For a generic quasi-symmetric representation X of a reductive group G, Halpern-Leistner and Sam show that the derived category of coherent sheaves on a GIT (stacky) quotient of X is equivalent to magic windows, which are certain triangulated subcategories of the derived categories of coherent sheaves on the quotient stack [X/G]. In this talk, we explain that the equivalences of magic windows,...

We give a nontrivial example of a cotilting complex that induces a compactly generated t-structure in the unbounded derived category of a commutative noetherian ring, and explain its relation with big Cohen-Macaulay modules and Cohen-Macaulay approximations. This talk is partly based on joint work with Michal Hrbek and Jan Stovicek (arXiv:2207.01309).

We consider an example of derived McKay correspondence between non-commutative deformations in the case of surface singularities of type An.

We construct a versal NC deformation of the commutative crepant resolution and compare it with the versal NC deformation of the non-commutative crepant resolution. We show the derived McKay correspondence in the case n=1.

In this talk I will show how to construct examples of G-Hilb when G is a trihedral group in SL(3,C). The calculations are based on the connection between trihedral boats and representations of the McKay quiver, allowing us to compute explicitly the exceptional locus of the crepant resolution G-Hlib C^3 \to C^3/G and Reid’s recipe for small cases.

My goal in this talk is to give an accessible discussion of an ongoing research project with David Kazhdan. In this work, we define L-function genera and use them in the spectral analysis of Eisenstein series and in related problems of enumerative geometry.

Crepant resolutions of 3-fold singularities may contain elaborate configurations of exceptional surfaces. Using toric cases as a guide, I review some known contributions of these configurations to the derived autoequivalence group of the resolution, in particular from work of Seidel-Thomas, and discuss work in progress with Luyu Zheng.

I will explain how most of the results we know so far about the McKay correspondence in dim = 2 and 3 can be conjecturally packaged up into a single mathematical object known as a perverse schober. These were proposed by Kapranov and Schechtman in 2014 as a categorification of an earlier notion of a perverse sheaf by Beilinson, Bernstein, and Deligne.

Hyperpolygon spaces are a family of symplectic singularities in all even dimensions generalising the D4 surface singularity. We present a work in which we describe the Cox rings of crepants resolutions of hyperpolygon spaces and give a method for enumerating ALL such crepant resolutions (including non-projective resolutions).

I will explain how to get noncommutative resolutions of some finite dimensional commutative algebras via birational morphisms of smooth surfaces. The resolution is given by the null category which happens to be highest weight category, whose projective generator is the discrepancy sheaf.

The existence of non-commutative crepant resolutions (NCCRs) for certain classes is one of the most well-studied problems. In this talk, we discuss the construction of NCCRs of toric rings using their conic divisorial ideals and we give an NCCR of a special family of stable set rings, which are toric rings arising from graphs.

The notion of tilting objects is basic to study the structure of a given derived category. The class of silting objects gives a completion of the class of tilting objects from the point of view of mutation, and they correspond bĳectively with other important objects in the derived category. The subset of 2-term silting complexes enjoys especially nice properties, which is closely related to...

It is known that any projective crepant resolution of a three-dimensional Gorenstein toric singularity can be described as the moduli space of representations of the quiver associated to a consistent dimer model for some stability parameter. The space of stability parameters has the wall-and-chamber structure, that is, it is decomposed into chambers separated by walls. The moduli spaces...

Motivated by the theory of non-commutative resolutions and the results on Auslander correspondence, we study the category of reflexive modules over (commutative or non-commutative) Noetherian rings. One well-established sufficient condition for this category to behave well is that the ring should be (commutative) normal. We will explain that these nice behaviors are governed by the...