL_{n}(\\mathbb{Z})\\to GL_{n}(\\mathbb{Z}/m\\math
bb{Z}))\n\\]\nfor some $m\\in\\mathbb{Z}$. It was known already in the 19t
h century\nthat for $n=2$ the answer is negative\, and actually $GL_{2}(\\
mathbb{Z})$\nhas many finite index subgroups which do not come from congru
ence\nconsiderations. On the other hand\, quite surprisingly\, it was prov
ed\nin the sixties by Mennicke and by Bass-Lazard-Serre that for $n\\geq 3
$\nthe answer to the CSP is affirmative. This breakthrough led to a rich\n
theory of the CSP for general arithmetic groups.\n\n$$ $$\n\nViewing $GL_{
n}(\\mathbb{Z})\\cong Aut(\\mathbb{Z}^{n})$ as the automorphism\ngroup of
$\\Gamma=\\mathbb{Z}^{n}$\, one can generalize the CSP to automorphism\ngr
oups as follows: Let $\\Gamma$ be a finitely generated group\; does\nevery
finite index subgroup of $Aut(\\Gamma)$ contain a principal\ncongruence s
ubgroup of the form \n\\[\n\\ker(Aut(\\Gamma)\\rightarrow Aut(\\Gamma/M))\
n\\]\nfor some finite index characteristic subgroup $M\\leq\\Gamma$? Consi
dering\nthis generalization\, there are very few results when $\\Gamma$ is
\nnon-abelian. For example\, only in 2001 Asada proved\, using concepts\nf
rom Algebraic Geometry\, that $Aut(F_{2})$ has an affirmative answer\nto t
he CSP\, when $F_{2}$ is the free group on two generators. For\n$Aut(F_{n}
)$ when $n\\geq 3$ the problem is still unsettled.\n\n$$ $$\n\nIn the talk
I will give a survey of some recent results regarding\nthe case where $\\
Gamma$ is non-abelian. We will see that when $\\Gamma$\nis a nilpotent gro
up the CSP for $Aut(\\Gamma)$ is completely determined\nby the CSP for ari
thmetic groups. We will also see that when $\\Gamma$\nis a finitely genera
ted free metabelian group the picture changes\nand we have a dichotomy bet
ween $n=2\,3$ and $n\\geq 4$.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5133/10/
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