McKay correspondence, Tilting theory and related topics

Title and Abstract

29. McKay correspondence for Hilb^n(C^3), categorical DT theory and geometric Langlands

Yukinobu Toda (Tokyo)

18/12/2023, 10:00

The McKay correspondence for Hilb^n(C^2) is its derived equivalence with C^{2n}/S_n, proven by Bridgeland-King-Reid and Haiman. In this talk, I will explan how to give its version for Hilb^n(C^3) using categorical DT theory and its categorical wall-crossing formula. It involves semiorthogonal decomposition with factors categorical Hall products of quasi-BPS categories, which we conjecture to...

16. The invariant Hilbert scheme of the Cox realization

Ayako Kubota (Waseda)

18/12/2023, 11:00

The invariant Hilbert scheme is a moduli space of schemes which are stable under an action of a reductive algebraic group. By a suitable choice of the parameter, it becomes a candidate for a resolution of singularities of an affine quotient variety via the so-called Hilbert-Chow morphism. In this talk, we will focus on the Cox realization as a way to represent an affine singularity as a...

12. Mutations of noncommutative crepant resolutions in geometric invariant theory

Yuki Hirano (Tokyo U. Agri & Tech)

18/12/2023, 11:40

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,...

34. F-blowups of quotient singularities and toric singularities

Takehiko Yasuda (Osaka)

18/12/2023, 14:00

The F-blowup gives a caonical way to construct a birational transform of a singular variety in positive characteristic. There is also a non-commutative counterpart of this construction. There are several natural questions concering the F-blowup. Does this blowup give resolution of singularities? If not, does it improve singularities? Is it close to be a resolution of singularities in a certain...

6. Hilbert schemes and the Le Bruyn-Procesi theorem

Alastair Craw (Bath)

18/12/2023, 15:30

For a positive integer n and a finite subgroup \Gamma in SL(2,C), I’ll describe work in preparation with Ryo Yamagishi which shows that the Hilbert scheme of n-points on C^2/\Gamma is reduced. In fact, it’s isomorphic to a Nakajima quiver variety, so it has symplectic singularities and it admits a unique crepant resolution. This strengthens previous joint work of mine with Gammelgaard, Gyenge...

23. Cotilting complexes over commutative noetherian rings

Tsutomu Nakamura (Mie)

18/12/2023, 16:30

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).

8. An example of crepant resolution in characteristic 2 and its duality

Linghu Fan (Tokyo)

18/12/2023, 17:10

In positive characteristic, few examples of crepant resolutions of modular quotient singularities are known. In this talk, I will introduce a crepant resolution of the quotient singularity given by the permutation action of the alternating group of degree 4 in characteristic 2. In addition, I will present some special properties of this resolution by considering its algebraic and geometric structure.

21. The E10 diagram, symmetric polytopes and Enriques surfaces

Shigeru Mukai (Kyoto)

19/12/2023, 10:00

The affine E8 diagram parametrizes the irreducible representations of the binary icosahedral groups by McKay. The next diagram E10, or T237, parametrizes a set of symmetric polyhedra consisting of octa, dodeca, icosa and several zonohedra. This was observed in studying polarization types of Enriques surfaces. In this talk I will present type III degenerations of Enriques surfaces...

28. The Euler characteristic of the Fujiki-Oka resolution via continued fractions

Yusuke Sato (Kogakuin)

19/12/2023, 11:00

Let G be a finite subgroup of SL(n, C). If a quotient variety C n/G has a crepant resolution, then its Euler characteristic is equal to the number of conjugacy classes of G, which is a weak version of the McKay correspondence. In this talk, we generalize this correspondence to a finite cyclic group of GL(n, C). We construct this correspondence using certain toric resolutions obtained through...

30. Invariant theory of skew polynomial algebras

Kenta Ueyama (Shinshu)

19/12/2023, 11:40

Traces are a classical tool in (commutative) invariant theory. When studying invariant subalgebras of noncommutative algebras, traces also play an essential role. In this talk, for skew polynomial algebras, we provide a generalization of the classical formula that expresses the trace series of an automorphism as the reciprocal of the reverse characteristic polynomial of it. We then use this...

33. Birational geometry of quiver moduli through Cox rings

Ryo Yamagishi (Bath)

19/12/2023, 14:00

Moduli spaces of quiver representations appear in various contexts of mathematics. These spaces depend on the choice of stability parameters and, in many cases, variation of the parameters induces birational transformations such as flops. In this talk I will introduce a new method to investigate such transformations using the Cox rings. We will then see that this method works well especially...

15. On derived McKay correspondence between non-commutative deformations

Yujiro Kawamata (Tokyo)

19/12/2023, 15:30

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.

24. Explicit examples of trihedral G-Hilbert schemes

Alvaro Nolla de Celis (Madrid)

19/12/2023, 16:30

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.

25. L-genera and applications

Andrei Okounkov (Columbia)

20/12/2023, 10:00

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.

10. Spherical objects in dimension two and three

Wahei Hara (Tokyo)

20/12/2023, 11:00

During this talk we discuss the classification problem of spherical “like” objects in various geometric settings including the minimal resolution of an ADE surface singularity and a 3-fold flopping contraction. The classification of spherical objects is related to questions about the autoequivalence groups or Bridgeland stability conditions, but in 3-fold settings this is not always a correct...

20. Distribution of bricks and one-parameter families of stable modules

Kaveh Mousavand (OIST)

20/12/2023, 11:40

Let A be a finite dimensional associative algebra A over an algebraically closed field k. A (left) A-module M is called a brick if the endomorphism algebra of M over A is isomorphic to k. Bricks (also known as Schur representations) play decisive roles in the algebraic and geometric aspects of representation theory of algebras, including in the stability conditions, wall-and-chamber...

31. 3d McKay correspondence and 5d SCFTs

Yi-Nan Wang (Peking)

20/12/2023, 14:00

In theoretical physics, a large class of superconformal field theories (SCFTs) can be constructed by putting superstring/M-theory on canonical singularities. In this talk, I'm going to mainly discuss the cases of M-theory on C3 orbifold singularities, which lead to a large class of 5d SCFTs. Many physical properties of the SCFT, such as rank, flavor rank and 1-form symmetry can be read off...

7. Exceptional surfaces in 3-folds and derived symmetries

Will Donovan (Tsinghua)

20/12/2023, 15:30

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.

17. McKay correspondence and perverse schobers

Thimothy Logvinenko (Cardiff)

20/12/2023, 16:30

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.

35. Combinatorial actions on quivers and geometric interpretations

Amihay Hanany

20/12/2023, 17:10

The Coulomb branch of 3d N=4 gauge theory is a new construction of symplectic singularities.
This talk will cover operations on quivers that start with a given symplectic singularity and ends with a new symplectic singularity, while the relation between the two is understood geometrically.
These operations include
— Quiver subtraction where the resulting singularity is a degeneration of a...

32. (Derived) Deformations of Crepant Curves

Michael Wemyss (Glasgow)

21/12/2023, 10:00

Motivated by various contraction conjectures, I will describe the full A_infty structure associated to a general (-3,1)-curve C inside a smooth CY 3-fold. As a corollary, the noncommutative deformation theory of C can be described as a superpotential algebra derived from what we call free necklace polynomials, establishing a suitably interpreted string theory prediction due to Ferrari,...

13. Hyperpolygon spaces and their crepant resolutions

Austin Hubbard (Bath)

21/12/2023, 11:00

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).

11. Towards derived Reid’s recipe for dimer models

Liana Heuberger (Bath)

21/12/2023, 11:40

Reid's recipe is an equivalent of the McKay correspondence in dimension three. It marks interior line segments and lattice points in the fan of the G-Hilbert scheme with characters of irreducible representations of G. In joint work with Craw and Tapia Amador, we generalise this by marking the toric fan of a crepant resolution of any affine Gorenstein singularity, in a way that is compatible...

14. The Donovan-Wemyss Conjecture via the Derived Auslander-Iyama Correspondence

Gustavo Jesso (Lund)

21/12/2023, 14:00

The Donovan-Wemyss Conjecture predicts that the isomorphism type of an isolated compound Du Val singularity R that admits a crepant resolution is completely determined by the derived-equivalence class of any of its contraction algebras. Crucial results of August, Hua-Keller and Wemyss reduced the DW conjecture to a problem closely related the question of uniqueness of enhancements of the...

4. Nc resolutions of finite dimensional commutative algebras via smooth surface blow-ups

Alexey Bondal (Steklov/Tokyo)

21/12/2023, 15:30

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.

18. Non-commutative crepant resolutions of a special family of stable set rings

Koji Matsushita (Osaka)

21/12/2023, 16:30

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.

26. $(1,2)$-symmetric subgroups of SL$(4,\mathbb{C})$

Miles Reid (Warwick)

21/12/2023, 17:10

The topic is finite diagonal subgroups $A\subset$SL$(4,\mathbb{C})$ and their \hbox{$A$-Hilbert} schemes. As a dimension reducing assumption, I impose the additional $(1,2)$-symmetric condition. The case to bear in mind is $\frac1r(1,1,a,b)$ with $r = a+b+2$. The ``junior end and all-even'' conditions for the quotient $X=\mathbb{A}^4/A$ to have a crepant resolution are known from Sarah Davis's...

5. On recent developments in quasi-hereditary structures

Aaron Chan (Nagoya)

22/12/2023, 10:00

The notion of quasi-hereditary algebras were introduced by Cline-Parshall-Scott, and there is an abundance of examples arising in algebraic Lie theory and non-commutative resolution of singularities. This notion is defined with respect to a poset structure on the set of simple modules. In this talk, we will survey some recent developments in enumerating these structures, and in particular,...

27. Classifying torsionfree classes of the category of coherent sheaves and their Serre subcategories

Shunya Saito (Nagoya)

22/12/2023, 11:00

The classification of subcategories is one of the long-studied topics in the representation theory of algebras. The most classical result is Gabriel's classification of Serre subcategories (i.e., subcategories closed under taking subobjects, quotients, and extensions). He classified the Serre subcategories of the category of coherent sheaves on a noetherian scheme by using...

3. TF equivalence, silting theory and canonical decompositions

Sota Asai (Tokyo)

22/12/2023, 11:40

This talk is based on joint work with Osamu Iyama. The representation theory of a finite dimensional algebra $A$ deals with the category $\mathsf{mod} A$ of finitely generated $A$-modules. One of the main topics is torsion pairs in $\mathsf{mod} A$. Functorially finite torsion pairs have been well-studied, but they are too few among all torsion pairs. Thus, we are now studying a wider class...

19. Tilting theory and g-fans

Yuya Mizuno (Osaka Metropolitan)

22/12/2023, 14:00

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...

22. Projective crepant resolutions of toric cDV singularities and dimer combinatorics

Yusuke Nakajima (Kyoto Sangyo)

22/12/2023, 15:30

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...

9. Reflexive modules and Auslander-Gorenstein rings

Norihiro Hanihara (Tokyo)

22/12/2023, 16:30

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...