Title and abstract

  • Alvaro Nolla de Celis (UAM, Spain) How to calculate G-Hilb for dihedral groups in SL(3,C)

Abstract: It is well known that the G-equivariant Hilbert scheme G-Hilb(C^3) is a crepant resolution of the quotient singularity C^3/G. This space is well understood in the (toric) case when the group G is abelian, but when G is not abelian there are very few cases where an explicit description of it is known. In this talk I will explain with examples how to construct G-Hilb(C^3) for dihedral subgroups in SL(3,C), giving the main ingredients towards the description of the whole family.


  • Aaron Chan (Nagoya, Japan) Curves on marked surfaces and complexes of Brauer graph algebras

Abstract: In a paper by Khovanov and Seidel, they study autoequivalences of some "Fukaya-like categories" of Milnor fibres associated to type A singularities. Their strategy is to consider a certain triangulated category generated by "braid twists" on a disc (with marked points). This category is equivalent to the bounded homotopy category of finitely generated projective module over a finite dimensional algebra, where certain complexes can be easily described using combinatorics on the associated disc. It is well-known to many representation theorists that this algebra is (related to) a special case of the so-called Brauer graph algebras. In this talk, we are going to explain how Brauer graph algebras are related to marked surface, and how this relation allows us to extend the idea of Khovanov-Seidel to this more general context. This is an on-going joint work with Takahide Adachi.


  • Alastair Craw (Bath, UK) Birational geometry of symplectic quotient singularities

Abstract: For a finite subgroup G of SL(2,C) and for n>1, I'll describe the movable cone of the Hilbert scheme of n points on the minimal resolution of the Kleinian singularity C^2/G. The key is to show that every projective, crepant resolution of C^{2n}/G_n (here, G_n is the wreath product of G with S_n) is a fine moduli space of \theta-stable modules over the framed preprojective algebra of G, where \theta is a choice of generic stability condition. These results provide higher-dimensional analogues of Kronheimer’s work on the minimal resolution of Kleinian singularities. This is joint work with Gwyn Bellamy.


  • Yusuke Nakajima (Kavli IPMU, Japan) Representation theory of three dimensional Gorenstein toric singularities

Abstract: The classical McKay correspondence gives the close connection between the minimal resolution of rational double points and the Dynkin diagram of type ADE. Also, the relationships between these objects and some non-commutative algebras had been discovered in the last few decades. For example, it is known that the minimal resolution of a rational double point is derived equivalent to the skew group algebra, in which case this algebra is called a non-commutative crepant resolution, and the skew group algebra is Morita equivalent to the preprojective algebra of the extended Dynkin quiver of type ADE. Furthermore, the stable category of Z-graded maximal Cohen-Macaulay modules over a rational double point is equivalent to the bounded derived category of finitely generated modules over the path algebra of the Dynkin quiver of type ADE. In my talk, I will explain about some analogues of the above correspondences for three dimensional Gorenstein toric singularities by using dimer models and the associated quiver. This talk is based on the paper arXiv:1806.05331.


  • Will Donovan (Tsinghua, China ) Perverse sheaves of categories on mirror moduli spaces

Abstract: The moduli spaces associated to the A-side and B-side of mirror symmetry carry much-studied bundles with connections, which may be related by a mirror map. I explain progress towards categorification of such structures in examples associated to surface and 3-fold singularities, and explain applications to homological mirror symmetry. This talk will discuss joint work with T Kuwakagi, and joint work with M Wemyss.


  • Yusuke Sato (Nagoya, Japan) G-Hilb(C^4) and crepant resolutions of certain abelian groups in SL(4,C)

Abstract: Let G be a finite subgroup of SL(n,C), then the quotient C^n/G has a Gorenstein canonical singularity.If n=2 or 3, C^n/G has a crepant resolution, and G-Hilb(C^n) is a crepant resolution. However, in the case n=4, C^4/G does not always have crepant resolution, and the relationship between G-Hilb(C^n) and crepant resolutions is not well known. In this talk, we will show several examples where G-Hilb(C^4) is blow-up of certain crepant resolutions, or G-Hilb(C^4) has a singularity.


  • Yasuaki Gyoda (Nagoya, Japan) Duality between mutations and rear mutations in cluster algebras

Abstract: The rear mutations are new transformations in cluster algebra theory. They are the transformations between rational expressions of cluster variables in terms of the initial cluster under the change of the initial cluster. In this talk, we discuss the duality between the (usual) mutations and the rear mutations through the C-matrices, the G-matrices and the F-polynomials. In particular, we introduce the F-matrices, which is the maximal degree matrices of the F-polynomials, and show that they have the self-duality which is analogous to the duality between the C- and G-matrices by Nakanishi and Zelevinsky. This is a joint work with Shogo Fujiwara.


  • Sota Asai (Nagoya, Japan) The wall-chamber structures of the real-valued Grothendieck groups

Abstract: We consider a finite-dimensional algebra $A$ over a field and the real-valued Grothendieck group of the category of finite-dimensional projective $A$-modules. The real-valued Grothendieck group can be identified with a Euclidean space, and Br\"{u}stle--Smith--Treffinger defined a wall-chamber structure of the real-valued Grothendieck group by associating a wall to each finite-dimensional $A$-module via the semistability condition introduced by King. In this talk, I will observe the wall-chamber structure from the point of view of the numerical torsion pairs defined for each element of the real-valued Grothendieck group by Baumann--Kamnitzer--Tingley, and explain my new result that the chambers of the wall-chamber structure bijectively correspond to the 2-term silting objects of the perfect derived category.


  • Wahei Hara (Waseda, Japan) On derived equivalences for Abuaf's 5-dimensional flop

Abstract: In this talk, we discuss one concrete example of a 5-dimensional flop, which was found by Abuaf, and was first studied by Segal. The exceptional loci of this flop are P^3 and Q_3, and in particular they are not isomorphic. This point is an interesting feature of this new flop. Segal proved that both sides of this flop are derived equivalent as conjectured by Bondal and Orlov. The aim of this talk is to do further studies for derived categorical aspects and non-commutative aspects of this flop.


  • Ayako Kubota (Waseda, Japan) Invariant Hilbert scheme resolution of Popov’s SL(2)-varieties

Abstract: Three dimensional affine normal quasihomogeneous SL(2)-varieties were completely classified by Popov in 1973, and using this classification, related works have been conducted since then. One of these is that of Batyrev and Haddad, which shows that every such SL(2)-variety admits a GIT quotient description. This enables us to study its singularities by means of the invariant Hilbert scheme, and in this talk we discuss its desingularization given by the associated Hilbert—Chow morphism and describe the main component of the considering invariant Hilbert scheme.


  • Ryo Yamagishi (Kyoto, Japan) Minimal models of quotient singularities

Abstract: Crepant resolutions of a quotient singularity do not exist in general. However, their slight generalizations, minimal models, always exist by a result from birational geometry. In the talk, I will explain a construction of minimal models using a Cox ring and the (partial) McKay correspondence established by Ito and Reid.