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BEGIN:VEVENT
SUMMARY:Avner Ash (Boston College)
DTSTART:20211011T140000Z
DTEND:20211011T144500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/1
DESCRIPTION:Title: Homology of arithmetic groups and Galois representations\nby Avne
r Ash (Boston College) as part of BIRS workshop: Cohomology of Arithmetic
Groups: Duality\, Stability\, and Computations\n\n\nAbstract\nI give a few
examples of how Galois representations can help in the understanding and
computation of the homology of congruence subgroups of $\\mathrm{GL}_n(\\m
athbb{Z})$. Then I sketch a current project of mine with Darrin Doud in
which we hope to prove the following: If $\\rho=\\sigma_1 \\oplus \\sigma
_2$ is an $n$-dimensional odd mod $p$ Galois representation\, with $\\sigm
a_1$ and $\\sigma_2$ irreducible odd Galois representations that are attac
hed to Hecke eigenclasses in the homology of the predicted congruence subg
roups\, with predicted weights\, then $\\rho$ is attached to a Hecke eigen
classes in the homology of the predicted congruence subgroup of $\\mathrm{
GL}_n(\\mathbb{Z})$\, with predicted weight. Here\, "predicted" refers to
the Serre-type conjecture of Ash–Doud–Pollack–Sinnott. We assume t
hat $p$ is greater than $n+1$ and that the Serre conductor of $\\rho$ is s
quare-free.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Patzt (Copenhagen University/ University of Oklahoma)
DTSTART:20211011T150000Z
DTEND:20211011T154500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/2
DESCRIPTION:Title: Rognes' connectivity conjecture and the Koszul dual of Steinberg\
nby Peter Patzt (Copenhagen University/ University of Oklahoma) as part of
BIRS workshop: Cohomology of Arithmetic Groups: Duality\, Stability\, and
Computations\n\n\nAbstract\nIn this talk\, I will explain how a homotopy
equivalence\nbetween certain $E_k$-buildings both proves Rognes' connectiv
ity\nconjecture for fields and computes the Koszul dual of Steinberg. Rogn
es'\nconnectivity conjecture states that the common basis complex is highl
y\nconnected. This is relevant as the equivariant homology of this complex
\nappears in a rank filtration spectral sequence computing the homology of
\nthe $K$-theory spectrum. The Steinberg modules appear in various context
s\,\nimportantly as the dualizing modules of special linear groups of numb
er\nrings. They can be put together to form a ring. When considered\nequiv
ariantly over the general linear groups of fields\, one can show\nthat thi
s ring is Koszul and we compute its Koszul dual. Results in this\ntalk inc
lude joint work with Jeremy Miller\, Rohit Nagpal\, and Jennifer\nWilson.\
n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Kupers
DTSTART:20211011T163000Z
DTEND:20211011T171500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/3
DESCRIPTION:Title: On homological stability for GL_n(Z)\nby Alexander Kupers as part
of BIRS workshop: Cohomology of Arithmetic Groups: Duality\, Stability\,
and Computations\n\n\nAbstract\nI will explain what is known about homolog
ical stability for the general linear groups of the integers. In particula
r\, I will discuss a recent result\, joint work with Jeremy Miller and Pet
er Patzt\, that improves the homological stability range to slope 1. It bu
ilds on machinery developed with Soren Galatius and Oscar Randal-Williams\
, and is closely related to homology with coefficients in the Steinberg mo
dule.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mathilde Gerbelli-Gauthier
DTSTART:20211011T210000Z
DTEND:20211011T212000Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/4
DESCRIPTION:Title: Growth of cohomology in towers and endoscopy\nby Mathilde Gerbell
i-Gauthier as part of BIRS workshop: Cohomology of Arithmetic Groups: Dual
ity\, Stability\, and Computations\n\n\nAbstract\nHow fast do Betti number
s grow in a congruence tower of compact arithmetic manifolds? The dimensio
n of the middle degree of cohomology is proportional to the volume of the
manifold\, but away from the middle the growth is known to be sub-linear.
I’ll discuss this question from the point of view of automorphic forms\,
and outline how the phenomenon of endoscopy can be used to explain the sl
ow rates of growth and to compute upper bounds.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathalie Wahl (University of Copenhagen)
DTSTART:20211012T140000Z
DTEND:20211012T144500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/5
DESCRIPTION:Title: Stability in the homology of classical groups\nby Nathalie Wahl (
University of Copenhagen) as part of BIRS workshop: Cohomology of Arithmet
ic Groups: Duality\, Stability\, and Computations\n\n\nAbstract\nGeneral l
inear groups\, symplectic groups\, unitary groups and orthogonal groups ha
ve long been known to satisfy homological stability under appropriate cond
itions. In joint work with David Sprehn\, we improved the earlier known ho
mological stability ranges for $\\mathrm{Sp}_{2n}(\\mathbb{F})$\, $\\mathr
m{O}_{n\,n}(\\mathbb{F})$ and $\\mathrm{U}_{2n}(\\mathbb{F})$ over any fie
ld $\\mathbb{F}$ other than $\\mathbb{F}_2$\, following a strategy of Quil
len for general linear groups $\\mathrm{GL}_n(\\mathbb{F})$. Under more re
stricted assumptions\, we deduce a stability theorem for the orthogonal gr
oup $\\mathrm{O}_n(\\mathbb{F})$. I'll present these results\, focussing o
n what these groups have in common\, and presenting this maybe less well-k
nown strategy of Quillen that gives a slope 1 stability range for $\\mathr
m{GL}_n(\\mathbb{F})$.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Putman
DTSTART:20211012T150000Z
DTEND:20211012T154500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/6
DESCRIPTION:Title: The Steinberg representation is irreducible\nby Andrew Putman as
part of BIRS workshop: Cohomology of Arithmetic Groups: Duality\, Stabilit
y\, and Computations\n\n\nAbstract\nWe prove that the Steinberg representa
tion of $\\mathrm{GL}_n$ (or\, more generally\, a connected reductive grou
p) over an infinite field is irreducible. For finite fields\, this is a cl
assical theorem of Steinberg and Curtis. This is joint work with Andrew S
nowden.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathan Broaddus (The Ohio State University)
DTSTART:20211012T163000Z
DTEND:20211012T171500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/7
DESCRIPTION:Title: Level structures and images of the Steinberg module for surfaces with
marked points\nby Nathan Broaddus (The Ohio State University) as part
of BIRS workshop: Cohomology of Arithmetic Groups: Duality\, Stability\,
and Computations\n\n\nAbstract\nThe moduli space $\\mathcal{M}$ of complex
curves of fixed topology is an\norbifold classifying space for surface bu
ndles. As such the cohomology\nrings of $\\mathcal{M}$ and its various orb
ifold covers give characteristic classes\nfor surface bundles. I will disc
uss the Steinberg module which is\ncentral to the duality present in these
cohomology rings. I will then\nexplain current joint work with T. Brendle
and A. Putman on surfaces\nwith marked points which expands on results of
N. Fullarton and A.\nPutman for surfaces without marked points. We show t
hat certain\nfinite-sheeted orbifold covers $\\mathcal{M}[l]$ of $\\mathca
l{M}$ have large nontrivial\n$Q$-cohomology in their cohomological dimensi
on.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Calegari (University of Chicago)
DTSTART:20211012T190000Z
DTEND:20211012T194500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/8
DESCRIPTION:Title: The stable cohomology of SL(F_p)\nby Frank Calegari (University o
f Chicago) as part of BIRS workshop: Cohomology of Arithmetic Groups: Dual
ity\, Stability\, and Computations\n\n\nAbstract\nLet $p$ be a prime. One
can make sense of various “compatible” algebraic representations of $\
\mathrm{SL}_N(\\mathbb{F}_p)$ as $p$ is fixed and as $N$ varies (for examp
le\, the standard representation\, or the adjoint representation\, or the
trivial representation). It turns out that the cohomology groups of these
representations are stable as $N$ gets large. So what are they? We discuss
a conjectural answer to this. We also discuss how this relates to a conje
ctural computation of $H^i(\\mathrm{SL}_N(\\mathbb{F}_p)\,\\mathbb{F}_p)$
for $i$ fixed and $N$ going off to infinity which should be true for “al
most all $p$”.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Yasaki
DTSTART:20211012T203000Z
DTEND:20211012T211500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/9
DESCRIPTION:Title: Cohomology of Congruence Subgroups\, Steinberg Modules\, and Real Qua
dratic Fields\nby Dan Yasaki as part of BIRS workshop: Cohomology of A
rithmetic Groups: Duality\, Stability\, and Computations\n\n\nAbstract\nGi
ven a real quadratic field\, there is a naturally defined Hecke-stable sub
space of the cohomology of a congruence subgroup of $\\mathrm{SL}_3(\\math
bb{Z})$. We investigate this subspace and make conjectures about its dep
endence on the real quadratic field and the relationship to boundary cohom
ology. This is joint work with Avner Ash.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melody Chan (Brown University)
DTSTART:20211013T140000Z
DTEND:20211013T144500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/10
DESCRIPTION:Title: The top-weight rational cohomology of $\\mathcal{A}_g$\nby Melod
y Chan (Brown University) as part of BIRS workshop: Cohomology of Arithmet
ic Groups: Duality\, Stability\, and Computations\n\n\nAbstract\nIn joint
work with Madeline Brandt\, Juliette Bruce\, Margarida Melo\,\nGwyneth Mor
eland\, and Corey Wolfe\, we recently identified new\ntop-weight rational
cohomology classes for moduli spaces $\\mathcal{A}_g$ of\nabelian varietie
s\, by using computations of Voronoi complexes for\n$\\mathrm{GL}(g\,\\mat
hbb{Z})$ of Elbaz-Vincent--Gangl--Soulé. In this talk\, I will try to\ne
xplain these results from the beginning\, surveying some of the main\ntech
niques and ingredients.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Gunnells (University of Massachusetts)
DTSTART:20211013T150000Z
DTEND:20211013T154500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/11
DESCRIPTION:Title: Modular symbols over function fields\nby Paul Gunnells (Universi
ty of Massachusetts) as part of BIRS workshop: Cohomology of Arithmetic Gr
oups: Duality\, Stability\, and Computations\n\n\nAbstract\nModular symbol
s\, due to Birch and Manin\, provide a very\nconcrete way to compute with
classical holomorphic modular forms.\nLater modular symbols were extended
to $\\mathrm{GL}(n)$ by Ash and Rudolph\, and\nsince then such symbols and
variations have played a central role in\ncomputational investigation of
the cohomology of arithmetic groups\nover number fields\, and in particula
r in explicitly computing the\nHecke action on cohomology. $$ \\qquad \\\\
[-2em]$$ \n\nA theory of modular symbols for $\\mathrm{GL}(2)$ over the ra
tional function field\nwas developed by Teitelbaum and later by Armana. I
n this talk we extend\nthis construction to $\\mathrm{GL}(n)$ and show how
it can be used to compute Hecke\noperators on cohomology. This is joint
work with Dan Yasaki.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark McConnell (Princeton University)
DTSTART:20211013T163000Z
DTEND:20211013T171500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/12
DESCRIPTION:Title: Binary Quadratic Forms and Hecke Operators for $\\mathrm{SL}(2\,\\ma
thbb{Z})$\nby Mark McConnell (Princeton University) as part of BIRS wo
rkshop: Cohomology of Arithmetic Groups: Duality\, Stability\, and Computa
tions\n\n\nAbstract\nRobert MacPherson and I developed an algorithm for co
mputing the Hecke operators on the cohomology $H^d$ of arithmetic subgroup
s of $\\mathrm{SL}(n)$ defined over any division algebra\, for all $d$ and
all $n$. It extends Voronoi's notion of perfect forms by introducing tem
pered perfect forms. To find the tempered perfect forms\, our code must c
ompute the facets of a large convex polytope of $n(n+1)/2$ dimensions\, wh
ich is slow even for $n = 3$ or $4$. The talk will report on recent work\
, in the classical case of $\\mathrm{SL}(2\,\\mathbb{Z})$\, where we have
succeeded in identifying the tempered perfect forms directly. The story c
omes down to binary quadratic forms in the spirit of Lagrange and Gauss\,
together with some modern class field theory. This is joint work with Eri
k Bahnson and Kyrie McIntosh.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oscar Randal-Williams (University of Cambridge)
DTSTART:20211014T140000Z
DTEND:20211014T144500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/13
DESCRIPTION:Title: $E_\\infty$-algebras and general linear groups\nby Oscar Randal-
Williams (University of Cambridge) as part of BIRS workshop: Cohomology of
Arithmetic Groups: Duality\, Stability\, and Computations\n\n\nAbstract\n
I will discuss joint work with S. Galatius and A. Kupers in which we inves
tigate the homology of general linear groups over a ring $A$ by considerin
g the collection of all their classifying spaces as a graded $E_\\infty$-a
lgebra. I will first explain diverse results that we obtained in this inve
stigation\, which can be understood without reference to $E_\\infty$-algeb
ras but which seem unrelated to each other: I will then explain how the po
int of view of cellular $E_\\infty$-algebras unites them.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bena Tshishiku
DTSTART:20211014T150000Z
DTEND:20211014T154500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/14
DESCRIPTION:Title: Unstable cohomology of arithmetic groups and geometric cycles\nb
y Bena Tshishiku as part of BIRS workshop: Cohomology of Arithmetic Groups
: Duality\, Stability\, and Computations\n\n\nAbstract\nWe construct unsta
ble cohomology classes of nonuniform arithmetic subgroups of $\\mathrm{SO}
(p\,q)$ using ideas of Millson-Raghunathan and more recent work of Avramid
i and Nguyen-Phan. The classes we construct are dual to maximal periodic f
lats in the locally symmetric space. One motivation for this result is to
produce characteristic classes for certain manifold bundles that are not i
n the algebra generated by the stable (MMM) classes.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benson Farb (University of Chicago)
DTSTART:20211014T163000Z
DTEND:20211014T171500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/15
DESCRIPTION:Title: Rigidity of moduli spaces\nby Benson Farb (University of Chicago
) as part of BIRS workshop: Cohomology of Arithmetic Groups: Duality\, Sta
bility\, and Computations\n\n\nAbstract\nAlgebraic geometry contains an ab
undance of miraculous constructions. Examples include ``resolving the qua
rtic''\; the existence of 9 flex points on a smooth plane cubic\; the Jaco
bian of a genus $g$ curve\; and the 27 lines on a smooth cubic surface. In
this talk I will explain some ways to systematize and formalize the idea
that such constructions are special: conjecturally\, they should be the on
ly ones of their kind. I will state a few of these many (mostly open) conj
ectures. They can be viewed as forms of rigidity (a la Mostow and Margulis
) for various moduli spaces and maps between them.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew Emerton (University of Chicago)
DTSTART:20211014T190000Z
DTEND:20211014T194500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/16
DESCRIPTION:by Matthew Emerton (University of Chicago) as part of BIRS wor
kshop: Cohomology of Arithmetic Groups: Duality\, Stability\, and Computat
ions\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng (Massachusetts Institute of Technology)
DTSTART:20211014T200000Z
DTEND:20211014T202000Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/17
DESCRIPTION:Title: The Galois action on symplectic K-theory\nby Tony Feng (Massachu
setts Institute of Technology) as part of BIRS workshop: Cohomology of Ari
thmetic Groups: Duality\, Stability\, and Computations\n\n\nAbstract\nI wi
ll talk about some connections between the cohomology of arithmetic groups
\, $K$-theory\, and number theory. One reason for these connections is the
fact that there is a natural Galois action on the cohomology of symplecti
c groups of integers\, which turns out to provide Galois representations i
mportant in the Langlands correspondence. The same mechanism leads to a Ga
lois action on a symplectic variant of K-theory of the integers. In joint
work with Soren Galatius and Akshay Venkatesh\, we compute this Galois act
ion and find that it also enjoys a certain universality.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Brück
DTSTART:20211014T210000Z
DTEND:20211014T212000Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/18
DESCRIPTION:Title: High-dimensional rational cohomology of $\\operatorname{SL}_n(\\math
bb{Z})$ and $\\operatorname{Sp}_{2n}(\\mathbb{Z})$\nby Benjamin Brück
as part of BIRS workshop: Cohomology of Arithmetic Groups: Duality\, Stab
ility\, and Computations\n\n\nAbstract\nBy results of Lee-Szarba and Churc
h-Putman\, the rational cohomology of $\\operatorname{SL}_n(\\mathbb{Z})$
vanishes in "codimensions zero and one"\, i.e. $H^{{n \\choose 2} -i}(\\op
eratorname{SL}_n(\\mathbb{Z})\;\\mathbb{Q}) = 0$ for $i\\in \\{0\,1\\}$ an
d $n \\geq i+2$\, where ${n \\choose 2}$ is the virtual cohomological dime
nsion of $\\operatorname{SL}_n(\\mathbb{Z})$. I will talk about work in pr
ogress on two generalisations of these results: The first project is joint
work with Miller-Patzt-Sroka-Wilson. We show that the rational cohomology
of $\\operatorname{SL}_n(\\mathbb{Z})$ vanishes in codimension two\, i.e.
$H^{{n \\choose 2} -2}(\\operatorname{SL}_n(\\mathbb{Z})\;\\mathbb{Q}) =
0$ for $n \\geq 4$. The second project is joint with Patzt-Sroka. Its aim
is to study whether the rational cohomology of the symplectic group $\\ope
ratorname{Sp}_{2n}(\\mathbb{Z})$ vanishes in codimension one\, i.e. whethe
r $H^{n^2 -1}(\\operatorname{Sp}_{2n}(\\mathbb{Z})\;\\mathbb{Q}) = 0$ for
$n \\geq 2$.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Orsola Tommas (University of Padova)
DTSTART:20211015T140000Z
DTEND:20211015T144500Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/19
DESCRIPTION:Title: Stability results for toroidal compactifications of $\\mathcal{A}_g$
\nby Orsola Tommas (University of Padova) as part of BIRS workshop: Co
homology of Arithmetic Groups: Duality\, Stability\, and Computations\n\n\
nAbstract\nIn this talk\, we will discuss the geometry of the moduli space
$\\mathcal{A}_g$ of\nprincipally polarized abelian varieties of dimension
$g$ and its\ncompactifications. As is well known\, in degree $k < g$ the
rational\ncohomology of $\\mathcal{A}_g$\, which coincides with the cohomo
logy of the symplectic\ngroup\, is freely generated by the odd Chern class
es of the Hodge bundle\nby a classical result of Borel. Work of Charney an
d Lee provides an\nanalogous result for the stable cohomology of the minim
al\ncompactification of $\\mathcal{A}_g$\, the Satake compactification.\nH
owever\, for most geometric applications it is more natural to work with\n
the toroidal compactifications of $\\mathcal{A}_g$. We will report on join
t work with\nSam Grushevsky and Klaus Hulek on the toroidal compactificati
ons of $\\mathcal{A}_g$\,\nand describe stability results for the perfect
cone compactification and\nthe matroidal partial compactification and thei
r combinatorial features.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fabian Hebestreit (University of Bonn)
DTSTART:20211015T150000Z
DTEND:20211015T152000Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/20
DESCRIPTION:Title: The stable cohomology of symplectic groups over the integers\nby
Fabian Hebestreit (University of Bonn) as part of BIRS workshop: Cohomolo
gy of Arithmetic Groups: Duality\, Stability\, and Computations\n\n\nAbstr
act\nI will report on joint work with M. Land and T. Nikolaus in which we
compute the stable part of the cohomology of both symplectic groups and or
thogonal groups with vanishing signature over the integers at regular prim
es\, in particular at the prime 2. Our approach is by identifying the stab
le cohomology with that of a certain Grothendieck-Witt space\, whose homot
opy type can be analysed using recent advances in hermitian $K$-theory.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jordan Ellenberg (University of Wisconsin)
DTSTART:20211015T153000Z
DTEND:20211015T155000Z
DTSTAMP:20260422T185557Z
UID:BIRS-21w5011/21
DESCRIPTION:Title: Legendre symbols and secondary stability\nby Jordan Ellenberg (U
niversity of Wisconsin) as part of BIRS workshop: Cohomology of Arithmetic
Groups: Duality\, Stability\, and Computations\n\n\nAbstract\nThis talk i
s really a problem proposal. Mark Shusterman and I have been talking abo
ut the problem of controlling sums of Legendre symbols $\\left(\\frac{f}{g
}\\right)$ as $f$ and $g$ range over squarefree polynomials of degree $m$
and $n$ over $\\mathbb{F}_q$\, with $m$ and $n$ growing while the finite f
ield $\\mathbb{F}_q$ stays the same. This can be expressed as a problem
about the trace of Frobenius acting on the etal cohomology of a space whos
e complex points are a $K(\\pi\,1)$ for a certain finite-index subgroup of
a colored braid group\; it seems to me that the behavior we expect to see
for these averages would follow from a good result on secondary homolog
ical stability for these subgroups. The question is whether the assemble
d topological might of this workshop can help figure out whether such a st
atement is true and provable with current methods.\n
LOCATION:https://researchseminars.org/talk/BIRS-21w5011/21/
END:VEVENT
END:VCALENDAR