BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Osamu Iyama (Nagoya)
DTSTART:20200817T130000Z
DTEND:20200817T134500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/1
DESCRIPTION:Title: Simple-minded systems in cluster categories and general
ized cluster complexes\nby Osamu Iyama (Nagoya) as part of Cluster alg
ebras 2020\n\n\nAbstract\nThe set S of simple modules over an algebra sati
sfies three basic properties in the derived category: Schur’s Lemma\, va
nishing of negative extensions\, and generating. For a positive integer d\
, Coelho-Simoes introduced the notion of d-simple-minded systems (d-SMS) b
y using analogous 3 conditions depending on d. A typical example of d-SMS
is given by the set of simple dg modules over a d-self-injective dg algebr
a in the singularity category [Riedtmann for d=1 (1980)\, Jin for any d\,
(2019)].\nIn this talk\, I will discuss d-SMSs in the (−d)-cluster categ
ory C of the path algebra kQ of a Dynkin quiver Q. We show that there is a
bijection between d-SMSs in C and positive maximal simplices of the gener
alized cluster complex \\Delta^d(Q) of Fomin-Reading. In particular\, the
number of d-SMSs in C is given by the positive Fuss-Catalan number C_d^+(Q
). To prove this\, we give a bijection between d-SMSs in C and silting obj
ects in the derived category D^b(kQ) whose cohomologies are concentrated i
n degrees 0\,1\,…\,d-1. This bijection is based on the silting-t-structu
re correspondence\, and holds true also for any acyclic quiver by an indep
endent result of Coelho-Simoes-Pauksztello-Ploog. We also use Buan-Reiten-
Thomas' and Zhu's results on generalized cluster complexes.\nThis is a joi
nt work with Haibo Jin (arXiv:2002.09952).\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiuping Su (Bath)
DTSTART:20200817T140000Z
DTEND:20200817T144500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/2
DESCRIPTION:Title: Categorifying flow polynomials and Newton-Okounkov bodi
es\nby Xiuping Su (Bath) as part of Cluster algebras 2020\n\n\nAbstrac
t\nIn this talk I will define an invariant in the Grassmannian cluster cat
egory CM(A)\, apply the invariant to study flow polynomials defined on a n
etwork chart and explain its link to the Newton-Okounkov body constructed
by Rietsch-Williams. \n\nThis talk is based on an ongoing project joint wi
th B T Jensen and A King.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Goncharov (Yale)
DTSTART:20200818T000000Z
DTEND:20200818T004500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/4
DESCRIPTION:Title: Cluster nature of the second universal motivic Chern cl
ass\nby Alexander Goncharov (Yale) as part of Cluster algebras 2020\n\
n\nAbstract\nLet G be a split\, simply-connected\, semi-simple group over
Q. Denote by BG the classifying space of the group G.\n\nThe degree 4\, we
ight 2\, integral motivic cohomology group of BG is isomorphic to Z. The g
enerator C_2\, known as the second universal motivic Chern class\, plays a
n important role in arithmetic\, algebraic geometry\, representation theor
y and mathematical physics.\n\nFor the group GL(m)\, there is an explicit
construction of a cocycle for the generator C_2 which I found in 1993. A d
ecade later it was used in our work with V. Fock on higher Teichmuller the
ory. However a similar explicit construction for any group G was not known
till now. \n\nI will explain that such a construction follows from\, and
in fact is essentially equivalent to\, the main result of our recent work
with Linhui Shen on the cluster structure of the moduli space A(G\,S)\, cl
osely related to the space of G-local systems on a decorated surface S\, p
lus "epsilon". \n\nApplications include the following:\n\ni) An explicit c
ombinatorial construction of the second Chern class of a G-bundle on a man
ifold.\n\nii) An explicit construction of the determinant line bundle on t
he affine Grassmannian\, and of the Kac-Moody group.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linhui Shen (Michigan State)
DTSTART:20200818T140000Z
DTEND:20200818T144500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/5
DESCRIPTION:Title: Legendrian links with infinitely many exact Lagrangian
fillings\nby Linhui Shen (Michigan State) as part of Cluster algebras
2020\n\n\nAbstract\nThe space $R^3$ is equipped with the standard contact
structure from the 1-form $\\alpha = y dx -dz$. A Legendrian link in $R^3$
is a link $\\Lambda$ such that the restriction of $\\alpha$ to $\\Lambda$
vanishes. In this talk\, we focus on Legendrian links that are obtained a
s the rainbow closure of positive braids. We show that if the quiver assoc
iated to any positive braid is not mutation equivalent to a finite type qu
iver\, then the corresponding Legendrian link has infinitely many exact La
grangian fillings. The main technique of this proof includes cluster algeb
ras and Chekanov-Eliashberg differential graded algebras. This is joint wo
rk in progress with Honghao Gao and Daping Weng.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dylan Allegretti (U. British Columbia)
DTSTART:20200818T150000Z
DTEND:20200818T160000Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/6
DESCRIPTION:Title: Exact WKB analysis and Riemann-Hilbert problems\nby
Dylan Allegretti (U. British Columbia) as part of Cluster algebras 2020\n
\n\nAbstract\nExact WKB analysis is a tool for constructing exact solution
s of Schrödinger's equation. Recently\, Iwaki and Nakanishi discovered a
surprising link between exact WKB analysis and cluster algebras. In this t
alk\, I will explain how these ideas can be used to solve a certain Rieman
n-Hilbert problem posed by Bridgeland in the context of Donaldson-Thomas t
heory.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Se-jin Oh (Ewha Womans U.)
DTSTART:20200819T130000Z
DTEND:20200819T134500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/7
DESCRIPTION:Title: Monoidal categorification and quantum affine algebras\nby Se-jin Oh (Ewha Womans U.) as part of Cluster algebras 2020\n\n\nAb
stract\nThe notion of monoidal categorification\, introduced by Hernandez
and Leclerc is a one of the tools for proving the (quantum) Laurent positi
vity and phenomenon of cluster algebras. In the cowork with Kashiwara\, Ki
m and Park\, we proved that lots of categories over quantum affine algebra
s provide monoidal categorification of cluster algebras by introducing new
integer-valued invariants and using several known results. In this talk\,
I will introduce these results and show several quivers related to the re
sult. If time permits\, I will show that the result can be generalized to
quantum version by using the recent result with Fujita.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sabin Cautis (U. British Columbia)
DTSTART:20200820T000000Z
DTEND:20200820T004500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/8
DESCRIPTION:Title: Categorical structure of Coulomb branches of 4D N=2 gau
ge theories\nby Sabin Cautis (U. British Columbia) as part of Cluster
algebras 2020\n\n\nAbstract\nWe will discuss the categorical structure of
Coulomb branches. For concreteness we focus on the massless case which is
just the category of coherent sheaves on the affine Grassmannian (the cohe
rent Satake category).\n\nThese categories are conjecturally governed by a
cluster algebra structure. We describe a solution of this conjecture in t
he case of general linear groups and discuss its extension to more general
Coulomb branches of 4D N=2 gauge theories. This is joint work with Harold
Williams.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peng Shan (YMC Tsinghua)
DTSTART:20200820T130000Z
DTEND:20200820T134500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/9
DESCRIPTION:Title: Coherent categorification of quantum loop sl(2)\nby
Peng Shan (YMC Tsinghua) as part of Cluster algebras 2020\n\n\nAbstract\n
We explain an equivalence of categories between a module category of quive
r Hecke algebras associated with the Kronecker quiver and a category of eq
uivariant perverse coherent sheaves on the nilpotent cone of type A. This
provides a link between two different monoidal categorifications of the op
en quantum unipotent cell of affine type A_1\, one given by Kang-Kashiwara
-Kim-Oh-Park in terms of quiver Hecke algebras\, the other given by Cautis
-Williams in terms of equivariant perverse coherent sheaves on affine Gras
smannians. This is a joint work with Michela Varagnolo and Eric Vasserot.\
n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joel Kamnitzer (Toronto)
DTSTART:20200820T140000Z
DTEND:20200820T144500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/10
DESCRIPTION:Title: The theta basis and the Mirkovic-Vilonen basis\nby
Joel Kamnitzer (Toronto) as part of Cluster algebras 2020\n\n\nAbstract\n
When Fomin and Zelevinsky invented cluster algebras\, one motivation was t
o understand canonical bases in representation theory. More recently\, Gro
ss\, Hacking\, Keel\, and Kontsevich constructed the theta basis in a larg
e class of cluster algebras. On the other hand\, in representation theory\
, we have the Mirkovic-Vilonen basis\, constructed using the geometric Sat
ake correspondence. I will explain our attempts to relate the Mirkovic-Vil
onen basis to cluster algebras and to the theta basis.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gus Schrader (Columbia)
DTSTART:20200821T010000Z
DTEND:20200821T014500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/11
DESCRIPTION:Title: An alternative polynomial representation of spherical
DAHA\nby Gus Schrader (Columbia) as part of Cluster algebras 2020\n\n\
nAbstract\nThe spherical subalgebra of Cherednik's double affine Hecke alg
ebra of type A has a polynomial representation in which the algebra acts o
n a space of symmetric Laurent polynomials by rational q-difference operat
ors. This representation has many useful applications e.g. to the theory o
f Macdonald polynomials. I'll present an alternative polynomial representa
tion of the spherical DAHA\, in which the algebra acts on a space of non-s
ymmetric Laurent polynomials by Laurent polynomial q-difference operators.
This latter representation turns out to be compatible with a natural clus
ter algebra structure\, in such a way that the action of the modular group
on DAHA is given by cluster transformations. Based on joint work in progr
ess with Philippe di Francesco\, Rinat Kedem\, and Alexander Shapiro.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anastasia Volovich (Brown U.)
DTSTART:20200821T130000Z
DTEND:20200821T134500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/12
DESCRIPTION:Title: Cluster Algebras\, Plabic Graphs and Scattering Amplit
udes\nby Anastasia Volovich (Brown U.) as part of Cluster algebras 202
0\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Song He (ITP Beijing)
DTSTART:20200821T140000Z
DTEND:20200821T144500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/13
DESCRIPTION:Title: Generalized associahedra and scattering of particles a
nd strings\nby Song He (ITP Beijing) as part of Cluster algebras 2020\
n\n\nAbstract\nI will review recent works connecting generalized associahe
dra and cluster algebras to scattering amplitudes of (certain generalizati
ons) of particles and strings. Tree amplitudes of a cubic scalar theory ar
e given by "canonical forms" of associahedra in kinematic space. The rule
for constructing the associahedron can be abstracted away to a certain “
walk” associated with any acyclic quiver\, remarkably yielding a finite
polytope for the case of Dynkin quivers\, and for classical types they giv
e scalar amplitudes through one-loop order. Furthermore\, we introduce "cl
uster configuration spaces" and associated open and closed "cluster string
integrals" for any Dynkin diagram\, which for type A reduces to usual mod
uli spaces and string amplitudes. Both the geometries and integrals enjoy
remarkable factorization properties at finite α′ (inverse of string ten
sion)\, obtained simply by removing nodes of the Dynkin diagram\, and as
α′ → 0 the integrals reduce to canonical forms of our generalized ass
ociahedra.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Hernandez (Paris)
DTSTART:20200824T130000Z
DTEND:20200824T134500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/14
DESCRIPTION:Title: Deformations of Grothendieck rings and toroidal cluste
r algebras\nby David Hernandez (Paris) as part of Cluster algebras 202
0\n\n\nAbstract\nThe Grothendieck ring K(C) of a category C of finite-dime
nsional representations of a quantum affine algebra admit natural quantum
deformations obtained from perverse sheaves on quiver varieties\, from def
ormed W-algebras or from vertex operators on quantum Heisenberg algebras.
We review and discuss the interplay with the cluster algebra structures in
troduced with Bernard Leclerc. We also discuss how cluster algebras with s
everal quantum parameters (toroidal cluster algebras) appear in this conte
xt (based in part on joint works with Laura Fedele\, Bernard Leclerc and H
ironori Oya).\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Misha Gekhtman (Notre Dame)
DTSTART:20200824T140000Z
DTEND:20200824T144500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/15
DESCRIPTION:Title: Generalized cluster structures related to the Drinfeld
double of GL(n)\nby Misha Gekhtman (Notre Dame) as part of Cluster al
gebras 2020\n\n\nAbstract\nI will present a unifying approach to a constru
ction of several generalized cluster structures of geometric type. Example
s include the Drinfeld double of GL(n)\, spaces of periodic difference ope
rators and generalized cluster structures in GL(n) compatible with a certa
in subclass of Belavin-Drinfeld Poisson-Lie brackets. Based on a joint wor
k with M. Shapiro and A. Vainshtein.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Toshiya Yurikusa (Tohoku)
DTSTART:20200825T010000Z
DTEND:20200825T014500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/16
DESCRIPTION:Title: Cluster algebras with dense g-vector fans\nby Tosh
iya Yurikusa (Tohoku) as part of Cluster algebras 2020\n\n\nAbstract\nClus
ter variables (monomials) have a numerical invariant\, called g-vector. Th
e g-vectors in a cluster algebra form a simplicial polyhedral fan\, called
g-vector fan. We give a classification of cluster algebras with dense g-v
ector fans except for some types.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tsukasa Ishibashi (RIMS Kyoto)
DTSTART:20200825T130000Z
DTEND:20200825T134500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/17
DESCRIPTION:Title: Algebraic entropy of sign-stable mutation loops\nb
y Tsukasa Ishibashi (RIMS Kyoto) as part of Cluster algebras 2020\n\n\nAbs
tract\nA mutation loop is a certain equivalence class of a sequence of mut
ations and permutations of indices. They form a group called the cluster m
odular group\, which can be regarded as a combinatorial generalization of
the mapping class groups of marked surfaces.\nWe introduce a new property
of mutation loops which we call the “sign stability” as a generalizati
on of the pseudo-Anosov property of a mapping class.\nA sign-stable mutati
on loop has a numerical invariant which we call the “cluster stretch fac
tor”\, in analogy with the stretch factor of a pA mapping class. We comp
ute the algebraic entropies of the cluster A- and X-transformations induce
d by a sign-stable mutation loop\, and conclude that these two are estimat
ed by the logarithm of the cluster stretch factor. This talk is based on a
joint work with Shunsuke Kano.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Milen Yakimov (Louisiana State)
DTSTART:20200825T140000Z
DTEND:20200825T144500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/18
DESCRIPTION:Title: Green groupoids of 2-Calabi-Yau categories and derived
actions\nby Milen Yakimov (Louisiana State) as part of Cluster algebr
as 2020\n\n\nAbstract\nThis is a joint work with Peter Jorgensen (Aarhus U
niversity). Starting with work of Seidel and Thomas\, there has been a gre
at interest in the construction of faithful actions of various classes of
groups on derived categories (braid groups\, fundamental groups of hyperpl
ane arrangements\, mapping class groups). We will present a general constr
uction of this sort in the setting of algebraic 2-Calabi–Yau triangulate
d categories. To every 2-Calabi-Yau category C (satisfying standard finite
ness conditions) we associate a groupoid\, the green grouped of C\, define
d in an intrinsic homological way. In the case of cluster categories\, thi
s can be formulated purely combinatorially using the exchange graph of the
cluster algebra and Keller's notion of maximal green sequence. For every
Frobenius model of the 2-Calabi-Yau category C\, we construct a categorica
l representation of the green groupoid of C. This action is proved to be f
aithful under certain general assumptions.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peigen Cao (Paris)
DTSTART:20200826T130000Z
DTEND:20200826T134500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/19
DESCRIPTION:Title: The valuation pairing on an upper cluster algebra\
nby Peigen Cao (Paris) as part of Cluster algebras 2020\n\n\nAbstract\nThi
s talk is based on the joint work with B. Keller and F. Qin. It is known t
hat many (upper) cluster algebras are not unique factorization domains. In
order to study the local factorization properties of upper cluster algebr
as\, we introduce the valuation pairing on any upper cluster algebra $\\ma
thcal U$. To each pair $(A_{k\;t}\,M)$ consisting of a cluster variable $A
_{k\;t}$ and an element $M$ in $\\mathcal U$\, it associates the largest i
nteger $s$ (maybe infinity) such that $M/A_{k\;t}^s$ still belongs to $\\m
athcal U$. Using the valuation pairing we prove that any full rank upper c
luster algebra has the following locally unique factorization property: Fo
r each seed $t$ of $\\mathcal U$\, any non-zero element $M$ can be uniquel
y factorized as $M=N\\cdot L$\, where $N$ is a cluster monomial in $t$ and
$L$ is an element in $\\mathcal U$ not divisible by any cluster variable
in $t$.\n\nWe give many applications to $d$-vectors\, $F$-polynomials\, fa
ctoriality of upper cluster algebras and combinatorics of cluster Poisson
variables. To be more precisely\, firstly\, we prove that a full rank uppe
r cluster algebra $\\mathcal U$ with initial seed $t_0$ is factorial if an
d only if each exchange binomial of $t_0$ is irreducible in the correspond
ing polynomial ring. In particular\, full rank and primitive upper cluster
algebras are factorial\, which include principal coefficients upper clust
er algebras as a special case. Secondly\, we show how to express the $d$-c
ompatibility degree and $d$-vectors using the valuation pairing. Thirdly\,
we prove that if a cluster monomial $M$ contains no initial cluster varia
bles\, then $M$ is uniquely determined by its $F$-polynomial $F_M$. Fourth
ly\, we prove that the $F$-polynomials of non-initial cluster variables ar
e irreducible.\n\nThanks to the results on $F$-polynomials\, we give sever
al equivalent characterizations when two cluster Poisson variables are equ
al. As the first application\, we prove that the cluster Poisson variables
of a cluster Poisson algebra $\\mathcal X$ parameterize the $\\mathscr A$
-exchange pairs of an upper cluster algebra $\\mathcal U$ of the same type
with $\\mathcal X$. As the second application\, we prove that the $\\math
scr X$-seeds of $\\mathcal X$ whose Poisson clusters containing particular
cluster Poisson variables forms a connected subgraph of the exchange grap
h of $\\mathcal X$.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Fraser (Minneapolis)
DTSTART:20200826T140000Z
DTEND:20200826T144500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/20
DESCRIPTION:Title: Cyclic symmetry loci in Grassmannians\nby Chris Fr
aser (Minneapolis) as part of Cluster algebras 2020\n\n\nAbstract\nThe Gra
ssmannian Gr(k\,n) admits an action by a finite cyclic group of order n vi
a the cyclic shift automorphism. We study cyclic symmetry loci\, i.e. the
set of points in the Grassmannian fixed by a given iterate of the cyclic s
hift map. We give a simple geometric description of the fixed point locus\
, give a cell decomposition of its set of totally nonnegative points\, and
discuss the existence of generalized cluster charts (in the sense of Chek
hov and Shapiro).\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melissa Sherman-Bennett (UC Berkeley)
DTSTART:20200826T150000Z
DTEND:20200826T154500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/21
DESCRIPTION:Title: Many cluster structures on Schubert varieties\nby
Melissa Sherman-Bennett (UC Berkeley) as part of Cluster algebras 2020\n\n
\nAbstract\nWe give a family of cluster structures on Schubert (and more g
enerally\, positroid) varieties in the Grassmannian. Each of the cluster s
tructures in this family has seeds given by face labels of relabeled plabi
c graphs\, which are plabic graphs whose boundary is labeled by a permutat
ion of 1\, ...\, n. For Schubert varieties\, all cluster structures in thi
s family quasi-coincide\, meaning they differ only by rescaling by frozen
variables and their cluster monomials coincide. As part of our results\, w
e show the "target" and "source" cluster structures on Schubert varieties
quasi-coincide\, confirming a conjecture of Muller and Speyer. One proof t
ool of independent interest is a permuted version of the Muller-Speyer twi
st map\, which we use to prove many (open) positroid varieties are isomorp
hic.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Yue Yu (Paris Saclay)
DTSTART:20200827T130000Z
DTEND:20200827T134500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/22
DESCRIPTION:Title: Cluster algebra via non-archimedean geometry\nby T
ony Yue Yu (Paris Saclay) as part of Cluster algebras 2020\n\n\nAbstract\n
I will explain the enumeration of non-archimedean analytic curves in clust
er varieties. We can construct a canonical scattering diagram via the enum
eration of infinitesimal non-archimedean cylinders and prove the Kontsevic
h-Soibelman consistency. Moreover\, we prove a comparison theorem with the
combinatorial constructions of Gross-Hacking-Keel-Kontsevich. This has se
veral nice implications\, such as the broken-line convexity conjecture\, a
geometric proof of the positivity in the Laurent phenomenon\, removal of
the EGM assumption in GHKK\, and the independence of the mirror algebra on
the choice of cluster structure\, as conjectured by GHKK. This is part of
my joint work with Keel\, arXiv:1908.09861.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Travis Mandel (Oklahoma)
DTSTART:20200827T140000Z
DTEND:20200827T144500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/23
DESCRIPTION:Title: Quantum theta bases\nby Travis Mandel (Oklahoma) a
s part of Cluster algebras 2020\n\n\nAbstract\nI will discuss joint work w
ith Ben Davison in which we construct the quantum analog of the Gross-Hack
ing-Keel-Kontsevich theta bases. We prove that these quantum theta bases
satisfy the expected properties\, particularly universal and strong positi
vity and atomicity with respect to the quantum scattering atlas. The cons
truction uses quantum scattering diagrams\, and the positivity proof relie
s on deep ideas from the DT-theory of quiver representations. For (quantu
m) cluster algebras from surfaces\, upcoming joint work with Fan Qin will
show that (quantum) theta bases agree with the (quantum) bracelet bases.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Man-wai Cheung (Harvard)
DTSTART:20200828T130000Z
DTEND:20200828T134500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/24
DESCRIPTION:Title: Polytopes\, wall crossings\, and cluster varieties
\nby Man-wai Cheung (Harvard) as part of Cluster algebras 2020\n\n\nAbstra
ct\nCluster varieties are log Calabi-Yau varieties which are a union of al
gebraic tori glued by birational "mutation" maps. Partial compactification
s of the varieties\, studied by Gross\, Hacking\, Keel\, and Kontsevich\,
generalize the polytope construction of toric varieties. However\, it is n
ot clear from the definitions how to characterize the polytopes giving com
pactifications of cluster varieties. We will show how to describe the comp
actifications easily by broken line convexity. As an application\, we will
see the non-integral vertex in the Newton Okounkov body of Gr(3\,6) comes
from broken line convexity. Further\, we will also see certain positive p
olytopes will give us hints about the Batyrev mirror in the cluster settin
g. The mutations of the polytopes will be related to the almost toric fibr
ation from the symplectic point of view. Finally\, we can see how to exten
d the idea of gluing of tori in Floer theory which then ended up with the
Family Floer Mirror for the del Pezzo surfaces of degree 5 and 6. The talk
will be based on a series of joint works with Bossinger\, Lin\, Magee\, N
ajera-Chavez\, and Vianna.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lara Bossinger (UNAM Oaxaca)
DTSTART:20200828T140000Z
DTEND:20200828T144500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/25
DESCRIPTION:Title: Families of Gröbner degenerations\, Grassmannians and
universal cluster algebras\nby Lara Bossinger (UNAM Oaxaca) as part o
f Cluster algebras 2020\n\n\nAbstract\nLet V be the weighted projective va
riety defined by a weighted homogeneous ideal J and C a maximal cone in th
e Gröbner fan of J with m rays. We construct a flat family over affine m-
space that assembles the Gröbner degenerations of V associated with all f
aces of C. This is a multi-parameter generalization of the classical one-p
arameter Gröbner degeneration associated to a weight. We show that our fa
mily can be constructed from Kaveh--Manon's recent work on the classificat
ion of toric flat families over toric varieties: it is the pullback of a t
oric family defined by a Rees algebra with base X_C (the toric variety ass
ociated to C) along the universal torsor $\\mathbb A^m \\to X_C$. \nI will
explain how to apply this construction to the Grassmannians Gr(2\,n) (wit
h Plücker embedding) and Gr(3\,6) (with "cluster embedding"). In each cas
e there exists a unique maximal Gröbner cone whose associated initial ide
al is the Stanley--Reisner ideal of the cluster complex. We show that the
corresponding cluster algebra with universal coefficients arises as the al
gebra defining the flat family associated to this cone. Further\, for Gr(2
\,n) we show how Escobar--Harada's mutation of Newton--Okounkov bodies can
be recovered as tropicalized cluster mutation. This is joint work with Fa
temeh Mohammadi and Alfredo Nájera Chávez.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Jordan (Edinburgh)
DTSTART:20200819T140000Z
DTEND:20200819T144500Z
DTSTAMP:20260404T131143Z
UID:ClusterAlgebras2020/26
DESCRIPTION:Title: Cluster quantization from factorization homology\n
by David Jordan (Edinburgh) as part of Cluster algebras 2020\n\n\nAbstract
\nFactorization homology is a powerful universal invariant of (possibly\,
stratified) manifolds\, introduced in a series of papers by Ayala\, Franci
s\, Lurie\, and Tanaka. Many constructions in quantum topology receive new
interpretations\, stronger functoriality/TFT properties\, and much greate
r generality when they can be recast in the framework of factorization hom
ology. \n\nIn this talk I'll explain some forthcoming work joint with Ian
Le\, Gus Schrader and Sasha Shapiro\, in which we compute factorization ho
mology of stratified surfaces\, with coefficients taken in: quantum SL_2 f
or the bulk region of the surface\; it's quantum Borel subgroup for defect
lines\, and its quantum Cartan quotient for boundary regions. The resulti
ng ``quantum decorated character stacks" admit interesting systems of q-to
ric charts which echo Fock--Goncharov's cluster quantizations of decorated
character varieties. Remarkably\, these charts emerge not as a primary co
nstruction/definition as in Fock--Goncharov's work -- but rather as a cons
equence of the universal properties of factorization homology.\n
LOCATION:https://stable.researchseminars.org/talk/ClusterAlgebras2020/26/
END:VEVENT
END:VCALENDAR