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BEGIN:VEVENT
SUMMARY:Michael Shapiro (Michigan State University)
DTSTART;VALUE=DATE-TIME:20200901T150000Z
DTEND;VALUE=DATE-TIME:20200901T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/1
DESCRIPTION:Title: Non
-commutative Networks on a Cylinder\nby Michael Shapiro (Michigan Stat
e University) as part of Online Cluster Algebra Seminar (OCAS)\n\nAbstract
: TBA\n
LOCATION:https://researchseminars.org/talk/OCAS/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Li Li (Oakland University)
DTSTART;VALUE=DATE-TIME:20200908T150000Z
DTEND;VALUE=DATE-TIME:20200908T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/2
DESCRIPTION:Title: A p
roof of two conjectures on Markov Numbers\nby Li Li (Oakland Universit
y) as part of Online Cluster Algebra Seminar (OCAS)\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/OCAS/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lara Bossinger (UNAM-Oaxaca)
DTSTART;VALUE=DATE-TIME:20200915T150000Z
DTEND;VALUE=DATE-TIME:20200915T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/3
DESCRIPTION:Title: Und
erstanding universal coefficients of Grassmannians through Groebner theory
\nby Lara Bossinger (UNAM-Oaxaca) as part of Online Cluster Algebra Se
minar (OCAS)\n\n\nAbstract\nIn this talk I will present recent results of
a joint work with Fatemeh Mohammadi and Alfredo Nájera Chávez. For a po
larized weighted projective variety V(J) we introduce a flat family that c
ombines all Groebner degenerations of V associated to a maximal cone in th
e Groebner fan of J. It turns out that this family can alternatively be ob
tained as a pull-back of a toric family (in the sense of Kaveh--Manon's cl
assification of such).\nThe most surprising application of this constructi
on is its relation to cluster algebras with universal coefficients. To dem
onstrate this connection we analyze the cases of the Grassmannians Gr(2\,n
) and Gr(3\,6) in depth.\nFor Gr(2\,n) we fix its Pluecker embedding and f
or Gr(3\,6) what we call its "cluster embedding". In both cases we identif
y a specific maximal cone C in the Groebner fan of the defining ideal such
that the algebra defining the flat family mentioned above is canonically
isomorphic to the corresponding cluster algebra with universal coefficient
s.\n
LOCATION:https://researchseminars.org/talk/OCAS/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fan Qin (Shanghai Jiao Tong University)
DTSTART;VALUE=DATE-TIME:20200922T150000Z
DTEND;VALUE=DATE-TIME:20200922T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/4
DESCRIPTION:Title: Dua
l canonical bases and triangular bases of quantum cluster algebras\nby
Fan Qin (Shanghai Jiao Tong University) as part of Online Cluster Algebra
Seminar (OCAS)\n\n\nAbstract\nOne of the main motivations for cluster alg
ebras was to create a combinatorial framework to study the dual canonical
bases. Correspondingly\, it has been long expected that the quantum cluste
r monomials (certain monomials of generators) belong to the dual canonical
bases (of quantum unipotent subgroups) up to scalar multiples. We discuss
how to use the triangular bases to show this conjecture in full generalit
y. Moreover\, we show that the (double) triangular bases verify an analog
of Leclerc’s conjecture for dual canonical bases.\n
LOCATION:https://researchseminars.org/talk/OCAS/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Esther Banaian (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20200929T150000Z
DTEND;VALUE=DATE-TIME:20200929T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/5
DESCRIPTION:Title: Fri
eze Patterns from Dissections\nby Esther Banaian (University of Minnes
ota) as part of Online Cluster Algebra Seminar (OCAS)\n\n\nAbstract\nFinit
e frieze patterns of positive integers were shown by Conway and Coxeter t
o be in bijection with triangulated polygons. Baur\, Parsons\, and Tschabo
ld generalized this result\, showing that infinite\nfrieze patterns of pos
itive integers are in bijection with triangulated annuli and once-punctur
ed discs. More recently\, Holm and Jørgensen investigated frieze patterns
arising from dissected polygons.\nThe frieze patterns of Holm and Jørge
nsen involve algebraic integers of the form 2cos(pi/p) for an integer p.
We combine these generalizations and present results on frieze patterns fr
om dissected\nannuli\, using these same algebraic integers. We also discu
ss how some of these frieze patterns from dissections can be connected to
generalized cluster algebras\, in the sense of Chekhov and Shapiro.\nThis
is based on joint work with Jiuqi (Lena) Chen and with Elizabeth Kelley.
\n
LOCATION:https://researchseminars.org/talk/OCAS/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melissa Sherman-Bennett (UC Berkely and Harvard University)
DTSTART;VALUE=DATE-TIME:20201006T150000Z
DTEND;VALUE=DATE-TIME:20201006T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/6
DESCRIPTION:Title: Man
y cluster structures on positroid varieties\nby Melissa Sherman-Bennet
t (UC Berkely and Harvard University) as part of Online Cluster Algebra Se
minar (OCAS)\n\n\nAbstract\nEarly in the history of cluster algebras\, Sco
tt showed that the homogeneous coordinate ring of the Grassmannian is a cl
uster algebra\, with seeds given by Postnikov's plabic graphs for the\nGr
assmannian. Recently the analogous statement has been proved for open Schu
bert varieties (Leclerc\, Serhiyenko-SB-Williams) and more generally\, for
open positroid varieties (Galashin-Lam). I'll\ndiscuss joint work with Ch
ris Fraser\, in which we give a family of cluster structures on open Schub
ert (and more generally\, positroid) varieties. Each of the cluster struct
ures in this family has seeds given by face labels of relabeled plabic gra
phs\, which are plabic graphs whose boundary is labeled by a permutation o
f 1\, ...\, n. For Schubert varieties\, all cluster structures in this fam
ily\nquasi-coincide\, meaning they differ only by rescaling by frozen vari
ables and their cluster monomials coincide. In particular\, all relabeled
plabic graphs for a Schubert variety give rise to seeds in the "usual" clu
ster algebra structure on the coordinate ring. As part of our results\, we
show the "target" and "source" cluster structures on Schubert varieties q
uasi-coincide\, confirming a conjecture of Muller and Speyer. One proof to
ol of independent interest is a permuted version of the Muller-Speyer twis
t map\, which we use to prove many (open) positroid varieties are isomorph
ic.\n
LOCATION:https://researchseminars.org/talk/OCAS/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Jørgensen (Aarhus University)
DTSTART;VALUE=DATE-TIME:20201013T150000Z
DTEND;VALUE=DATE-TIME:20201013T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/7
DESCRIPTION:Title: The
green groupoid and its action on derived categories (joint work with Mile
n Yakimov)\nby Peter Jørgensen (Aarhus University) as part of Online
Cluster Algebra Seminar (OCAS)\n\n\nAbstract\nWe introduce the green group
oid $\\mathcal{G}$ of a $2$-Calabi-Yau triangulated category $\\mathcal{C}
$. It is an augmentation of the mutation graph of $\\mathcal{C}$\, which
is defined by means of silting theory.\n\nThe green groupoid $\\mathcal{G}
$ has certain key properties:\n\n1. If $\\mathcal{C}$ is the stable catego
ry of a Frobenius category $\\mathcal{E}$\, then $\\mathcal{G}$ acs on the
derived categories of the endomorphism rings $\\mathcal{E}(m\,m)$ where $
m$ is a maximal rigid object.\n\n2. $\\mathcal{G}$ can be obtained geometr
ically from the $g$-vector fan of $\\mathcal{C}$.\n\n3. If the $g$-vector
fan of $\\mathcal{C}$ is a hyperplane arrangement $\\mathcal{H}$\, then $\
\mathcal{G}$ specialises to the Deligne groupoid of $\\mathcal{H}$\, and $
\\mathcal{G}$ acts faithfully on the derived categories of the endomorphis
m rings $\\mathcal{E}(m\,m)$.\n\nThe situation in (3) occurs if $\\Sigma_{
\\mathcal{C}}^2$\, the square of the suspension functor\, is the identity.
It recovers results by Donovan\, Hirano\, and Wemyss where $\\mathcal{E}
$ is the category of maximal Cohen-Macaulay modules over a suitable singul
arity.\n
LOCATION:https://researchseminars.org/talk/OCAS/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Lam (University of Michigan)
DTSTART;VALUE=DATE-TIME:20201020T150000Z
DTEND;VALUE=DATE-TIME:20201020T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/8
DESCRIPTION:Title: Clu
ster configuration spaces of finite type\nby Thomas Lam (University of
Michigan) as part of Online Cluster Algebra Seminar (OCAS)\n\n\nAbstract\
nI will talk about a "cluster configuration space" $M_D$\,\ndepending on a
finite Dynkin diagram $D$. The space $M_D$ is an affine\nalgebraic varie
ty that is defined using only the compatibility degree\nof the correspondi
ng finite-type cluster algebra. In the case that $D$\nis of type $A$\, we
recover the configuration space $M_{0\,n}$ of $n$\n(distinct) points in $
P^1$. There are many relations to finite-type\ncluster theory\, but an es
pecially close connection to the finite-type\ncluster algebra with univers
al coefficients.\n
LOCATION:https://researchseminars.org/talk/OCAS/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gregg Musiker (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20201117T160000Z
DTEND;VALUE=DATE-TIME:20201117T170000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/9
DESCRIPTION:Title: Dou
ble-dimer configurations and quivers of dP3 (del Pezzo) type\nby Gregg
Musiker (University of Minnesota) as part of Online Cluster Algebra Semin
ar (OCAS)\n\n\nAbstract\nIn this talk\, I will describe our work extending
combinatorial interpretations for so called toric cluster variables as wa
s previously studied by myself and Tri Lai. In [LM 2017] and [LM 2020]\, m
ost toric cluster variables were shown to have Laurent expansions agreeing
with partition functions of dimers on subgraphs cut out by six-sided cont
ours. However\, the case of cluster variables\nparameterized by six-sided
contours with a self-intersection eluded our techniques. In this talk we d
iscuss our research rectifying this issue by using Helen Jenne’s condens
ation results for the\ndouble-dimer model [J 2019]. While we focus on quiv
ers of dP3 type of Model 1 and Model 4\, we anticipate our techniques will
extend to certain additional cluster algebras related to brane tilings.
\nThis is joint work with Helen Jenne and Tri Lai.\n
LOCATION:https://researchseminars.org/talk/OCAS/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karin Baur (University of Leeds)
DTSTART;VALUE=DATE-TIME:20201103T160000Z
DTEND;VALUE=DATE-TIME:20201103T170000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/10
DESCRIPTION:Title: Fl
ips in triangulations and matchings\nby Karin Baur (University of Leed
s) as part of Online Cluster Algebra Seminar (OCAS)\n\n\nAbstract\nPlane p
erfect matchings of $2n$ points in convex position are known to be in\nbij
ection with triangulations of convex polygons of size $n + 2$\; they are\n
both counted by the Catalan numbers.\nWe explain how to give a direct bije
ction and how it can be extended to a\nbijection between monochromatic mat
chings on $k$ colours and tilings by\n$(k+2)$-gons. Edge flips are a class
ic operation to perform local changes in\nboth sets. We use the above bije
ction to determine the two types of edge\nflips are related. We use this t
o give an algebraic interpretation of the\nflip graph of triangulations in
terms of elements of the corresponding\nTemperley-Lieb algebra.\nThis is
joint work with\nO. Aichholzer\, L. Donner (Andritsch)\, B. Vogtenhuber.\n
LOCATION:https://researchseminars.org/talk/OCAS/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dylan Allegretti (University of British Columbia)
DTSTART;VALUE=DATE-TIME:20201110T160000Z
DTEND;VALUE=DATE-TIME:20201110T170000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/11
DESCRIPTION:Title: St
ability conditions and cluster varieties\nby Dylan Allegretti (Univers
ity of British Columbia) as part of Online Cluster Algebra Seminar (OCAS)\
n\n\nAbstract\nIn the first part of the talk\, I will describe a construct
ion in low-dimensional topology that takes a holomorphic quadratic differe
ntial on a surface and produces a $PGL(2)$-local system. This\nconstructio
n provides a local homeomorphism from the moduli space of quadratic differ
entials to the moduli space of local systems. In the second part of the ta
lk\, I will propose a categorical\ngeneralization of this construction. In
this generalization\, the space of quadratic differentials is replaced by
a complex manifold parametrizing Bridgeland stability conditions on a cer
tain\n3-Calabi-Yau triangulated category\, while the space of local system
s is replaced by a cluster variety. I will describe a local homeomorphism
from the space of stability conditions to the cluster\nvariety in a large
class of examples and explain how it preserves the structures of these two
spaces.\n
LOCATION:https://researchseminars.org/talk/OCAS/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christof Geiss
DTSTART;VALUE=DATE-TIME:20201208T160000Z
DTEND;VALUE=DATE-TIME:20201208T170000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/12
DESCRIPTION:by Christof Geiss as part of Online Cluster Algebra Seminar (O
CAS)\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/OCAS/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathan Reading (North Carolina State University)
DTSTART;VALUE=DATE-TIME:20210119T160000Z
DTEND;VALUE=DATE-TIME:20210119T170000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/13
DESCRIPTION:Title: Th
eta functions and the mutation fan\nby Nathan Reading (North Carolina
State University) as part of Online Cluster Algebra Seminar (OCAS)\n\n\nAb
stract\nThe setting for this work is the cluster scattering diagram define
d by\nGross\, Hacking\, Keel\, and Kontsevich (GHKK). The cluster scatte
ring\ndiagram is a collection of walls (codimension-1 cones plus some\nadd
itional algebraic data). Theta functions (one for each g-vector)\ninclud
e the cluster monomials and form a basis for the cluster algebra\n(or ofte
n something larger). Explicit constructions of cluster\nscattering diagr
ams and explicit computations of theta functions are\nhopelessly complicat
ed in general\, but I believe that eventually there\nwill be combinatorial
models in all mutation-finite types. I'll\nmention work with Salvatore
Stella on combinatorial models in affine\ntype\, and work with Greg Muller
and Shira Viel on the surfaces case.\nBut I will spend most of the time d
iscussing a result (with Stella)\nthat I think will make it possible to co
mplete these combinatorial\nconstructions of theta functions.\n\nThe mutat
ion fan encodes the piecewise-linear geometry of matrix\nmutation. The r
esult is: If you take a product of theta functions\nwhose g-vectors are
all in one cone of the mutation fan\, the product\nexpands as a sum of the
ta functions whose g-vectors are all in one\ncone of the mutation fan. T
he result seems natural and in some sense\nunsurprising\, but it requires
some work and it is quite useful. The\nresult requires two serious chang
es in point of view from the GHKK\nsetup: Taking a different point of vi
ew on what "mutation of\nscattering diagrams" means and demoting "frozen v
ariables" to the\nstatus of coefficients.\n
LOCATION:https://researchseminars.org/talk/OCAS/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bruce Sagan (Michigan State University)
DTSTART;VALUE=DATE-TIME:20201124T160000Z
DTEND;VALUE=DATE-TIME:20201124T170000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/14
DESCRIPTION:Title: On
a rank-unimodality conjecture of Morier-Genoud and Ovsienko\nby Bruce
Sagan (Michigan State University) as part of Online Cluster Algebra Semin
ar (OCAS)\n\n\nAbstract\nLet $\\alpha=(a\,b\,\\ldots)$ be a composition\,
that is\, a finite sequence of positive integers. Consider the associated
partially ordered set $F(\\alpha)$\, called a fence\, whose covering rela
tions are\n$$
\nx_1\
\lhd x_2 \\lhd \\ldots\\lhd x_{a+1}\\rhd x_{a+2}\\rhd \\ldots\\rhd x_{a+b+
1}\\lhd x_{a+b+2}\\lhd \\ldots\\ .
\n$$\nWe
study the associated distributive lattice $L(\\alpha)$ consisting of all l
ower order ideals of $F(\\alpha)$.\nThese lattices are important in the th
eory of cluster algebras and their rank generating functions can be used t
o define $q$-analogues of rational numbers.\nWe make progress on a recent
conjecture of Morier-Genoud and Ovsienko that $L(\\alpha)$ is rank unimoda
l.\nAll terms from the theory of partially ordered sets will be carefully
defined. This is joint work with Thomas McConville and Clifford Smyth.\n
LOCATION:https://researchseminars.org/talk/OCAS/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elizabeth Kelley (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20201201T160000Z
DTEND;VALUE=DATE-TIME:20201201T170000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/15
DESCRIPTION:Title: Th
eta basis for reciprocal generalized cluster algebras\nby Elizabeth Ke
lley (University of Minnesota) as part of Online Cluster Algebra Seminar (
OCAS)\n\n\nAbstract\nCluster algebras are characterized by binomial exchan
ge relations. A natural generalization of these algebras\, introduced by C
hekhov and Shapiro\, relaxes this restriction and allows the\nexchange pol
ynomials to have arbitrarily many terms. Following the work of Gross\, Ha
cking\, Keel\, and Kontsevich\, we give the construction of scattering dia
grams for the subclass of generalized cluster\nalgebras with reciprocal ex
change coefficients. We then define the theta basis for these algebras and
show that the fixed data of the left companion algebra is\, up to isomorp
hism\, Langlands dual to\nthat of the right companion algebra (and vice ve
rsa). This is joint work with Man-Wai Cheug and Gregg Musiker.\n
LOCATION:https://researchseminars.org/talk/OCAS/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daping Weng (Michigan State University)
DTSTART;VALUE=DATE-TIME:20210126T160000Z
DTEND;VALUE=DATE-TIME:20210126T170000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/16
DESCRIPTION:Title: Sy
mplectic Structures on Augmentation Varieties\nby Daping Weng (Michiga
n State University) as part of Online Cluster Algebra Seminar (OCAS)\n\n\n
Abstract\nIn a recent joint project with H. Gao and L. Shen\, we introduce
a cluster K2 structure on\nthe augmentation variety of the Chekanov-Elias
hberg dga for the rainbow closure of any positive\nbraid with marked point
decorations. This cluster K2 structure naturally equips the complex\naugm
entation variety with a holomorphic presymplectic 2-form. Using a result o
f Goncharov and\nKenyon on surface bipartite graphs\, we prove that this h
olomorphic presymplectic 2-form becomes\nsymplectic after we reduce the nu
mber of marked points to a single marked per link component (plus\nsome mo
dification).\n
LOCATION:https://researchseminars.org/talk/OCAS/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hyun Kyu Kim (Ewha Womans University)
DTSTART;VALUE=DATE-TIME:20210216T160000Z
DTEND;VALUE=DATE-TIME:20210216T170000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/17
DESCRIPTION:Title: $A
_2$-laminations as basis for ${\\rm PGL}_3$ cluster variety for surface\nby Hyun Kyu Kim (Ewha Womans University) as part of Online Cluster Alg
ebra Seminar (OCAS)\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/OCAS/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leonid Chekhov (Steklov Mathematical Institute and Michigan State
University)
DTSTART;VALUE=DATE-TIME:20210202T160000Z
DTEND;VALUE=DATE-TIME:20210202T170000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/18
DESCRIPTION:Title: Da
rboux coordinates for symplectic groupoid and cluster algebras\nby Leo
nid Chekhov (Steklov Mathematical Institute and Michigan State University)
as part of Online Cluster Algebra Seminar (OCAS)\n\n\nAbstract\nThe talk
is based on Arxiv:2003:07499\, joint work with Misha Shapiro. We use Fock-
-Goncharov higher Teichmüller space variables to derive Darboux coordina
te representation for entries of general symplectic leaves of the $\\mathc
al A_n$ groupoid of upper-triangular matrices and\, in a more general sett
ing\, of higher-dimensional symplectic leaves for algebras governed by the
quantum reflection equation with the trigonometric $R$-matrix. This resul
t can be generalized to any planar directed network on disc with separated
sinks and sources. For the groupoid of upper-triangular matrices\, we rep
resent braid-group transformations via sequences of cluster mutations in t
he special $\\mathbb A_n$-quiver. We prove the groupoid relations for quan
tum transport matrices and\, as a byproduct\, obtain quantum commutation r
elations having the Goldman bracket as their semiclassical limit. Time per
mitting\, I will also describe a generalization of this construction to af
fine Lie-Poisson algebras and to quantum loop algebras (Arxiv:2012:10982).
\n
LOCATION:https://researchseminars.org/talk/OCAS/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bernhard Keller (Université de Paris)
DTSTART;VALUE=DATE-TIME:20210223T160000Z
DTEND;VALUE=DATE-TIME:20210223T170000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/19
DESCRIPTION:Title: Pr
ogress on Leclerc's conjecture via Ménard's and Qin's theorems\nby Be
rnhard Keller (Université de Paris) as part of Online Cluster Algebra Sem
inar (OCAS)\n\n\nAbstract\nIn 2014\, Leclerc conjectured the existence of
cluster structures for all open Richardson\nvarieties $R_{v\,w}$\, i.e. in
tersections of a Schubert cell $C_w$ with an opposite\nSchubert cell $C^v$
in a simple complex algebraic group which is simply connected and\nof sim
ply laced type. Using representations of preprojective algebras\, he gave
a candidate\nseed for this structure and proved that the conjecture holds
when $v$ is less than or\nequal to $w$ in the weak right order. This holds
in particular for open Schubert varieties\nin the Grassmannian. In this c
ase\, Leclerc's seed was identified with a seed given by a\nplabic graph b
y Serhiyenko--Sherman-Bennett--Williams (02/2019). This identification was
\ngeneralized to open positroid varieties by Galashin--Lam (06/2019)\, who
moreover proved\nLeclerc's conjecture for this class\, confirming a conje
cture that had been known to\nthe experts since Scott's work (2006) and wa
s put down in writing by Muller--Speyer (2017).\n\nIn his upcoming thesis\
, using representations of preprojective algebras\,\nEtienne Ménard provi
des an algorithm for the explicit computation of an initial seed\n(expecte
d to agree with Leclerc's) in arbitrary type and shows that the correspond
ing\nconjectural cluster structure is a cluster reduction of Geiss--Lecler
c--Schröer's on the\nSchubert cell $C_w$. We will explain how this last r
esult yields progress on Leclerc's conjecture\nfor Ménard's seed thanks t
o Fan Qin's generic basis theorem and previous work by Muller\,\nPlamondon
\, Geiss--Leclerc--Schröer\, Palu\, K--Reiten\, ... .\nThis is a report o
n joint work with Peigen Cao.\n
LOCATION:https://researchseminars.org/talk/OCAS/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sira Gratz (University of Glasgow)
DTSTART;VALUE=DATE-TIME:20210330T150000Z
DTEND;VALUE=DATE-TIME:20210330T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/20
DESCRIPTION:Title: Gr
assmannians\, Cluster Algebras and Hypersurface Singularities\nby Sira
Gratz (University of Glasgow) as part of Online Cluster Algebra Seminar (
OCAS)\n\n\nAbstract\nGrassmannians are objects of great combinatorial and
geometric beauty\, which arise in myriad contexts. Their coordinate rings
serve as a classical example of cluster algebras\, and their combinatorics
is intimately related to algebraic and geometric concepts such as to repr
esentations of algebras and hypersurface singularities. \n\nIn this talk\,
we take these ideas to the limit to explore the a priori simple question:
What happens if we allow infinite clusters? In particular\, we discuss th
e notion of a cluster algebra of infinite rank (based on joint work with G
rabowski)\, and of a Grassmannian category of infinite rank (based on join
t work with August\, Cheung\, Faber and Schroll).\n
LOCATION:https://researchseminars.org/talk/OCAS/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Fraser (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20210209T160000Z
DTEND;VALUE=DATE-TIME:20210209T170000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/21
DESCRIPTION:Title: Cl
uster combinatorics of $SL_k$ skein algebras of punctured surfaces\nby
Chris Fraser (University of Minnesota) as part of Online Cluster Algebra
Seminar (OCAS)\n\n\nAbstract\nBy work of several authors\, the space of de
corated $G$-local\nsystems on a bordered marked surface is a cluster varie
ty. When $G$ is\n$SL_2$\, the associated cluster algebras are the cluster
algebras from\nsurfaces. We will present algebraic and combinatorial resul
ts and\nconjectures probing this family of cluster algebras when $G = SL_k
$\, in\nthe spirit of previous work of Fomin-Shapiro-Thurston\,\nFomin-Pyl
yavskyy\, and Goncharov-Shen. The main ingredients\ngeneralize tagged arc
s and tagged triangulations from the $SL_2$ case.\nJoint with Pavlo Pyl
yavskyy.\n
LOCATION:https://researchseminars.org/talk/OCAS/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elie Casbi (MPIM)
DTSTART;VALUE=DATE-TIME:20210316T160000Z
DTEND;VALUE=DATE-TIME:20210316T170000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/22
DESCRIPTION:Title: Eq
uivariant multiplicities via representations of quantum affine algebras\nby Elie Casbi (MPIM) as part of Online Cluster Algebra Seminar (OCAS)\n
\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/OCAS/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Trey Trampel (University of Notre Dame)
DTSTART;VALUE=DATE-TIME:20210323T150000Z
DTEND;VALUE=DATE-TIME:20210323T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/23
DESCRIPTION:Title: Ro
ot of unity quantum cluster algebras and discriminants\nby Trey Trampe
l (University of Notre Dame) as part of Online Cluster Algebra Seminar (OC
AS)\n\n\nAbstract\nWe will define the notion of a root of unity quantum cl
uster algebra\, which is not necessarily a specialization of a quantum clu
ster algebra. Through these algebras\, we connect the subjects of cluster
algebras and discriminants. Motivation for discriminants will be given in
terms of their applications to representation theory. We show that the roo
t of unity quantum cluster algebras are polynomial identity algebras\, and
we identify a large canonical central subalgebra. This central subalgebra
is shown to be isomorphic to the underlying classical cluster algebra of
geometric type. These central subalgebras can be thought of as a generaliz
ation of De Concini-Kac-Procesi's canonical central subalgebras for quantu
m groups at roots of unity. In particular\, we recover their structure in
the case of quantum Schubert cells. We prove a general theorem on the form
of discriminants\, which is given as a product of frozen cluster variable
s. From this we derive specific formulas in examples\, such as for all roo
t of unity quantum Schubert cells for any symmetrizable Kac-Moody algebra.
This is joint work with Bach Nguyen and Milen Yakimov.\n
LOCATION:https://researchseminars.org/talk/OCAS/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Greg Muller (University of Oklahoma)
DTSTART;VALUE=DATE-TIME:20210406T150000Z
DTEND;VALUE=DATE-TIME:20210406T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/24
DESCRIPTION:Title: Sp
aces of quasiperiodic sequences\nby Greg Muller (University of Oklahom
a) as part of Online Cluster Algebra Seminar (OCAS)\n\n\nAbstract\nA "quas
iperiodic space" is a vector space of sequences which are periodic up to a
\nconstant factor. The moduli of such vector spaces are 1-dimensional exte
nsions of\nGrassmannians\, and there are analogous positroid stratificatio
ns of the former. I\nwill demonstrate that these "quasiperiodic positroid
varieties" have a Y-type cluster\nstructure that is mirror dual to the X-t
ype cluster structure on (the Plucker cone\nover) the corresponding positr
oid variety. This structure is defined by extending a\nversion of Postniko
v's boundary measurement map to the quasiperiodic case. Time\npermitting\,
I will discuss an alternative construction of this boundary measurement\n
map\, which uses the twist to construct a linear recurrence whose solution
s are the\nspace in question. This provides a generalization of MGOST's co
nnection between\nlinear recurrences\, friezes\, and the Gale transform. A
motivating goal of this\nproject is to understand the tropical points of
these quasiperiodic positroid\nvarieties\, as they parametrize the canonic
al basis of theta functions on (the Plucker\ncone over) the corresponding
positroid variety.\n
LOCATION:https://researchseminars.org/talk/OCAS/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Barbieri (University of Milano Statale)
DTSTART;VALUE=DATE-TIME:20210309T160000Z
DTEND;VALUE=DATE-TIME:20210309T170000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/25
DESCRIPTION:Title: Fr
om special functions to stability conditions\nby Anna Barbieri (Univer
sity of Milano Statale) as part of Online Cluster Algebra Seminar (OCAS)\n
\n\nAbstract\nThe Gamma function studied by Bernoulli appear all over math
ematics and in particular whenever we study special contour integrals. We
will review a class of special functions called Barnes multiple Gamma func
tions that generalize the Gamma function and we will see how they appear i
n the study of a class of Bridgeland stability conditions with a very simp
le Donaldson-Thomas (DT) theory. This goes through solving a Riemann-Hilbe
rt-Birkhoff boundary value problem induced by the wall-crossing formula fo
r DT counting invariants\, and involving factors that look like cluster tr
ansformations. Based on a joint work with T. Bridgeland and J. Stoppa.\n
LOCATION:https://researchseminars.org/talk/OCAS/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Étienne Ménard (Université de Caen)
DTSTART;VALUE=DATE-TIME:20210413T150000Z
DTEND;VALUE=DATE-TIME:20210413T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/26
DESCRIPTION:Title: Cl
uster algebra associated to open Richardson varieties : an algorithm to co
mpute initial seed\nby Étienne Ménard (Université de Caen) as part
of Online Cluster Algebra Seminar (OCAS)\n\n\nAbstract\nIn his paper of 20
16\, Leclerc wanted to study the total nonnegativity criteria on flag vari
ety in the same way as Fomin and Zelevinsky studied in '99 the total nonne
gativity on $GL_n(mathbb{R})$ by stratification via double Bruhat cells. I
n this setting he wanted to study the cluster algebra structure on the ope
n Richardson varieties stratifying the flag variety.\n\nBut in order to st
udy this cluster algebra he used an additive categorification of the open
Richardson variety $mathcal{R}_{v\,w}$ by the category $mathcal{C}_{v\,w}$
. He proved that there is a cluster structure (in the sense of Buan\, Iyam
a\, Reiten\, Scott) but hadn't given a way to explicitly build a seed for
this cluster structure.\n\nMy PhD work was to design a prove an algorithm
to explictly build such a seed starting from a seed for the cluster struct
ure on the category $mathcal{C}_wsupset mathcal{C}_{v\,w}$. I will explain
the principle\, the concrete usage of this algorithm and draw a sketch of
the proof.\n\nIf time allows it\, I will also introduce the Sage implemen
tation I have written during my PhD.\n
LOCATION:https://researchseminars.org/talk/OCAS/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dani Kaufman (University of Maryland)
DTSTART;VALUE=DATE-TIME:20210420T150000Z
DTEND;VALUE=DATE-TIME:20210420T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/27
DESCRIPTION:Title: Mu
tation Invariant Functions On Cluster Algebras\nby Dani Kaufman (Unive
rsity of Maryland) as part of Online Cluster Algebra Seminar (OCAS)\n\n\nA
bstract\nExamples of functions of cluster variables which remain unchanged
after mutation arise naturally when studying cluster algebras. They appea
r as nontrivial elements of upper cluster algebras\, elements of a theta b
asis\, trace functions\, cluster characters\, and Diophantine equations wh
ose solutions are parameterized by a cluster algebra. Interestingly\, one
often finds that the same mutation invariant function can be interpreted i
n several distinct ways\, but it is not immediately clear why this would b
e.\n \nI will give a concise definition of a mutation invariant function i
n terms of an action of the cluster modular group\, and give many more int
eresting examples. I will also discuss a classification of invariants for
Dehn twists on surface cluster algebras\, and more generally for "cluster
Dehn twists" on mutation finite cluster algebras. This is the primary resu
lt of my recent PhD thesis. \n \nIt is my hope that this classification a
llows us to begin to see why the same types functions appear in many disti
nct guises\; each of these constructions (theta basis\, trace functions\,
cluster characters\, etc.) produce functions which are manifestly mutation
invariant.\n
LOCATION:https://researchseminars.org/talk/OCAS/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peigen Cao (Université de Paris)
DTSTART;VALUE=DATE-TIME:20210427T150000Z
DTEND;VALUE=DATE-TIME:20210427T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/28
DESCRIPTION:by Peigen Cao (Université de Paris) as part of Online Cluster
Algebra Seminar (OCAS)\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/OCAS/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nick Ovenhouse (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20210511T150000Z
DTEND;VALUE=DATE-TIME:20210511T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/29
DESCRIPTION:Title: Ex
pansion Formulas for Decorated Super Teichmüller Space\nby Nick Ovenh
ouse (University of Minnesota) as part of Online Cluster Algebra Seminar (
OCAS)\n\n\nAbstract\nIt is well-known that cluster variables in cluster al
gebras coming from surfaces can be thought of as "lambda-length" coordinat
es on decorated Teichmuller spaces. In the case of a polygon (a disk with
marked points on the boundary)\, there is a combinatorial formula for the
terms in the Laurent expansion of cluster variables\, due to Schiffler\, i
n terms of "T-paths". Recently\, Penner and Zeitlin introduced Decorated S
uper Teichmuller Spaces\, and presented a modified version of the Ptolemy
exchange relation. In joint work with Gregg Musiker and Sylvester Zhang\,
we give a version of the "T-path" formula for the super lambda-lengths. We
also present connections with super frieze patterns introduced by Ovsienk
o\, Morier-Genoud\, and Tabachnikov.\n
LOCATION:https://researchseminars.org/talk/OCAS/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lauren Williams (Harvard University)
DTSTART;VALUE=DATE-TIME:20210525T150000Z
DTEND;VALUE=DATE-TIME:20210525T160000Z
DTSTAMP;VALUE=DATE-TIME:20230926T002830Z
UID:OCAS/30
DESCRIPTION:Title: Th
e amplituhedron and cluster algebras\nby Lauren Williams (Harvard Univ
ersity) as part of Online Cluster Algebra Seminar (OCAS)\n\n\nAbstract\nTh
e amplituhedron is the image of the positive Grassmannian under a linear m
ap induced by a totally positive matrix. Special cases of the amplituhedro
n include the positive Grassmannian\, cyclic polytopes in projective space
\, and the bounded complex of the cyclic hyperplane arrangement.\n\nWhile
at first glance the amplituhedron seems complicated\, it has many beautifu
l properties. I will explain how ideas from oriented matroids\, total posi
tivity\, and cluster algebras leads to new results about the amplituhedron
.\n\n\nBased on joint work with Matteo Parisi and Melissa Sherman-Bennett.
\n
LOCATION:https://researchseminars.org/talk/OCAS/30/
END:VEVENT
END:VCALENDAR