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DTSTART:20180325T030000
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DTSTART:20171029T020000
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DTSTAMP:20200706T085708Z
UID:5936c170405c5030110904@ist.ac.at
DTSTART:20180118T130000
DTEND:20180118T150000
DESCRIPTION:Speaker: Alexander Minets\nhosted by Tamas Hausel\nAbstract: Gi
ven a (hereditary) finitary abelian category $\\mathcal C$\, one can assoc
iate to it the Hall algebra $H_\\mathcal C$\, which encodes the combinator
ial information about extensions between objects in $\\mathcal C$. When $\
\mathcal C$ is the category of finite-dimensional representations of a qui
ver over a finite field\, these algebras are rather well understood and pr
ovide a useful viewpoint on quantum groups. Much less is known when $\\mat
hcal C=\\operatornameCohX$ is the category of coherent sheaves over a smoo
th projective curve $X$\, even though they seem to encode interesting geom
etric and arithmetic data associated to curves\, such as the number of cus
pidal functions. In particular\, there is no comprehensive representation
theory of such algebras.My talk will consist of two parts. The first hour
will primarily serve as a motivation for the second part\, and will be ded
icated to an overview of the classical theory of Hall algebras. We will re
call their relation to quantum groups as evidenced by Ringel's theorem\, a
nd some structural results in the case when $\\mathcal C=\\operatornameCoh
X$. In the second hour\, I will shift the gears somewhat and introduce the
cohomological version of Hall algebras (so-called CoHAs). I will then dis
cuss some recent results providing an action of CoHA of Higgs torsion shea
ves on homology groups of certain moduli spaces. These moduli spaces are h
eavily inspired by Nakajima quiver varieties\, and are closely related to
Hilbert schemes of points and moduli of sheaves on $T^*X$. If time permits
\, I will speculate about possible analogs of these CoHAs for an arbitrary
smooth surface $S$ and/or extension of these results to Higgs bundles.
LOCATION:Big Seminar room Ground floor / Office Bldg West (I21.EG.101)\, IS
T Austria
ORGANIZER:jdeanton@ist.ac.at
SUMMARY:Cohomological Hall algebras of Higgs torsion sheaves and moduli of
stable triples
URL:https://talks-calendar.app.ist.ac.at/events/993
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