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SUMMARY:Eugene Gorsky (UC Davis)
DTSTART;VALUE=DATE-TIME:20200824T180000Z
DTEND;VALUE=DATE-TIME:20200824T190000Z
DTSTAMP;VALUE=DATE-TIME:20201101T005957Z
UID:UMassRep/2
DESCRIPTION:Title: Parabolic Hilbert schemes on singular curves and repres
entation theory\nby Eugene Gorsky (UC Davis) as part of UMass Amherst Repr
esentation theory seminar\n\n\nAbstract\nI will construct representations
of various interesting algebras (such as rational Cherednik algebras and q
uantized Gieseker varieties) using the geometry of parabolic Hilbert schem
es of points on plane curve singularities. A connection to Coulomb branch
algebras of Braverman\, Finkelberg and Nakajima will be also outlined. The
talk is based on a joint work with Jose Simental and Monica Vazirani.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dima Arinkin (UWisconsin-Madison)
DTSTART;VALUE=DATE-TIME:20200914T200000Z
DTEND;VALUE=DATE-TIME:20200914T210000Z
DTSTAMP;VALUE=DATE-TIME:20201101T005957Z
UID:UMassRep/3
DESCRIPTION:Title: Singular support of categories\nby Dima Arinkin (UWisco
nsin-Madison) as part of UMass Amherst Representation theory seminar\n\n\n
Abstract\nIn many situations\, geometric objects on a space have some kind
of singular support\, which refines the usual support.\nFor instance\, fo
r smooth X\, the singular support of a D-module (or a perverse sheaf) on X
is as a conical subset\nof the cotangent bundle\; there is also a version
of this notion for coherent sheaves on local complete intersections.\nI w
ould like to describe a higher categorical version of this notion.\n\nLet
X be a smooth variety\, and let Z be a closed conical isotropic subset of
the cotangent bundle of X. I will define a\n2-category associated with Z\;
its objects may be viewed as `categories over X with singular support in
Z'. In particular\, if Z is\nthe zero section\, this gives the notion of c
ategories over Z in the usual sense.\n\nThe project is motivated by the lo
cal geometric Langlands correspondence\; time permitting\,\nI hope to sket
ch the relation with the Langlands correspondence at the end of the talk.\
n
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BEGIN:VEVENT
SUMMARY:David Nadler (UC Berkeley)
DTSTART;VALUE=DATE-TIME:20200921T200000Z
DTEND;VALUE=DATE-TIME:20200921T210000Z
DTSTAMP;VALUE=DATE-TIME:20201101T005957Z
UID:UMassRep/4
DESCRIPTION:Title: Verlinde formulas in Betti Geometric Langlands\nby Davi
d Nadler (UC Berkeley) as part of UMass Amherst Representation theory semi
nar\n\n\nAbstract\nI will discuss recent progress in "gluing" automorphic
categories of sheaves found in arxiv:2003.11477 and joint work with Zhiwei
Yun. Roughly speaking\, the geometry involves the wonderful compactificat
ion/Vinberg degeneration of loop groups. I will focus on the case of curve
s of genus one and its relation to the Drinfeld cocenter/topological Hoch
schild homology category of the affine Hecke category.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martha Precup (Washington University)
DTSTART;VALUE=DATE-TIME:20201019T200000Z
DTEND;VALUE=DATE-TIME:20201019T210000Z
DTSTAMP;VALUE=DATE-TIME:20201101T005957Z
UID:UMassRep/5
DESCRIPTION:Title: The cohomology of nilpotent Hessenberg varieties and th
e dot action representation\nby Martha Precup (Washington University) as p
art of UMass Amherst Representation theory seminar\n\n\nAbstract\nIn 2015\
, Brosnan and Chow\, and independently Guay-Paquet\, proved the Shareshian
--Wachs conjecture\, which links the combinatorics of chromatic symmetric
functions to the geometry of Hessenberg varieties via a permutation group
action on the cohomology ring of regular semisimple Hessenberg varieties.
This talk will give a brief overview of that story and discuss how the dot
action can be computed in all Lie types using the Betti numbers of certai
n nilpotent Hessenberg varieties. As an application\, we obtain new geomet
ric insight into certain linear relations satisfied by chromatic symmetric
functions\, known as the modular law. This is joint work with Eric Sommer
s.\n
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BEGIN:VEVENT
SUMMARY:Mikhail Mazin (Kansas State University)
DTSTART;VALUE=DATE-TIME:20200928T180000Z
DTEND;VALUE=DATE-TIME:20200928T190000Z
DTSTAMP;VALUE=DATE-TIME:20201101T005957Z
UID:UMassRep/6
DESCRIPTION:Title: Equivariant K-theory of the partial flag varieties.\nby
Mikhail Mazin (Kansas State University) as part of UMass Amherst Represen
tation theory seminar\n\n\nAbstract\nBack in 1990 Beilinson\, Lusztig\, an
d MacPherson used convolution algebras of diagonal orbits in the double pa
rtial flag varieties over finite fields to provide a geometric framework f
or the quantum groups in type A. In 1998 Vasserot used equivariant K-theor
y of the Steinberg subvarieties in the cotangent bundle of the double part
ial flag varieties to provide a geometric framework for the affine quantum
group.\n\nIn a joint project with Sergey Arkhipov\, we define an algebra
$\\mathcal{A}_n$ that plays the role of a $q=0$ degeneration of the affine
quantum group of type $A_n$\, and use the equivariant K-theory of the dou
ble partial flag variety with $n$ steps to provide a geometric framework f
or it. Our algebra is defined via generators and relations. Then for each
dimension $d$ of the ambient space\, we show that there is a natural surje
ctive map $\\mathcal{A}_n\\to A(n\,d)$\, were $A(n\,d)$ is the equivariant
K-theory of the double partial flag variety with n step in $\\mathbb{C}^
d$ equipped with the convolution product.\n
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BEGIN:VEVENT
SUMMARY:Joel Kamnitzer (University of Toronto)
DTSTART;VALUE=DATE-TIME:20201005T180000Z
DTEND;VALUE=DATE-TIME:20201005T190000Z
DTSTAMP;VALUE=DATE-TIME:20201101T005957Z
UID:UMassRep/7
DESCRIPTION:Title: BFN Springer theory\nby Joel Kamnitzer (University of T
oronto) as part of UMass Amherst Representation theory seminar\n\n\nAbstra
ct\nGiven a representation of a reductive group\,\nBraverman-Finkelberg-Na
kajima have defined a remarkable Poisson\nvariety called the Coulomb branc
h. Their construction of this space\nwas motivated by considerations from
supersymmetric gauge theories and\nsymplectic duality. The coordinate ring
of this Coulomb branch is\ndefined as a kind of cohomological Hall algebr
a.\n\nWe develop a theory of Springer fibres related to\nBraverman-Finkelb
erg-Nakajima's construction. We use these Springer\nfibres to construct m
odules for\n(quantized) Coulomb branch algebras. In doing so\, we partial
ly prove a\nconjecture of Baumann-Kamnitzer-Knutson and give evidence for\
nconjectures of Hikita\, Nakajima\, and Kamnitzer-McBreen-Proudfoot. We\n
also prove a relation between BFN Springer fibres and quasimap spaces\n
END:VEVENT
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SUMMARY:Mee Seong Im (United States Naval Academy)
DTSTART;VALUE=DATE-TIME:20200831T180000Z
DTEND;VALUE=DATE-TIME:20200831T190000Z
DTSTAMP;VALUE=DATE-TIME:20201101T005957Z
UID:UMassRep/9
DESCRIPTION:Title: Nakajima quiver varieties and irreducible components of
Springer fibers\nby Mee Seong Im (United States Naval Academy) as part of
UMass Amherst Representation theory seminar\n\n\nAbstract\nSpringer fiber
s and Nakajima quiver varieties are amongst the most important objects in
geometric representation theory. While Springer fibers can be used to geom
etrically construct and classify irreducible representations of Weyl group
s\, Nakajima quiver varieties play a key role in the geometric representat
ion theory of Kac--Moody Lie algebras.\nI will begin by first recalling so
me background on the objects of interest mentioned above. I will then conn
ect Springer fibers and quiver varieties by realizing the irreducible comp
onents of two-row Springer fibers inside a suitable Nakajima quiver variet
y and describing the resulting subvariety in terms of explicit quiver repr
esentations.\n\nNext\, consider certain fixed-point subvarieties of these
quiver varieties\, which were studied by Henderson--Licata and Li with the
goal of developing the geometric representation theory for certain coidea
l subalgebras. By applying this machinery\, I will give an explicit algebr
aic description of the irreducible components of all two-row Springer fibe
rs for classical types\, thereby generalizing results of Fung and Stroppel
--Webster in type A.\n\nThis is joint with C.-J. Lai and A. Wilbert.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lin Chen (Harvard)
DTSTART;VALUE=DATE-TIME:20201123T190000Z
DTEND;VALUE=DATE-TIME:20201123T200000Z
DTSTAMP;VALUE=DATE-TIME:20201101T005957Z
UID:UMassRep/10
DESCRIPTION:by Lin Chen (Harvard) as part of UMass Amherst Representation
theory seminar\n\nAbstract: TBA\n
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SUMMARY:Sam Raskin (University of Texas at Austin)
DTSTART;VALUE=DATE-TIME:20201102T210000Z
DTEND;VALUE=DATE-TIME:20201102T220000Z
DTSTAMP;VALUE=DATE-TIME:20201101T005957Z
UID:UMassRep/11
DESCRIPTION:Title: Geometric Langlands for l-adic sheaves\nby Sam Raskin (
University of Texas at Austin) as part of UMass Amherst Representation the
ory seminar\n\n\nAbstract\nIn celebrated work\, Beilinson-Drinfeld formula
ted a categorical analogue of the Langlands program for unramified automor
phic forms. Their conjecture has appeared specialized to the setting of al
gebraic D-modules: non-holonomic D-modules play a prominent role in known
constructions. \n\nIn this talk\, we will discuss a categorical conjecture
suitable in other geometric settings\, including l-adic sheaves. One of t
he main constructions is a suitable moduli space of local systems. We will
also discuss applications to unramified automorphic forms for function fi
elds. This is joint work with Arinkin\, Gaitsgory\, Kazhdan\, Rozenblyum\,
and Varshavsky.\n
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BEGIN:VEVENT
SUMMARY:Tudor Dimofte (UC Davis)
DTSTART;VALUE=DATE-TIME:20201026T180000Z
DTEND;VALUE=DATE-TIME:20201026T190000Z
DTSTAMP;VALUE=DATE-TIME:20201101T005957Z
UID:UMassRep/12
DESCRIPTION:Title: 3d mirror symmetry and HOMFLY-PT homology\nby Tudor Dim
ofte (UC Davis) as part of UMass Amherst Representation theory seminar\n\n
\nAbstract\nSince the original physical prediction of triply-graded HOMFLY
-PT link homology by Gukov-Schwarz-Vafa\, and its mathematical definition
by Khovanov-Rozansky\, many other (conjectural) constructions of HOMFLY-PT
link homology have appeared --- with different algebraic and geometric or
igins\, and manifesting different properties. One recent proposal of Oblom
kov-Rozansky (closely related to work of Gorsky-NeguČ›-Rasmussen) associat
ed to a link L a coherent sheaf E_L on a Hilbert scheme\, whose cohomology
reproduces HOMFLY-PT homology. Another proposal\, by Gorsky-Oblomkov-Rasm
ussen-Shende\, computes HOMFLY-PT homology of algebraic knots via Borel-Mo
ore homology of affine Springer fibers. I will explain how the first (Hilb
ert scheme) construction is realized in the "B" twist of a 3d supersymmetr
ic gauge theory\, and then carefully apply 3d mirror symmetry to discover
a variant of the second (Springer fiber) construction. I will also indicat
e how both 3d gauge theory setups are related to the original work of Guko
v-Schwarz-Vafa based using M-theory on the resolved conifold. (Preprint so
on to appear\, with N. Garner\, J. Hilburn\, A. Oblomkov\, and L. Rozansky
).\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nick Rozenblyum (University of Chicago)
DTSTART;VALUE=DATE-TIME:20201109T190000Z
DTEND;VALUE=DATE-TIME:20201109T200000Z
DTSTAMP;VALUE=DATE-TIME:20201101T005957Z
UID:UMassRep/13
DESCRIPTION:by Nick Rozenblyum (University of Chicago) as part of UMass Am
herst Representation theory seminar\n\nAbstract: TBA\n
END:VEVENT
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SUMMARY:Alexander Braverman (University of Toronto)
DTSTART;VALUE=DATE-TIME:20201012T200000Z
DTEND;VALUE=DATE-TIME:20201012T210000Z
DTSTAMP;VALUE=DATE-TIME:20201101T005957Z
UID:UMassRep/14
DESCRIPTION:Title: Category O via Zastava spaces\nby Alexander Braverman (
University of Toronto) as part of UMass Amherst Representation theory semi
nar\n\n\nAbstract\nIn my talk I will recall basic results about category O
for\nfinite-dimensional and affine Lie algebras - such as Kazhdan-Lusztig
\nconjecture\, Jantzen conjecture etc. I will then describe a new\ngeometr
ic approach to proving these conjectures via so called Zastava\nspaces. de
veloped in my recent paper with Finkelberg and Nakajima. In\nthat paper we
give a new proof of the Kazhdan-Lusztig conjecture for\nsemi-simple Lie a
lgebras\, I will describe how it should be possible to\nextend this to Jan
tzen conjectures and to the affine case.\n
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