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SUMMARY:Evgeny Feigin (National Research University Higher School of Econo
mics)
DTSTART:20201002T150000Z
DTEND:20201002T160000Z
DTSTAMP:20260404T095652Z
UID:LieTheory/1
DESCRIPTION:Title: Veronese embeddings\, arc schemes and global Demazure modules
\nby Evgeny Feigin (National Research University Higher School of Economic
s) as part of CRM-Regional Conference in Lie Theory\n\nLecture held in Vir
tual.\n\nAbstract\nVeronese curve of degree d (also known as rational norm
al curve) can be realized as an embedding of the complex projective line i
nto a d-dimensional projective space. The equations cutting out the image
of this embedding can be written down explicitly and the homogeneous coor
dinate ring has an explicit description in terms of representations of the
complex Lie algebra sl(2). To pass to the corresponding arc scheme\, one
replaces the field of complex numbers with the ring of formal Taylor serie
s in one variable. We describe the reduced ideal of the arc scheme and the
homogeneous coordinate ring in terms of representation theory of the curr
ent algebra of sl(2).\nThe whole picture generalizes to the case of an arb
itrary simple Lie algebra. The analogues of the rational normal curves ar
e the Veronese embeddings of the flag varieties for the corresponding Lie
group. We identify the homogeneous coordinate ring of the reduced arc sche
me of the Veronese embedding with the direct sum of the global Demazure mo
dules of the current algebra (the higher level analogues of the global We
yl modules). Joint work with Ilya Dumanski. Geometric flows of $G_2$ and
Spin(7)-structures\n
LOCATION:https://stable.researchseminars.org/talk/LieTheory/1/
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BEGIN:VEVENT
SUMMARY:Yvan Saint-Aubin (Université de Montréal)
DTSTART:20201002T170000Z
DTEND:20201002T180000Z
DTSTAMP:20260404T095652Z
UID:LieTheory/2
DESCRIPTION:Title: The strucutre of the periodic spin chain XXZ seen as a module ove
r the affine Temperley-Lieb algebra\nby Yvan Saint-Aubin (Université
de Montréal) as part of CRM-Regional Conference in Lie Theory\n\nLecture
held in Virtual.\n\nAbstract\nhttp://www.crm.umontreal.ca/2020/LieAutomne2
0/pdf/Saint-Aubin.pdf\n
LOCATION:https://stable.researchseminars.org/talk/LieTheory/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Creutzig (University of Alberta)
DTSTART:20201002T183000Z
DTEND:20201002T193000Z
DTSTAMP:20260404T095652Z
UID:LieTheory/3
DESCRIPTION:Title: VOAs: From representation theory to physics\nby Thomas Creutz
ig (University of Alberta) as part of CRM-Regional Conference in Lie Theor
y\n\nLecture held in Virtual.\n\nAbstract\nVertex operator algebras (VOAs)
often serve as a bridge connecting interesting problems. I aim to explain
such an exciting connection to non-experts. The problem are representat
ions of affine Lie superalgebras and related W-superalgebras and their con
nection to geometry and physics.\nI will introduce VOA-analogues of the sp
ace of functions on a compact Lie group. Then I will explain what these VO
As tell us about equivalences of representation categories of different VO
As and how they are motivated from dualities in four-dimensional gauge the
ories.\n
LOCATION:https://stable.researchseminars.org/talk/LieTheory/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martina Lanini (Università degli Studi di Roma “Tor Vergata”)
DTSTART:20201003T150000Z
DTEND:20201003T160000Z
DTSTAMP:20260404T095652Z
UID:LieTheory/4
DESCRIPTION:Title: Torus actions on cyclic quiver Grassmannians\nby Martina Lani
ni (Università degli Studi di Roma “Tor Vergata”) as part of CRM-Regi
onal Conference in Lie Theory\n\nLecture held in Virtual.\n\nAbstract\nI w
ill report on recent joint work with Alexander Puetz\, where we define and
investigate algebraic torus actions on quiver Grassmannians for nilpotent
representations of the equioriented cycle.\nThese quiver Grassmannians\,
equipped with such torus actions\, are equivariantly formal spaces\, and t
he corresponding moment graphs can be combinatorially described and exploi
ted to compute equivariant cohomology. Our construction generalises the ve
ry much investigated (maximal) torus actions on type A flag varieties.\n
LOCATION:https://stable.researchseminars.org/talk/LieTheory/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgia Benkart (University of Wisconsin-Madison)
DTSTART:20201003T170000Z
DTEND:20201003T180000Z
DTSTAMP:20260404T095652Z
UID:LieTheory/5
DESCRIPTION:Title: Fusion Rules\nby Georgia Benkart (University of Wisconsin-Mad
ison) as part of CRM-Regional Conference in Lie Theory\n\nLecture held in
Virtual.\n\nAbstract\nFusion rules encode information about tensoring diff
erent types of modules \n(simple\, projective) with a finite-dimensional
module V\, and this information\ncan be recorded in a matrix that depends
on V. When the objects are complex \nsimple modules for a finite group\,
the resulting matrix (often called the McKay \nmatrix due to inspiration
from the McKay correspondence)\, has as its right \neigenvectors the colum
ns of the character table of the group\, and the eigenvalues\nare the char
acter of V evaluated on conjugacy class representatives. So in that\npart
icular case\, the eigenvectors are independent of V. We consider extensi
ons of \nsuch results to other settings where tensor products are defined
such as finite-dimensional\nHopf algebras (e.g. quantum groups at roots o
f unity\, restricted enveloping algebras\nof Lie algebras in prime charact
eristic\, and Drinfeld doubles). The eigenvectors and \neigenvalues have c
onnections with the characters of the Hopf algebra\, and in some \nexample
s\, many connections with Chebyshev polynomials of various kinds. Fusion r
ule \nmatrices have applications to chip-firing dynamics and to Markov cha
ins.\n\nIn this talk I will explain a joint work with Javier Aramayona\, J
ulio Aroca\, Rachel Skipper and Xiaolei Wu. We define a new family of grou
ps that are subgroups of the mapping class group $Map(\\Sigma_g)$ of a sur
face $\\Sigma_g$ of genus $g$ with a Cantor set removed and we will call t
hese groups Block Mapping Class Groups $B(H)$\, where $H$ is a subgroup of
$\\Sigma_g$ . More visually\, this family will be constructed by making a
tree-like surface gluing pair of pants and taking homeomorphisms that dep
end on $H$ with certain preservation properties (it will preserve what we
will call a block decomposition of this surface\, hence the name of our gr
oups). We will see that this family is closely related to Thompson’s gro
ups and that it has the property of being of type $F_n$ if and only if $H$
is. As a consequence\, for every $g\\in \\mathbb N \\cup \\{0\, \\infty\\
}$ and every $n\\ge 1$\, we construct a subgroup $G <\\Map(\\Sigma_g)$ tha
t is of type $F_n$ but not of type $F_{n+1}$\, and which contains the mapp
ing class group of every compact surface of genus less or equal to $\\g$ a
nd with non-empty boundary. As expected in this workshop\, the techniques
involve manipulating cube complexes\, as the Stein-Farley cube complex.\n
LOCATION:https://stable.researchseminars.org/talk/LieTheory/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Weekes (University of British Columbia)
DTSTART:20201003T183000Z
DTEND:20201003T193000Z
DTSTAMP:20260404T095652Z
UID:LieTheory/6
DESCRIPTION:Title: Coulomb branches and Yangians\nby Alex Weekes (University of
British Columbia) as part of CRM-Regional Conference in Lie Theory\n\nLect
ure held in Virtual.\n\nAbstract\nA classical result of Jørgensen and Thu
rston shows that the set of volumes of finite volume complete hyperbolic 3
-manifolds is a \nwell-ordered subset of the real numbers of order type w^
w\; moreover\, they showed that each volume can only be attained by finite
ly many isometry types of hyperbolic 3-manifolds.\nWe will discuss a group
-theoretic analogue of this result: If $\\Gamma$\nis a non-elementary hype
rbolic group\, then the set of exponential growth rates of $\\Gamma$ is w
ell-ordered\, the order type is at least w^w\, and each growth rate can on
ly be attained\n by finitely many finite generating sets (up to automorphi
sms)\, and further generalizations of these results.\nThe talk is intended
to be for a wider audience. All the notions that are mentioned in the abs
tract will be explained. It is based on a joint work with K. Fujiwara.\n\n
\n\n\n\nBraverman\, Finkelberg and Nakajima have recently given a mathemat
ical definition of the Coulomb branches associated to certain 3-dimensiona
l quantum field theories. They define Coulomb branches as affine algebraic
varieties\, and showed that many interesting varieties\narise in this way
.\n\nThe BFN construction also produces quantized Coulomb branches\, which
are non-commutative algebra. It is interesting to try to relate these non
-commutative algebras with more familiar ones\; one nice example \nthat ar
ises is the enveloping algebra of gl(n).\n\nI'll discuss how certain quant
ized Coulomb branches can be described using Yangians. This means that the
re are explicit generators for the quantized Coulomb branch (which is othe
rwise rather abstractly defined)\, a fact which has found application in d
escribing connections between Coulomb branches and cluster algebras. But g
oing the other way\, we may also learn more about Yangians and their modul
es by leveraging results from the Coulomb branch theory. In my talk\, I wi
ll overview recent progress on these topics.\n\n\nThe classical umkehr map
of Hopf assigns to a map of oriented manifolds\, $f:M \\to N\,$ `wrong-wa
y' homomorphisms in homology $f_!: H_*(N) \\to H_*(M)$ and in cohomology $
f^!:H^*(M) \\to H^*(N)\,$ the latter a version of `integration over the fi
bers'. Similar wrong-way maps\, sometimes known as transfer maps or Gysin
maps\, are defined for other generalized (co)homology theories as long as
the manifolds are suitably oriented and have had many applications. While
these maps are defined only for manifolds there has long been interest in
extending them to singular spaces. I'll discuss joint work with Markus Ba
nagl and Paolo Piazza in which we capitalize on recent work on the index t
heory of signature operators to give analytic definitions of transfer maps
in K-homology for stratified spaces and relate them to topological orient
ations.\n
LOCATION:https://stable.researchseminars.org/talk/LieTheory/6/
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