BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Michael Shapiro (Michigan State University)
DTSTART:20200901T150000Z
DTEND:20200901T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/1
DESCRIPTION:Title: Non-commutative Networks on a Cylinder\nby Michael Shapiro (Michig
an State University) as part of Online Cluster Algebra Seminar (OCAS)\n\nA
bstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Li Li (Oakland University)
DTSTART:20200908T150000Z
DTEND:20200908T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/2
DESCRIPTION:Title: A proof of two conjectures on Markov Numbers\nby Li Li (Oakland Un
iversity) as part of Online Cluster Algebra Seminar (OCAS)\n\nAbstract: TB
A\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lara Bossinger (UNAM-Oaxaca)
DTSTART:20200915T150000Z
DTEND:20200915T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/3
DESCRIPTION:Title: Understanding universal coefficients of Grassmannians through Groebner
theory\nby Lara Bossinger (UNAM-Oaxaca) as part of Online Cluster Alg
ebra Seminar (OCAS)\n\n\nAbstract\nIn this talk I will present recent resu
lts of a joint work with Fatemeh Mohammadi and Alfredo Nájera Chávez. F
or a polarized weighted projective variety V(J) we introduce a flat family
that combines all Groebner degenerations of V associated to a maximal con
e in the Groebner fan of J. It turns out that this family can alternativel
y be obtained as a pull-back of a toric family (in the sense of Kaveh--Man
on's classification of such).\nThe most surprising application of this con
struction is its relation to cluster algebras with universal coefficients.
To demonstrate 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 embeddin
g and for Gr(3\,6) what we call its "cluster embedding". In both cases we
identify a specific maximal cone C in the Groebner fan of the defining ide
al such that the algebra defining the flat family mentioned above is canon
ically isomorphic to the corresponding cluster algebra with universal coef
ficients.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fan Qin (Shanghai Jiao Tong University)
DTSTART:20200922T150000Z
DTEND:20200922T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/4
DESCRIPTION:Title: Dual canonical bases and triangular bases of quantum cluster algebras<
/a>\nby Fan Qin (Shanghai Jiao Tong University) as part of Online Cluster
Algebra Seminar (OCAS)\n\n\nAbstract\nOne of the main motivations for clus
ter algebras was to create a combinatorial framework to study the dual can
onical bases. Correspondingly\, it has been long expected that the quantum
cluster monomials (certain monomials of generators) belong to the dual ca
nonical bases (of quantum unipotent subgroups) up to scalar multiples. We
discuss how to use the triangular bases to show this conjecture in full ge
nerality. Moreover\, we show that the (double) triangular bases verify an
analog of Leclerc’s conjecture for dual canonical bases.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Esther Banaian (University of Minnesota)
DTSTART:20200929T150000Z
DTEND:20200929T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/5
DESCRIPTION:Title: Frieze Patterns from Dissections\nby Esther Banaian (University of
Minnesota) as part of Online Cluster Algebra Seminar (OCAS)\n\n\nAbstract
\nFinite frieze patterns of positive integers were shown by Conway and Co
xeter to be in bijection with triangulated polygons. Baur\, Parsons\, and
Tschabold generalized this result\, showing that infinite\nfrieze patterns
of positive integers are in bijection with triangulated annuli and once-
punctured discs. More recently\, Holm and Jørgensen investigated frieze p
atterns arising from dissected polygons.\nThe frieze patterns of Holm and
Jørgensen involve algebraic integers of the form 2cos(pi/p) for an int
eger p. We combine these generalizations and present results on frieze pat
terns from dissected\nannuli\, using these same algebraic integers. We al
so discuss how some of these frieze patterns from dissections can be conne
cted to generalized cluster algebras\, in the sense of Chekhov and Shapir
o.\nThis is based on joint work with Jiuqi (Lena) Chen and with Elizabeth
Kelley.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melissa Sherman-Bennett (UC Berkely and Harvard University)
DTSTART:20201006T150000Z
DTEND:20201006T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/6
DESCRIPTION:Title: Many cluster structures on positroid varieties\nby Melissa Sherman
-Bennett (UC Berkely and Harvard University) as part of Online Cluster Alg
ebra Seminar (OCAS)\n\n\nAbstract\nEarly in the history of cluster algebra
s\, Scott showed that the homogeneous coordinate ring of the Grassmannian
is a cluster algebra\, with seeds given by Postnikov's plabic graphs for
the\nGrassmannian. Recently the analogous statement has been proved for op
en Schubert varieties (Leclerc\, Serhiyenko-SB-Williams) and more generall
y\, for open positroid varieties (Galashin-Lam). I'll\ndiscuss joint work
with Chris Fraser\, in which we give a family of cluster structures on ope
n Schubert (and more generally\, positroid) varieties. Each of the cluster
structures in this family has seeds given by face labels of relabeled pla
bic graphs\, which are plabic graphs whose boundary is labeled by a permut
ation of 1\, ...\, n. For Schubert varieties\, all cluster structures in t
his family\nquasi-coincide\, meaning they differ only by rescaling by froz
en variables and their cluster monomials coincide. In particular\, all rel
abeled plabic graphs for a Schubert variety give rise to seeds in the "usu
al" cluster algebra structure on the coordinate ring. As part of our resul
ts\, we show the "target" and "source" cluster structures on Schubert vari
eties quasi-coincide\, confirming a conjecture of Muller and Speyer. One p
roof tool of independent interest is a permuted version of the Muller-Spey
er twist map\, which we use to prove many (open) positroid varieties are i
somorphic.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Jørgensen (Aarhus University)
DTSTART:20201013T150000Z
DTEND:20201013T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/7
DESCRIPTION:Title: The green groupoid and its action on derived categories (joint work wi
th Milen Yakimov)\nby Peter Jørgensen (Aarhus University) as part of
Online Cluster Algebra Seminar (OCAS)\n\n\nAbstract\nWe introduce the gree
n groupoid $\\mathcal{G}$ of a $2$-Calabi-Yau triangulated category $\\mat
hcal{C}$. It is an augmentation of the mutation graph of $\\mathcal{C}$\,
which is defined by means of silting theory.\n\nThe green groupoid $\\mat
hcal{G}$ has certain key properties:\n\n1. If $\\mathcal{C}$ is the stable
category 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
geometrically 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 endo
morphism rings $\\mathcal{E}(m\,m)$.\n\nThe situation in (3) occurs if $\\
Sigma_{\\mathcal{C}}^2$\, the square of the suspension functor\, is the id
entity. It recovers results by Donovan\, Hirano\, and Wemyss where $\\mat
hcal{E}$ is the category of maximal Cohen-Macaulay modules over a suitable
singularity.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Lam (University of Michigan)
DTSTART:20201020T150000Z
DTEND:20201020T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/8
DESCRIPTION:Title: Cluster configuration spaces of finite type\nby Thomas Lam (Univer
sity of Michigan) as part of Online Cluster Algebra Seminar (OCAS)\n\n\nAb
stract\nI will talk about a "cluster configuration space" $M_D$\,\ndependi
ng on a finite Dynkin diagram $D$. The space $M_D$ is an affine\nalgebrai
c variety that is defined using only the compatibility degree\nof the corr
esponding 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) poin
ts in $P^1$. There are many relations to finite-type\ncluster theory\, bu
t an especially close connection to the finite-type\ncluster algebra with
universal coefficients.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gregg Musiker (University of Minnesota)
DTSTART:20201117T160000Z
DTEND:20201117T170000Z
DTSTAMP:20260404T110823Z
UID:OCAS/9
DESCRIPTION:Title: Double-dimer configurations and quivers of dP3 (del Pezzo) type\nb
y Gregg Musiker (University of Minnesota) as part of Online Cluster Algebr
a Seminar (OCAS)\n\n\nAbstract\nIn this talk\, I will describe our work ex
tending combinatorial interpretations for so called toric cluster variable
s as was previously studied by myself and Tri Lai. In [LM 2017] and [LM 20
20]\, most toric cluster variables were shown to have Laurent expansions a
greeing with partition functions of dimers on subgraphs cut out by six-sid
ed contours. However\, the case of cluster variables\nparameterized by six
-sided contours with a self-intersection eluded our techniques. In this ta
lk we discuss our research rectifying this issue by using Helen Jenne’s
condensation results for the\ndouble-dimer model [J 2019]. While we focus
on quivers of dP3 type of Model 1 and Model 4\, we anticipate our techniqu
es will extend to certain additional cluster algebras related to brane til
ings. \nThis is joint work with Helen Jenne and Tri Lai.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karin Baur (University of Leeds)
DTSTART:20201103T160000Z
DTEND:20201103T170000Z
DTSTAMP:20260404T110823Z
UID:OCAS/10
DESCRIPTION:Title: Flips in triangulations and matchings\nby Karin Baur (University
of Leeds) as part of Online Cluster Algebra Seminar (OCAS)\n\n\nAbstract\n
Plane perfect matchings of $2n$ points in convex position are known to be
in\nbijection with triangulations of convex polygons of size $n + 2$\; the
y are\nboth counted by the Catalan numbers.\nWe explain how to give a dire
ct bijection and how it can be extended to a\nbijection between monochroma
tic matchings on $k$ colours and tilings by\n$(k+2)$-gons. Edge flips are
a classic operation to perform local changes in\nboth sets. We use the abo
ve bijection to determine the two types of edge\nflips are related. We use
this to give an algebraic interpretation of the\nflip graph of triangulat
ions in terms of elements of the corresponding\nTemperley-Lieb algebra.\nT
his is joint work with\nO. Aichholzer\, L. Donner (Andritsch)\, B. Vogtenh
uber.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dylan Allegretti (University of British Columbia)
DTSTART:20201110T160000Z
DTEND:20201110T170000Z
DTSTAMP:20260404T110823Z
UID:OCAS/11
DESCRIPTION:Title: Stability conditions and cluster varieties\nby Dylan Allegretti (
University 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 co
nstruction in low-dimensional topology that takes a holomorphic quadratic
differential on a surface and produces a $PGL(2)$-local system. This\ncons
truction provides a local homeomorphism from the moduli space of quadratic
differentials to the moduli space of local systems. In the second part of
the talk\, I will propose a categorical\ngeneralization of this construct
ion. In this generalization\, the space of quadratic differentials is repl
aced by a complex manifold parametrizing Bridgeland stability conditions o
n a certain\n3-Calabi-Yau triangulated category\, while the space of local
systems is replaced by a cluster variety. I will describe a local homeomo
rphism from the space of stability conditions to the cluster\nvariety in a
large class of examples and explain how it preserves the structures of th
ese two spaces.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christof Geiss
DTSTART:20201208T160000Z
DTEND:20201208T170000Z
DTSTAMP:20260404T110823Z
UID:OCAS/12
DESCRIPTION:by Christof Geiss as part of Online Cluster Algebra Seminar (O
CAS)\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathan Reading (North Carolina State University)
DTSTART:20210119T160000Z
DTEND:20210119T170000Z
DTSTAMP:20260404T110823Z
UID:OCAS/13
DESCRIPTION:Title: Theta functions and the mutation fan\nby Nathan Reading (North Ca
rolina State University) as part of Online Cluster Algebra Seminar (OCAS)\
n\n\nAbstract\nThe setting for this work is the cluster scattering diagram
defined by\nGross\, Hacking\, Keel\, and Kontsevich (GHKK). The cluster
scattering\ndiagram is a collection of walls (codimension-1 cones plus so
me\nadditional algebraic data). Theta functions (one for each g-vector)\
ninclude the cluster monomials and form a basis for the cluster algebra\n(
or often something larger). Explicit constructions of cluster\nscatterin
g diagrams and explicit computations of theta functions are\nhopelessly co
mplicated in general\, but I believe that eventually there\nwill be combin
atorial models in all mutation-finite types. I'll\nmention work with Sal
vatore 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 discussing a result (with Stella)\nthat I think will make it possibl
e to complete these combinatorial\nconstructions of theta functions.\n\nTh
e mutation fan encodes the piecewise-linear geometry of matrix\nmutation.
The result is: If you take a product of theta functions\nwhose g-vect
ors are all in one cone of the mutation fan\, the product\nexpands as a su
m of theta functions whose g-vectors are all in one\ncone of the mutation
fan. The result seems natural and in some sense\nunsurprising\, but it r
equires some work and it is quite useful. The\nresult requires two serio
us changes in point of view from the GHKK\nsetup: Taking a different poi
nt of view on what "mutation of\nscattering diagrams" means and demoting "
frozen variables" to the\nstatus of coefficients.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bruce Sagan (Michigan State University)
DTSTART:20201124T160000Z
DTEND:20201124T170000Z
DTSTAMP:20260404T110823Z
UID:OCAS/14
DESCRIPTION:Title: On a rank-unimodality conjecture of Morier-Genoud and Ovsienko\nb
y Bruce Sagan (Michigan State University) as part of Online Cluster Algebr
a Seminar (OCAS)\n\n\nAbstract\nLet $\\alpha=(a\,b\,\\ldots)$ be a composi
tion\, that is\, a finite sequence of positive integers. Consider the ass
ociated partially ordered set $F(\\alpha)$\, called a fence\, whose coveri
ng relations 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 o
f all lower order ideals of $F(\\alpha)$.\nThese lattices are important in
the theory of cluster algebras and their rank generating functions can be
used to define $q$-analogues of rational numbers.\nWe make progress on a
recent conjecture of Morier-Genoud and Ovsienko that $L(\\alpha)$ is rank
unimodal.\nAll terms from the theory of partially ordered sets will be car
efully defined. This is joint work with Thomas McConville and Clifford Sm
yth.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elizabeth Kelley (University of Minnesota)
DTSTART:20201201T160000Z
DTEND:20201201T170000Z
DTSTAMP:20260404T110823Z
UID:OCAS/15
DESCRIPTION:Title: Theta basis for reciprocal generalized cluster algebras\nby Eliza
beth Kelley (University of Minnesota) as part of Online Cluster Algebra Se
minar (OCAS)\n\n\nAbstract\nCluster algebras are characterized by binomial
exchange relations. A natural generalization of these algebras\, introduc
ed by Chekhov and Shapiro\, relaxes this restriction and allows the\nexcha
nge polynomials to have arbitrarily many terms. Following the work of Gro
ss\, Hacking\, Keel\, and Kontsevich\, we give the construction of scatter
ing diagrams for the subclass of generalized cluster\nalgebras with recipr
ocal exchange coefficients. We then define the theta basis for these algeb
ras and show that the fixed data of the left companion algebra is\, up to
isomorphism\, Langlands dual to\nthat of the right companion algebra (and
vice versa). This is joint work with Man-Wai Cheug and Gregg Musiker.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daping Weng (Michigan State University)
DTSTART:20210126T160000Z
DTEND:20210126T170000Z
DTSTAMP:20260404T110823Z
UID:OCAS/16
DESCRIPTION:Title: Symplectic Structures on Augmentation Varieties\nby Daping Weng (
Michigan State University) as part of Online Cluster Algebra Seminar (OCAS
)\n\n\nAbstract\nIn a recent joint project with H. Gao and L. Shen\, we in
troduce a cluster K2 structure on\nthe augmentation variety of the Chekano
v-Eliashberg dga for the rainbow closure of any positive\nbraid with marke
d point decorations. This cluster K2 structure naturally equips the comple
x\naugmentation variety with a holomorphic presymplectic 2-form. Using a r
esult of Goncharov and\nKenyon on surface bipartite graphs\, we prove that
this holomorphic presymplectic 2-form becomes\nsymplectic after we reduce
the number of marked points to a single marked per link component (plus\n
some modification).\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hyun Kyu Kim (Ewha Womans University)
DTSTART:20210216T160000Z
DTEND:20210216T170000Z
DTSTAMP:20260404T110823Z
UID:OCAS/17
DESCRIPTION:Title: $A_2$-laminations as basis for ${\\rm PGL}_3$ cluster variety for su
rface\nby Hyun Kyu Kim (Ewha Womans University) as part of Online Clus
ter Algebra Seminar (OCAS)\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leonid Chekhov (Steklov Mathematical Institute and Michigan State
University)
DTSTART:20210202T160000Z
DTEND:20210202T170000Z
DTSTAMP:20260404T110823Z
UID:OCAS/18
DESCRIPTION:Title: Darboux coordinates for symplectic groupoid and cluster algebras\
nby Leonid Chekhov (Steklov Mathematical Institute and Michigan State Univ
ersity) as part of Online Cluster Algebra Seminar (OCAS)\n\n\nAbstract\nTh
e talk is based on Arxiv:2003:07499\, joint work with Misha Shapiro. We us
e Fock--Goncharov higher Teichmüller space variables to derive Darboux c
oordinate representation for entries of general symplectic leaves of the $
\\mathcal A_n$ groupoid of upper-triangular matrices and\, in a more gener
al setting\, of higher-dimensional symplectic leaves for algebras governed
by the quantum reflection equation with the trigonometric $R$-matrix. Thi
s result can be generalized to any planar directed network on disc with se
parated sinks and sources. For the groupoid of upper-triangular matrices\,
we represent braid-group transformations via sequences of cluster mutatio
ns in the special $\\mathbb A_n$-quiver. We prove the groupoid relations f
or quantum transport matrices and\, as a byproduct\, obtain quantum commut
ation relations having the Goldman bracket as their semiclassical limit. T
ime permitting\, I will also describe a generalization of this constructio
n to affine Lie-Poisson algebras and to quantum loop algebras (Arxiv:2012:
10982).\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bernhard Keller (Université de Paris)
DTSTART:20210223T160000Z
DTEND:20210223T170000Z
DTSTAMP:20260404T110823Z
UID:OCAS/19
DESCRIPTION:Title: Progress on Leclerc's conjecture via Ménard's and Qin's theorems
\nby Bernhard Keller (Université de Paris) as part of Online Cluster Alge
bra Seminar (OCAS)\n\n\nAbstract\nIn 2014\, Leclerc conjectured the existe
nce of cluster structures for all open Richardson\nvarieties $R_{v\,w}$\,
i.e. intersections of a Schubert cell $C_w$ with an opposite\nSchubert cel
l $C^v$ in a simple complex algebraic group which is simply connected and\
nof simply laced type. Using representations of preprojective algebras\, h
e 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. Thi
s holds in particular for open Schubert varieties\nin the Grassmannian. In
this case\, Leclerc's seed was identified with a seed given by a\nplabic
graph by Serhiyenko--Sherman-Bennett--Williams (02/2019). This identificat
ion was\ngeneralized to open positroid varieties by Galashin--Lam (06/2019
)\, who moreover proved\nLeclerc's conjecture for this class\, confirming
a conjecture that had been known to\nthe experts since Scott's work (2006)
and was put down in writing by Muller--Speyer (2017).\n\nIn his upcoming
thesis\, using representations of preprojective algebras\,\nEtienne Ménar
d provides an algorithm for the explicit computation of an initial seed\n(
expected to agree with Leclerc's) in arbitrary type and shows that the cor
responding\nconjectural cluster structure is a cluster reduction of Geiss-
-Leclerc--Schröer's on the\nSchubert cell $C_w$. We will explain how this
last result yields progress on Leclerc's conjecture\nfor Ménard's seed t
hanks to Fan Qin's generic basis theorem and previous work by Muller\,\nPl
amondon\, Geiss--Leclerc--Schröer\, Palu\, K--Reiten\, ... .\nThis is a r
eport on joint work with Peigen Cao.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sira Gratz (University of Glasgow)
DTSTART:20210330T150000Z
DTEND:20210330T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/20
DESCRIPTION:Title: Grassmannians\, Cluster Algebras and Hypersurface Singularities\n
by Sira Gratz (University of Glasgow) as part of Online Cluster Algebra Se
minar (OCAS)\n\n\nAbstract\nGrassmannians are objects of great combinatori
al and geometric beauty\, which arise in myriad contexts. Their coordinate
rings serve as a classical example of cluster algebras\, and their combin
atorics is intimately related to algebraic and geometric concepts such as
to representations of algebras and hypersurface singularities. \n\nIn this
talk\, we take these ideas to the limit to explore the a priori simple qu
estion: What happens if we allow infinite clusters? In particular\, we dis
cuss the notion of a cluster algebra of infinite rank (based on joint work
with Grabowski)\, and of a Grassmannian category of infinite rank (based
on joint work with August\, Cheung\, Faber and Schroll).\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Fraser (University of Minnesota)
DTSTART:20210209T160000Z
DTEND:20210209T170000Z
DTSTAMP:20260404T110823Z
UID:OCAS/21
DESCRIPTION:Title: Cluster combinatorics of $SL_k$ skein algebras of punctured surfaces<
/a>\nby Chris Fraser (University of Minnesota) as part of Online Cluster A
lgebra Seminar (OCAS)\n\n\nAbstract\nBy work of several authors\, the spac
e of decorated $G$-local\nsystems on a bordered marked surface is a cluste
r variety. When $G$ is\n$SL_2$\, the associated cluster algebras are the c
luster algebras from\nsurfaces. We will present algebraic and combinatoria
l results and\nconjectures probing this family of cluster algebras when $G
= SL_k$\, in\nthe spirit of previous work of Fomin-Shapiro-Thurston\,\nFo
min-Pylyavskyy\, and Goncharov-Shen. The main ingredients\ngeneralize tag
ged arcs and tagged triangulations from the $SL_2$ case.\nJoint with Pa
vlo Pylyavskyy.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elie Casbi (MPIM)
DTSTART:20210316T160000Z
DTEND:20210316T170000Z
DTSTAMP:20260404T110823Z
UID:OCAS/22
DESCRIPTION:Title: Equivariant multiplicities via representations of quantum affine alge
bras\nby Elie Casbi (MPIM) as part of Online Cluster Algebra Seminar (
OCAS)\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Trey Trampel (University of Notre Dame)
DTSTART:20210323T150000Z
DTEND:20210323T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/23
DESCRIPTION:Title: Root of unity quantum cluster algebras and discriminants\nby Trey
Trampel (University of Notre Dame) as part of Online Cluster Algebra Semi
nar (OCAS)\n\n\nAbstract\nWe will define the notion of a root of unity qua
ntum cluster algebra\, which is not necessarily a specialization of a quan
tum cluster algebra. Through these algebras\, we connect the subjects of c
luster algebras and discriminants. Motivation for discriminants will be gi
ven in terms of their applications to representation theory. We show that
the root of unity quantum cluster algebras are polynomial identity algebra
s\, and we identify a large canonical central subalgebra. This central sub
algebra is shown to be isomorphic to the underlying classical cluster alge
bra of geometric type. These central subalgebras can be thought of as a ge
neralization of De Concini-Kac-Procesi's canonical central subalgebras for
quantum groups at roots of unity. In particular\, we recover their struct
ure in the case of quantum Schubert cells. We prove a general theorem on t
he form of discriminants\, which is given as a product of frozen cluster v
ariables. From this we derive specific formulas in examples\, such as for
all root of unity quantum Schubert cells for any symmetrizable Kac-Moody a
lgebra. This is joint work with Bach Nguyen and Milen Yakimov.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Greg Muller (University of Oklahoma)
DTSTART:20210406T150000Z
DTEND:20210406T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/24
DESCRIPTION:Title: Spaces of quasiperiodic sequences\nby Greg Muller (University of
Oklahoma) as part of Online Cluster Algebra Seminar (OCAS)\n\n\nAbstract\n
A "quasiperiodic space" is a vector space of sequences which are periodic
up to a\nconstant factor. The moduli of such vector spaces are 1-dimension
al extensions of\nGrassmannians\, and there are analogous positroid strati
fications of the former. I\nwill demonstrate that these "quasiperiodic pos
itroid varieties" have a Y-type cluster\nstructure that is mirror dual to
the X-type cluster structure on (the Plucker cone\nover) the corresponding
positroid variety. This structure is defined by extending a\nversion of P
ostnikov's boundary measurement map to the quasiperiodic case. Time\npermi
tting\, I will discuss an alternative construction of this boundary measur
ement\nmap\, which uses the twist to construct a linear recurrence whose s
olutions are the\nspace in question. This provides a generalization of MGO
ST's connection between\nlinear recurrences\, friezes\, and the Gale trans
form. A motivating goal of this\nproject is to understand the tropical poi
nts of these quasiperiodic positroid\nvarieties\, as they parametrize the
canonical basis of theta functions on (the Plucker\ncone over) the corresp
onding positroid variety.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Barbieri (University of Milano Statale)
DTSTART:20210309T160000Z
DTEND:20210309T170000Z
DTSTAMP:20260404T110823Z
UID:OCAS/25
DESCRIPTION:Title: From special functions to stability conditions\nby Anna Barbieri
(University of Milano Statale) as part of Online Cluster Algebra Seminar (
OCAS)\n\n\nAbstract\nThe Gamma function studied by Bernoulli appear all ov
er mathematics and in particular whenever we study special contour integra
ls. We will review a class of special functions called Barnes multiple Gam
ma functions that generalize the Gamma function and we will see how they a
ppear in the study of a class of Bridgeland stability conditions with a ve
ry simple Donaldson-Thomas (DT) theory. This goes through solving a Rieman
n-Hilbert-Birkhoff boundary value problem induced by the wall-crossing for
mula for DT counting invariants\, and involving factors that look like clu
ster transformations. Based on a joint work with T. Bridgeland and J. Stop
pa.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Étienne Ménard (Université de Caen)
DTSTART:20210413T150000Z
DTEND:20210413T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/26
DESCRIPTION:Title: Cluster algebra associated to open Richardson varieties : an algorith
m to compute initial seed\nby Étienne Ménard (Université de Caen) a
s part of Online Cluster Algebra Seminar (OCAS)\n\n\nAbstract\nIn his pape
r of 2016\, Leclerc wanted to study the total nonnegativity criteria on fl
ag variety in the same way as Fomin and Zelevinsky studied in '99 the tota
l nonnegativity on $GL_n(mathbb{R})$ by stratification via double Bruhat c
ells. In this setting he wanted to study the cluster algebra structure on
the open Richardson varieties stratifying the flag variety.\n\nBut in orde
r to study this cluster algebra he used an additive categorification of th
e 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
\, Iyama\, Reiten\, Scott) but hadn't given a way to explicitly build a se
ed for this cluster structure.\n\nMy PhD work was to design a prove an alg
orithm to explictly build such a seed starting from a seed for the cluster
structure on the category $mathcal{C}_wsupset mathcal{C}_{v\,w}$. I will
explain the principle\, the concrete usage of this algorithm and draw a sk
etch of the proof.\n\nIf time allows it\, I will also introduce the Sage i
mplementation I have written during my PhD.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dani Kaufman (University of Maryland)
DTSTART:20210420T150000Z
DTEND:20210420T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/27
DESCRIPTION:Title: Mutation Invariant Functions On Cluster Algebras\nby Dani Kaufman
(University of Maryland) as part of Online Cluster Algebra Seminar (OCAS)
\n\n\nAbstract\nExamples of functions of cluster variables which remain un
changed after mutation arise naturally when studying cluster algebras. The
y appear as nontrivial elements of upper cluster algebras\, elements of a
theta basis\, trace functions\, cluster characters\, and Diophantine equat
ions whose solutions are parameterized by a cluster algebra. Interestingly
\, one often finds that the same mutation invariant function can be interp
reted in several distinct ways\, but it is not immediately clear why this
would be.\n \nI will give a concise definition of a mutation invariant fun
ction in terms of an action of the cluster modular group\, and give many m
ore interesting examples. I will also discuss a classification of invarian
ts for Dehn twists on surface cluster algebras\, and more generally for "c
luster Dehn twists" on mutation finite cluster algebras. This is the prima
ry result of my recent PhD thesis. \n \nIt is my hope that this classific
ation allows us to begin to see why the same types functions appear in man
y distinct guises\; each of these constructions (theta basis\, trace funct
ions\, cluster characters\, etc.) produce functions which are manifestly m
utation invariant.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peigen Cao (Université de Paris)
DTSTART:20210427T150000Z
DTEND:20210427T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/28
DESCRIPTION:by Peigen Cao (Université de Paris) as part of Online Cluster
Algebra Seminar (OCAS)\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nick Ovenhouse (University of Minnesota)
DTSTART:20210511T150000Z
DTEND:20210511T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/29
DESCRIPTION:Title: Expansion Formulas for Decorated Super Teichmüller Space\nby Nic
k Ovenhouse (University of Minnesota) as part of Online Cluster Algebra Se
minar (OCAS)\n\n\nAbstract\nIt is well-known that cluster variables in clu
ster algebras coming from surfaces can be thought of as "lambda-length" co
ordinates on decorated Teichmuller spaces. In the case of a polygon (a dis
k with marked points on the boundary)\, there is a combinatorial formula f
or the terms in the Laurent expansion of cluster variables\, due to Schiff
ler\, in terms of "T-paths". Recently\, Penner and Zeitlin introduced Deco
rated Super Teichmuller Spaces\, and presented a modified version of the P
tolemy exchange relation. In joint work with Gregg Musiker and Sylvester Z
hang\, we give a version of the "T-path" formula for the super lambda-leng
ths. We also present connections with super frieze patterns introduced by
Ovsienko\, Morier-Genoud\, and Tabachnikov.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lauren Williams (Harvard University)
DTSTART:20210525T150000Z
DTEND:20210525T160000Z
DTSTAMP:20260404T110823Z
UID:OCAS/30
DESCRIPTION:Title: The amplituhedron and cluster algebras\nby Lauren Williams (Harva
rd University) as part of Online Cluster Algebra Seminar (OCAS)\n\n\nAbstr
act\nThe amplituhedron is the image of the positive Grassmannian under a l
inear map induced by a totally positive matrix. Special cases of the ampli
tuhedron include the positive Grassmannian\, cyclic polytopes in projectiv
e space\, and the bounded complex of the cyclic hyperplane arrangement.\n\
nWhile at first glance the amplituhedron seems complicated\, it has many b
eautiful properties. I will explain how ideas from oriented matroids\, tot
al positivity\, and cluster algebras leads to new results about the amplit
uhedron.\n\n\nBased on joint work with Matteo Parisi and Melissa Sherman-B
ennett.\n
LOCATION:https://stable.researchseminars.org/talk/OCAS/30/
END:VEVENT
END:VCALENDAR