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SUMMARY:Eli Matzri (Bar-Ilan University)
DTSTART:20200908T151000Z
DTEND:20200908T153500Z
DTSTAMP:20260404T042147Z
UID:BIRS_20w5133/1
DESCRIPTION:Title: Polynomials over central division algebras (joint with Shira G
ilat)\nby Eli Matzri (Bar-Ilan University) as part of BIRS workshop: A
rithmetic Aspects of Algebraic Groups\n\n\nAbstract\nLet $F$ be a field wh
ich 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 degre
e $p$) splits into linear factors. As an application we show that a stand
ard Kummer space is a degree $p$ symbol algebra $D=(a\,b)_{p\,F}$ generate
s the multiplicative group $D^{\\times}$. We also show that $GL_p(F)$ is a
lso generated by any standard Kummer space.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5133/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raman Parimala (Emory University)
DTSTART:20200908T155000Z
DTEND:20200908T163500Z
DTSTAMP:20260404T042147Z
UID:BIRS_20w5133/2
DESCRIPTION:Title: The unramified Brauer group\nby Raman Parimala (Emory Univ
ersity) as part of BIRS workshop: Arithmetic Aspects of Algebraic Groups\n
\n\nAbstract\nIn this talk we shall explain a method to translate arithmet
ic information to algebraic data in the context of the study of the unrami
fied Brauer group of tori.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5133/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nivedita Bhaskhar (University of Southern California)
DTSTART:20200908T165500Z
DTEND:20200908T172000Z
DTSTAMP:20260404T042147Z
UID:BIRS_20w5133/3
DESCRIPTION:Title: SK1 triviality for l-torsion algebras over p-adic curves - a p
roof sketch\nby Nivedita Bhaskhar (University of Southern California)
as part of BIRS workshop: Arithmetic Aspects of Algebraic Groups\n\n\nAbst
ract\nAny central simple algebra $A$ over a field $K$ is a form of a matri
x algebra. Further $A/K$ comes equipped with a reduced norm map which is o
btained by twisting the determinant function. Every element in the commut
ator subgroup $[A^*\, A^*]$ has reduced norm 1 and hence lies in $SL_1(A)$
\, the group of reduced norm one elements of A. Whether the reverse inclus
ion holds was formulated as a question in 1943 by Tannaka and Artin in ter
ms of the triviality of the reduced Whitehead group $SK_1(A) := SL_1(A)/[
A^*\, A^*]$. \n\n$$ $$\n\nPlatonov negatively settled the Tannaka-Artin qu
estion by giving a counter example over a cohomological dimension (cd) 4 b
ase field. In the same paper however\, the triviality of $SK_1(A)$ was sho
wn for all algebras over cd at most 2 fields. In this talk\, we investigat
e the situation for $l$-torsion algebras over a class of cd 3 fields of so
me arithmetic 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 p
roving the triviality of the reduced Whitehead group for these algebras. T
he proof relies on the techniques of patching as developed by Harbater-Har
tmann-Krashen and exploits the arithmetic of these fields.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5133/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jinbo Ren (University of Virginia)
DTSTART:20200908T173500Z
DTEND:20200908T180000Z
DTSTAMP:20260404T042147Z
UID:BIRS_20w5133/4
DESCRIPTION:Title: Mathematical logic and its applications in arithmetics of alge
braic groups and beyond\nby Jinbo Ren (University of Virginia) as part
of BIRS workshop: Arithmetic Aspects of Algebraic Groups\n\n\nAbstract\nA
large family of classical arithmetic problems (in algebraic groups) inclu
ding\n\n$$ $$\n\n(a) Finding rational solutions of the so-called trigonome
tric Diophantine 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 $(
2\,\\sqrt{2(2-\\lambda)})$ and $(3\, \\sqrt{6(3-\\lambda)})$ are both tors
ion points of the elliptic curve $y^2=x(x-1)(x-\\lambda)$\;\n\n$$ $$\n\nca
n be regarded as special cases of the Zilber-Pink conjecture in Diophantin
e geometry. In this short talk\, I will explain how we use tools from math
ematical logic to attack this conjecture. In particular\, I will present a
series partial results toward the Zilber-Pink conjecture\, including thos
e proved by Christopher Daw and myself.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5133/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Srimathy Srinivasan (University of Colorado)
DTSTART:20200908T181500Z
DTEND:20200908T184000Z
DTSTAMP:20260404T042147Z
UID:BIRS_20w5133/5
DESCRIPTION:Title: A finiteness theorem for special unitary groups of quaternioni
c skew-hermitian forms with good reduction\nby Srimathy Srinivasan (Un
iversity of Colorado) as part of BIRS workshop: Arithmetic Aspects of Alge
braic Groups\n\n\nAbstract\nI will give a brief sketch of why the number o
f special unitary groups of quaternionic skew-hermitian forms with good r
eduction at a set of discrete valuations is finite for certain fields. Thi
s answers a conjecture of Chernousov\, Rapinchuk and Rapinchuk for groups
of this type.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5133/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew Stover (Temple University)
DTSTART:20200909T160000Z
DTEND:20200909T162500Z
DTSTAMP:20260404T042147Z
UID:BIRS_20w5133/6
DESCRIPTION:Title: Superrigidity in rank one\nby Matthew Stover (Temple Unive
rsity) as part of BIRS workshop: Arithmetic Aspects of Algebraic Groups\n\
n\nAbstract\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 prove arithmeticitiy of finite volume real or complex hyperbolic man
ifold containing infinitely many maximal totally geodesic submanifolds\, a
nswering a question due independently to Alan Reid and Curtis McMullen.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5133/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Chernousov (University of Alberta)
DTSTART:20200909T164000Z
DTEND:20200909T172500Z
DTSTAMP:20260404T042147Z
UID:BIRS_20w5133/7
DESCRIPTION:Title: On the Tits-Weiss conjecture on U-operators and the Kneser-Tit
s conjecture for some groups of type E_7 and E_8.\nby Vladimir Cherno
usov (University of Alberta) as part of BIRS workshop: Arithmetic Aspects
of Algebraic Groups\n\n\nAbstract\nJoint work with S. Alsaody and A. Pianz
ola. In the first part of the talk we remind the definition of an $R$-equi
valence (introduced by Manin)\, state the Tits-Weiss conjecture on generat
ion of structure groups of Albert algebras by $U$-operators and the Kneser
-Tits conjecture for isotropic groups. In the second part of the talk we f
ocus on computation of $R$-equivalence classes for groups of type $E_6$. A
s applications of our result we prove the Tits-Weiss conjecture and the Kn
eser-Tits conjecture for some isotropic groups of type $E_7$ and $E_8$.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5133/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zev Rosengarten (Hebrew University of Jerusalem)
DTSTART:20200909T174500Z
DTEND:20200909T181000Z
DTSTAMP:20260404T042147Z
UID:BIRS_20w5133/8
DESCRIPTION:Title: Rigidity for Unirational Groups\nby Zev Rosengarten (Hebre
w University of Jerusalem) as part of BIRS workshop: Arithmetic Aspects of
Algebraic Groups\n\n\nAbstract\nOne of the most fundamental results under
lying the theory of abelian varieties is "rigidity" -- that is\, that any
k-scheme morphism of abelian varieties which preserves identities is actua
lly a k-group homomorphism. This result depends crucially upon the propern
ess of such varieties. For affine groups in general\, there is no analogou
s rigidity statement. We will nevertheless show that such a rigidity resul
t holds for unirational groups (which are always affine) satisfying certai
n conditions\, and discuss several implications.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5133/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Charlotte Ure (University of Virginia)
DTSTART:20200909T182500Z
DTEND:20200909T185000Z
DTSTAMP:20260404T042147Z
UID:BIRS_20w5133/9
DESCRIPTION:Title: The Generic Clifford Algebra and its Brauer Class\nby Char
lotte Ure (University of Virginia) as part of BIRS workshop: Arithmetic As
pects of Algebraic Groups\n\n\nAbstract\nThe Clifford algebra is an object
intimately connected with the theory of quadratic forms and orthogonal gr
oups. This classical notion of Clifford algebras associated to quadratic f
orms can 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 algebra defines a nontrivial Brauer class in the Brauer group of a r
elative elliptic curve – the Jacobian of the universal genus one curve o
btained from the Clifford algebra. This is joint work in progress with Raj
esh Kulkarni.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5133/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David El-Chai Ben-Ezra (The Hebrew University)
DTSTART:20200909T190500Z
DTEND:20200909T193000Z
DTSTAMP:20260404T042147Z
UID:BIRS_20w5133/10
DESCRIPTION:Title: The Congruence Subgroup Problem for Automorphism Groups\n
by David El-Chai Ben-Ezra (The Hebrew University) as part of BIRS workshop
: Arithmetic Aspects of Algebraic Groups\n\n\nAbstract\nIn its classical s
etting\, the Congruence Subgroup Problem (CSP) asks whether every finite i
ndex subgroup of $GL_{n}(\\mathbb{Z})$ contains a principal congruence sub
group of the form \n\\[\n\\ker(GL_{n}(\\mathbb{Z})\\to GL_{n}(\\mathbb{Z}/
m\\mathbb{Z}))\n\\]\nfor some $m\\in\\mathbb{Z}$. It was known already in
the 19th 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
congruence\nconsiderations. On the other hand\, quite surprisingly\, it w
as proved\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\ntheory of the CSP for general arithmetic groups.\n\n$$ $$\n\nViewin
g $GL_{n}(\\mathbb{Z})\\cong Aut(\\mathbb{Z}^{n})$ as the automorphism\ngr
oup of $\\Gamma=\\mathbb{Z}^{n}$\, one can generalize the CSP to automorph
ism\ngroups as follows: Let $\\Gamma$ be a finitely generated group\; does
\nevery finite index subgroup of $Aut(\\Gamma)$ contain a principal\ncongr
uence subgroup of the form \n\\[\n\\ker(Aut(\\Gamma)\\rightarrow Aut(\\Gam
ma/M))\n\\]\nfor some finite index characteristic subgroup $M\\leq\\Gamma$
? Considering\nthis generalization\, there are very few results when $\\Ga
mma$ is\nnon-abelian. For example\, only in 2001 Asada proved\, using conc
epts\nfrom Algebraic Geometry\, that $Aut(F_{2})$ has an affirmative answe
r\nto the CSP\, when $F_{2}$ is the free group on two generators. For\n$Au
t(F_{n})$ when $n\\geq 3$ the problem is still unsettled.\n\n$$ $$\n\nIn t
he talk I will give a survey of some recent results regarding\nthe case wh
ere $\\Gamma$ is non-abelian. We will see that when $\\Gamma$\nis a nilpot
ent group the CSP for $Aut(\\Gamma)$ is completely determined\nby the CSP
for arithmetic groups. We will also see that when $\\Gamma$\nis a finitely
generated free metabelian group the picture changes\nand we have a dichot
omy between $n=2\,3$ and $n\\geq 4$.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5133/10/
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