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:20260422T225702Z 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:20200908T150000Z DTEND:20200908T160000Z DTSTAMP:20260422T225702Z 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:20200915T150000Z DTEND:20200915T160000Z DTSTAMP:20260422T225702Z 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:20200922T150000Z DTEND:20200922T160000Z DTSTAMP:20260422T225702Z 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:20200929T150000Z DTEND:20200929T160000Z DTSTAMP:20260422T225702Z 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:20201006T150000Z DTEND:20201006T160000Z DTSTAMP:20260422T225702Z 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:20201013T150000Z DTEND:20201013T160000Z DTSTAMP:20260422T225702Z 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:20201020T150000Z DTEND:20201020T160000Z DTSTAMP:20260422T225702Z 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:20201117T160000Z DTEND:20201117T170000Z DTSTAMP:20260422T225702Z 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:20201103T160000Z DTEND:20201103T170000Z DTSTAMP:20260422T225702Z 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:20201110T160000Z DTEND:20201110T170000Z DTSTAMP:20260422T225702Z 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:20201208T160000Z DTEND:20201208T170000Z DTSTAMP:20260422T225702Z 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:20210119T160000Z DTEND:20210119T170000Z DTSTAMP:20260422T225702Z 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:20201124T160000Z DTEND:20201124T170000Z DTSTAMP:20260422T225702Z 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:20201201T160000Z DTEND:20201201T170000Z DTSTAMP:20260422T225702Z 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:20210126T160000Z DTEND:20210126T170000Z DTSTAMP:20260422T225702Z 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:20210216T160000Z DTEND:20210216T170000Z DTSTAMP:20260422T225702Z 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:20210202T160000Z DTEND:20210202T170000Z DTSTAMP:20260422T225702Z 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:20210223T160000Z DTEND:20210223T170000Z DTSTAMP:20260422T225702Z 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:20210330T150000Z DTEND:20210330T160000Z DTSTAMP:20260422T225702Z 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:20210209T160000Z DTEND:20210209T170000Z DTSTAMP:20260422T225702Z 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:20210316T160000Z DTEND:20210316T170000Z DTSTAMP:20260422T225702Z 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:20210323T150000Z DTEND:20210323T160000Z DTSTAMP:20260422T225702Z 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:20210406T150000Z DTEND:20210406T160000Z DTSTAMP:20260422T225702Z 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:20210309T160000Z DTEND:20210309T170000Z DTSTAMP:20260422T225702Z 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:20210413T150000Z DTEND:20210413T160000Z DTSTAMP:20260422T225702Z 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:20210420T150000Z DTEND:20210420T160000Z DTSTAMP:20260422T225702Z 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:20210427T150000Z DTEND:20210427T160000Z DTSTAMP:20260422T225702Z 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:20210511T150000Z DTEND:20210511T160000Z DTSTAMP:20260422T225702Z 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:20210525T150000Z DTEND:20210525T160000Z DTSTAMP:20260422T225702Z 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