The category $\mathrm{CM}(B_{k,n})$ of Cohen-Macaulay modules over a quotient $B_{k,n}$ of a preprojective algebra provides a categorification of the cluster algebra structure on the coordinate ring of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb{C}^{n}$, [13]. Among the indecomposable modules in this category are the rank 1 modules which are in bijection with $k$-subsets of $\{1, 2, \dots, n\}$, and their explicit construction has been given by Jensen, King and Su. These are the building blocks of the category as any module in $\mathrm{CM}(B_{k,n})$ can be filtered by them. In this paper we give an explicit construction of rank 2 modules. With this, we give all indecomposable rank 2 modules in the cases when $k = 3$ and $k = 4$. In particular, we cover the tame cases and go beyond them. We also characterise the modules among them which are uniquely determined by their filtrations. For $k \geq 4$, we exhibit infinite families of non-isomorphic rank 2 modules having the same filtration.

This is an expository survey article on compound Du Val (cDV) singularities, with emphasis on recent homological approaches, including: noncommutative resolutions, tilting theory, contraction algebras, classification, derived categories, autoequivalences, stability conditions, deformation theory, and cluster tilting theory.

This paper presents two new explicit examples of Reid's recipe for non-abelian groups in $\operatorname{SL}(3, \mathbb{C})$, namely the dihedral group $\mathbb{D}_{5,2}$ and a trihedral group of order 39.

This paper is based on a talk at the conference The McKay correspondence, mutation and related topics from July 2020. We provide an introduction to joint work of the author with Søren Gammelgaard, Ádám Gyenge and Balázs Szendrői [CGGS21b] that constructs the reduced scheme underlying the Hilbert scheme of $n$ points on an ADE singularity as a Nakajima quiver variety for a particular stability parameter. After drawing a parallel with two well-known constructions of the Hilbert scheme of $n$ points in $\mathbb{A}^2$, we summarise results of the author and Gwyn Bellamy [BC20] before describing the main result by cornering a noncommutative algebra obtained from the preprojective algebra of the framed McKay graph.

For a finite abelian subgroup $G \subset SL_n(\mathbb{C})$, we study whether a given crepant resolution $X$ of the quotient variety $\mathbb{C}^n/G$ is obtained as a moduli space of $G$-constellations. In particular we show that, if $X$ admits a natural $G$-constellation family in the sense of Logvinenko over it with all fibers being indecomposable as $\mathbb{C}[\mathbb{C}^n]$-modules, then $X$ is isomorphic to the normalization of a fine moduli space of $G$-constellations.

We study wall-crossing phenomena in the McKay correspondence. Craw–Ishii show that every projective crepant resolution of a Gorenstein abelian quotient singularity arises as a moduli space of $\theta$-stable representations of the McKay quiver. The stability condition $\theta$ moves in a vector space with a chamber decomposition in which (some) wall-crossings capture flops between different crepant resolutions. We investigate where chambers for certain resolutions with Hilbert scheme-like moduli interpretations – iterated Hilbert schemes, or ‘Hilb of Hilb’ – sit relative to the principal chamber defining the usual $G$-Hilbert scheme. We survey relevant aspects of wall-crossing, pose our main conjecture, prove it for some examples and special cases, and discuss connections to other parts of the McKay correspondence.

M. Ishida and N. Iwashita classified three-dimensional canonical cyclic quotient singularities [11]. By using the classification, we shall discuss Hilbert basis resolutions via $\mathrm{Hilb}^{G}(\mathbb{C}^{3})$, Fujiki-Oka resolutions and iterated Fujiki-Oka resolutions. In particular, we shall prove that there exists a Hilbert basis resolution for any three-dimensional canonical cyclic quotient singularity.

It is known that the homogeneous coordinate ring of a Grassmannian has a cluster structure, which is induced from the combinatorial structure of a plabic graph. A plabic graph is a certain bipartite graph described on the disk, and there is a family of plabic graphs giving a cluster structure of the same Grassmannian. Such plabic graphs are related by the operation called square move which can be considered as the mutation in cluster theory. By using a plabic graph, we also obtain the Newton–Okounkov polytope which gives a toric degeneration of the Grassmannian. The purposes of this article is to survey these phenomena and observe the behavior of Newton–Okounkov polytopes under the operation called the combinatorial mutation of polytopes. In particular, we reinterpret some operations defined for Newton–Okounkov polytopes using the combinatorial mutation.

The wild McKay correspondence is a form of McKay correspondence in terms of stringy invariants that is generalized to arbitrary characteristics. It gives rise to an interesting connection between the geometry of wild quotient varieties and arithmetic on extensions of local fields. The principal purpose of this article is to collect open problems on the wild McKay correspondence, as well as those in related fields that the author believes are interesting or important. It also serves as a survey on the present state of these fields.

I study a sequence of singularities in dimension 4 and above, each given by a cone of rank 1 tensors of a certain signature, which have crepant resolutions whose exceptional loci are isomorphic to cartesian powers of the projective line. In each dimension $n$, these resolutions naturally correspond to vertices of an $(n - 2)$-simplex, and flops between them correspond to edges of the simplex. I show that each face of the simplex may then be associated to a certain relation between flop functors.

Brane systems provide a large class of gauge theories that arise in string theory. This paper demonstrates how such brane systems fit with a somewhat exotic geometric object, called the affine Grassmannian. This gives a strong motivation to study physical aspects of the affine Grassmannian. Explicit quivers are presented throughout the paper, and a quiver addition algorithm to generate the affine Grassmannian is introduced. An important outcome of this study is a set of quivers for new elementary slices.

The $g$-vector fans play an important role in studying cluster algebras and silting theory. We survey cluster algebras with dense $g$-vector fans and show that a connected acyclic cluster algebra has a dense $g$-vector fan if and only if it is either finite type or affine type. As an application, we classify finite dimensional hereditary algebras with dense $g$-vector fans.

We characterise subcategories of semistable modules for noncommutative minimal models of compound Du Val singularities, including the non-isolated case. We find that the stability is controlled by an infinite polyhedral fan that stems from tilting theory, and which can be computed from the Dynkin diagram combinatorics of the minimal models found in the work of Iyama–Wemyss. In the isolated case, we moreover find an explicit description of the deformation theory of the stable modules in terms of factors of the endomorphism algebras of 2-term tilting complexes. To obtain these results we generalise a correspondence between 2-term silting theory and stability, which is known to hold for finite dimensional algebras, to the much broader setting of algebras over a complete local Noetherian base ring.

We find a duality between two well-known trees, the Calkin–Wilf tree and the Stern–Brocot tree, derived from cluster algebra theory. The vertex sets of these trees are the set of positive rational numbers, and they have cluster structures induced by a one-punctured torus. In particular, the Calkin–Wilf tree is an example of the structure given by initial-seed mutations.

We pose a question asking if the main component of the invariant Hilbert scheme of the Cox realization of a given singularity is a resolution of singularities. As a concrete example, we consider the case where the singularity is the nilpotent cone in $\mathfrak{sl}(n)$, and we see that in this case the answer to the question is positive.