Takahide Adachi (Yamaguchi university):Silting-discrete algebras

Keller--Vossieck give the notion of silting complexes to study bounded t-structures on path algebras. Recently, Aihara--Iyama introduce the notion of silting mutation, which is an analog of mutation in cluster algebras. From the viewpoint of silting mutation theory, a silting-discrete algebra is important because it has a property that any silting complex is given by a finite sequence of irreducible silting mutations. In this talk, I will explain some properties of silting-discrete algebras.

Sota Asai (RIMS, Kyoto university):The wall-chamber structures of the real Grothendieck groups

Let A be a finite-dimensional algebra over a field. Then the real Grothendieck group K_0(proj A)_R is an Euclidean space. For each module M, we have the subset Θ_M consisting of all elements θ such that M is θ-semistable in the sense of King. By considering such subsets as walls, we have a wall-chamber structure on K_0(proj A)_R. I will talk about my main result that the chambers in the wall-chamber structure bijectively correspond to the 2-term silting objects of the perfect derived category K^b(proj A).

Lara Bossinger (UNAM, Unidad Oaxaca) : Groebner degenerations and Universal coefficients for cluster algebras

Given an ideal I inside a polynomial ring we consider a maximal cone C in its Groebner fan with associated monomial initial ideal. We construct a flat family of degenerations of the affine variety V(I) with special fibers in correspondence with faces of C. Applying this construction to the Pluecker ideal for the Grassmannian Gr(2,n) and a specific maximal cone in its Groebner Fan we prove that our construction yields the Grassmannian cluster algebra of type A_{n-3} with universal coefficients. This talk is based on joint work with F. Mohammadi and A. Nájera Chávez.

Peigen Cao (Université Paris) : Explanation for d-vectors and applications

In this talk we will give an explanation for the d-vectors of cluster algebras and talk about its applications.

Aaron Chan (Nagoya university):Torsion classes of gentle algebras

A gentle algebra is an algebra which encodes the topological information of a marked surface equipped with a maximal collection of pairwise non-crossing arcs. Through various results, some of which coming from the categorification of cluster algebras associated to marked surfaces, it is known that functorially finite torsion classes of a gentle algebra correspond to maximal collections of pairwise non-crossing arcs of the associated marked surface. In this talk, we will review this correspondence and explain how this can be generalised to a correspondence for torsion classes. This is a joint work with Laurent Demonet.

Man-Wai Cheung (Harvard University):Compactification for cluster varieties

Cluster varieties are log Calabi-Yau varieties and can be seen as gluing of algebraic torus by the mutation process. Partial compactifications of the varieties, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties. However, the existence of such compactifications is not clear for general setting. Together with Magee and Najera-Chavez, we have extended the idea of convex polytope from toric geometry to the cluster setting which we named as broken line convexity. With this description, we can easily write down the compactifications for all the cluster varieties. Further, for the case of finite type without frozen variables, with Magee, we have found the polytopes with extra properties which give us a hint to the Batyrev-Borisov mirror construction.

Ryo Fujita (Kyoto university) : Graded quiver varieties and singularities of normalized R-matrices for fundamental modules

The normalized R-matrices are realized as unique intertwining operators between tensor products of two finite-dimensional simple modules of the quantum affine algebras. They can be seen as matrix-valued rational functions in spectral parameters, whose denominators determine when the tensor product modules become reducible. In this talk, we present a unified formula expressing the denominators of the normalized R-matrices between the fundamental modules of type $ADE$. It has an interpretation in terms of representations of the Dynkin quivers and can be proved in a unified way using the geometry of Nakajima's graded quiver varieties. As a by-product, we obtain a geometric interpretation of generalized quantum affine Schur-Weyl duality functor in the sense of Kang-Kashiwara-Kim when it arises from a collection of fundamental modules.

Shogo Fujiwara (Nagoya University):Equivalent condition for sign-coherence of C-matrices

The sign-coherence of the C-matrices was conjectured by Fomin, Zelevinsky and shown by Gross et al. The sign-coherence is a fundamental property, such as used to indicate duality between C-matrices and G-matrices. In this talk, we define a one-variable polynomial from an (n-variable) F polynomial and give a proposition that is equivalent to the sign-coherence of the C-matrices regarding the properties of the polynomial.

Christopher Fraser (The University of Minnesota):Additive categorification of Grassmannian braiding

In previous work, I established an action of an extended affine braid group on the open positroid subvariety in the Grassmannian by regular automorphisms. This action permutes cluster monomials in the Grassmannian cluster algebra (with localized coefficients). Jensen-King and Su, building on work of Geiss-Leclerc-Schroer, gave a Frobenius categorification of the Grassmannian cluster algebra as a category of Cohen-Macaulay modules over a certain quotient of the preprojective algebra of a cycle quiver. We will explain how the braid group action can be lifted categorically using Siedel-Thomas' spherical twists, and state some conjectural generalizations. Joint with Bernhard Keller.

Yasuaki Gyoda (Nagoya university):Compatibility degree of cluster complexes

I will introduce a new function on the set of pairs of cluster variables, which we call it the compatibility degree (of cluster complexes). The compatibility degree which I deal with in this talk isa generalization of the ``classical" compatibility degree introduced by Fomin and Zelevinsky. The classical one defines the generalized associahedras, and it is used to give the classification of cluster algebras of finite type. Cao and Li generalized this degree to that of cluster complexes by using d-vectors. We give another generalization by using f-vectors on the basis of their studies. This is joint work with Changjian Fu.

Akishi Ikeda (Osaka university):Bi-graded Calabi-Yau completions of gentle algebras and their cluster categories

For a full formal arc system of a graded marked bordered surface, we can associate with the graded gentle algebra by the work of Haiden-Katzarkov-Kontsevich. In this talk, we introduce the bi-graded quivers with potential associated with the full formal arc systems of the marked bordered surfaces and their Ginzburg dg algebras. We show that their cluster categories are the original derived categories of graded gentle algebras. By collapsing the bi-grading into the single grading, we can obtain quivers with potential associated with the N-angulations of marked bordered surfaces and N-cluster categories. This construction naturally explains the equivalence of two constructions of cluster categories, Amiot-Guo-Keller's quotient constructions and Keller's orbit category construction. This is the joint work with Qiu Yu and Zhou Yu.

Tsukasa Ishibashi (The University of Tokyo):Algebraic entropy of sign-stable mutation loops

A mutation loop is a certain equivalence class of a sequence of mutations and permutations of indices. It induces several dynamical systems via cluster transformations, and they form a group called the cluster modular group which can be seen as a combinatorial generalization of the mapping class groups of marked surfaces.We introduce a new property of mutation loops called the sign stability as a generalization of the pseudo-Anosov property of a mapping class. A sign-stable mutation loop has a numerical invariant which we call the "cluster stretch factor", in analogy with the stretch factor of a pA mapping class. We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor. This talk is based on a joint work with Shunsuke Kano.

Shunsuke Kano (Tokyo Institute of Technology):Pseudo-Anosov mapping classes are sign-stable.

We introduced the sign stability of mutation loops as a generalization of the pseudo-Anosov property of mapping classes of on surfaces. In this talk, I will explain the equivalence between the pseudo-Anosov property and the sign stability for a mapping class. If time permits, I will explain the relationship between the signs of mutations and train track splittings. This talk is based on a joint work with Tsukasa Ishibashi.

Hyun Kyu Kim (Ewha Womans University):Deformation quantization of Teichmüller space by Bonahon-Wong quantum trace

A version of Teichmüller space of a Riemann surface is an example of a cluster X-variety. There is a basis of ring of regular functions on this cluster X-variety enumerated by tropical integer points of cluster A-variety, which are in one-to-one correspondence with certain integral laminations on the surface. The regular function associated to a lamination (which are collection of simple curves with integer weights) can be quantized to a non-commutative Laurent polynomial, by using Bonahon-Wong's quantum trace map on the skein algebra of the surface. This construction is based on a joint work with Dylan Allegretti. If time allows, I will discuss further properties of these quantized functions, and some open problems.

Myungho Kim (Kyung Hee University):Monoidal categorification of cluster algebras and Laurent phenomenon

The quantum unipotent coordinate ring is isomorphic to the Grothendieck ring of the category of finite dimensional graded modules over a quiver Hecke algebra. Furthermore the quantum coordinate ring has a quantum cluster algebra structure and every cluster monomial corresponds to the class of real simple module under the isomorphism. This is an example of the notion which is called the "monoidal categorification of cluster algebras" suggested by Hernandez and Leclerc. In this talk, we will see some consequences of the monoidal categorification of the quantum unipotent rings which are mainly due to the Laurent phenomenon of cluster algebras. This is a joint work with Masaki Kashiwara.

Ian Le (Northwestern University):Tropical pairings for Grassmannians and configurations of flags

Given two cluster varieties X and Y, one can define a pairing between the tropical points of these varieties. I will give some convenient formulas for these pairings in some instances where X and Y come from representation theory. I will suggest a relationship between these pairings and the cluster quantization of these varieties. This based on work with Chris Fraser and work in progress with Nick Early.

Yuma Mizuno (Tokyo Institute of Technology):T-systems and Y-systems in cluster algebras.

We study T-systems (and Y-systems) arising from cluster algebras. We find that these systems are characterized by triples of matrices, which we call T-data, that satisfy a certain symplectic relation. We show that any T-datum corresponding to a periodic T-system has simultaneous positivity, which can be considered as a generalization of the characterization of finite type Cartan matrices. As an application, we discuss the relation between periodic Y-systems and Nahm's conjecture.

Yuya Mizuno (Osaka prefecture university):Preprojective algebras, path algebras and sortable elements

In this talk, we first explain the notion of c-sortable elements (associated with a Dynkin quiver) ,which are some elements of Weyl group, defined by N.Reading from the viewpoint of lattice congruences. Then we can give a map which provides sortable elements. The map connects the Hasse quiver of the Weyl group (defined by the weak order) with the exchange graph of the cluster algebra. We will explain a module-theoretic counterpart of this relationship in terms of representation theory of preprojective algebras and path algebras. We also apply this theory in the study of torsion classes. This talk is based on joint work with H.Thomas.

Yusuke Nakajima (Kavli IPMU, the university of Tokyo):Cluster tilting theory for algebras arising from dimer models

In this talk, I will focus on dimer models described on the real 2-torus, and review representation-theoretic aspects of algebras arising from dimer models. More precisely, a dimer model gives rise to the quiver with potential. Under some assumptions, the Jacobian algebra of such a quiver with potential is a 3-Calabi-Yau algebra, and the center of this algebra is a 3-dimensional Gorenstein toric ring R. I will explain that the stable category of maximal Cohen-Macaulay R-modules is equivalent to the generalized 2-cluster category of a certain algebra if R is an isolated singularity. Also, I discuss mutations of cluster tilting objects in such categories.

Naoto Okubo (Aoyama Gakuin University):Mutation conbinatorics and q-Painleve systems

In this talk, we will explain mutation conbinatorics. We present q-Painleve systems by using cluster mutations. This talk is based on joint work with Tetsu Masuda and Teruhisa Tsuda.

Hironori Oya (Shibaura Institute of Technology):Newton-Okounkov polytopes of Schubert varieties arising from cluster structures and representation-theoretic polytopes

The theory of Newton-Okounkov bodies is a generalization of that of Newton polytopes for toric varieties. One of the ingredients for the definition of a Newton-Okounkov body is a valuation on the function field of a given projective variety. In this talk, we treat Newton-Okounkov bodies of Schubert varieties defined from specific valuations which generalize extended $g$-vectors in cluster theory. We show that they provide polytopes unimodularly equivalent to string polytopes and Nakashima-Zelevinsky polytopes, which are well-known polytopes in representation theory. Indeed, this framework allows us to connect string polytopes with Nakashima-Zelevinsky polytopes by tropicalized cluster mutations. This talk is based on a joint work with Naoki Fujita.

Melissa Sherman-Bennett (Harvard University):Many cluster structures on Schubert varieties

In previous work with Serhiyenko and Williams, we showed that (open) Schubert varieties have a cluster structure where some seeds are given by target labelings of Postnikov's plabic graphs. Galashin and Lam later showed that Schubert varieties also have a cluster structure coming from source labels of plabic graphs. These cluster structures are not the same; in particular, the cluster variables and frozen variables differ. I will discuss joint work with Chris Fraser, in which we show these two cluster structures are closely related. Every variable in the "source" cluster structure can be scaled by a Laurent monomial in the frozen variables to obtain a variable in the "target" cluster structure, and this rescaling preserves all exchange ratios. Along the way to proving this, we produce a sequence of cluster structures on Schubert varieties which interpolate between source and target. Time permitting, I will also discuss the more general setting of positroid varieties.

Toshiya Yurikusa (Nagoya university):Denseness of g-vector cones from triangulated surfaces and applications

We study g-vector cones associated with a triangulated surface, and obtain that the closure of their union is equal to the whole space. As applications, the exchange graph of cluster tilting objects in the corresponding cluster category has at most two components; the corresponding cluster scattering diagram and stability scattering diagram are identified or only differ in a central wall-crossing.