Talks and Schedule

Here is the preliminary schedule. Titles, abstracts, and posters below.

POSTERS:

Severin Barmeier (Cologne), Deformations of Fukaya categories of surfaces

Celine Fietz (Leiden), Categorical resolutions of A_2 singularities

Daigo Ito (UC Berkeley), Positivity of line bundles in derived categories  

Sebastian Opper (Charles University), Autoequivalences of triangulated categories via Hochschild cohomology 

Ian Selvaggi (SISSA), Good moduli spaces for twisted categories

Parth Shimpi (Glasgow), What's on my rational curve?

Apoorva Varshney (UCL), Length 2 flops and Stringy Kahler Moduli 

MONDAY

Balázs Szendrői, The projective coinvariant algebra 

The coinvariant algebra, the quotient of the coordinate ring of (A^1)^n=A^n by the ideal generated by positive degree invariant polynomials, plays a basic role in algebraic combinatorics and the representation theory of the symmetric group S_n, equipping its regular representation with a graded algebra structure. Using the coordinate ring of (P^1)^n in its Segre embedding, I will introduce a degeneration of the coinvariant algebra, the projective coinvariant algebra, which gives a bigraded structure on the regular representation of S_n with interesting Frobenius character that generalises a classical result of Lusztig and Stanley. I will also show how this algebra contains bigraded versions of partial coinvariant algebras, coming from coordinate rings of all possible Segre embeddings. Relations to Haiman's diagonal coinvariant algebra, and a certain equivariant Hilbert scheme, will also be discussed. 

Agnieszka Bodzenta, Saturation of surfaces

I will address the question of adding closed points to a normal separated surface X, i.e. considering open embeddings of X with complements of dimension zero. I will argue that to a surface X one can add only finitely many points to get its saturated model, i.e. a surface to which one cannot add any points. I will show that the saturated model is unique and functorial in X.  I will also describe how to construct it explicitly from the additive category of reflexive sheaves on X.

I will say that a surface is saturated if it is isomorphic to its saturated model. I will discuss saturated surfaces and, more generally, saturated algebraic spaces of dimension two. The talk is based on joint work with A. Bondal, T. Pełka and D. Weissmann.

Ben Davison, 2-Calabi-Yau categories and quantum groups

2-Calabi-Yau (2CY) categories appear in many branches of maths: as categories of coherent sheaves on K3 or abelian surfaces, as categories of Higgs bundles and local systems on Riemann surfaces, and as categories of modules over various flavours of preprojective algebras.  In the final example, there is a well-established link between the geometry of the moduli space of objects, and quantum deformations of the current Lie algebras of simple Lie algebras: quantum groups known as Yangians.  I will explain how to build Hopf algebras and quantum deformations directly from the geometry of moduli spaces of objects in the 3CY completions of 2CY categories, focusing on the construction of coproducts.  If time permits I will explain some applications to the enumerative invariants of K3 surfaces.  This talk will be based on joint work with Lucien Hennecart, Tasuki Kinjo, Olivier Schiffmann and Eric Vasserot.

Ed Segal, The symmetric square of the Kuznetsov component 

The derived category of a cubic 4-fold contains a semi-orthogonal component which famously behaves as a `non-commutative K3 surface'. The symmetric square of this component is thus a `non-commutative hyperkaehler 4-fold'. Galkin conjectured that it should be equivalent to the derived category of an actual hyperkaehler 4-fold: the Fano of lines in the cubic. I will explain my recent proof of this conjecture with Kimoi Kemboi.

TUESDAY

Arend Bayer, Non-commutative abelian surfaces, Kummer-type Hyperkaehler varieties, and Weil-type abelian fourfolds

The derived category D(A) of an abelian surface A has a six-dimensional space of deformations, but A has only a 3-dimensional space of deformations as an algebraic variety. 

Kummer-type HK varieties have 4-dimensional moduli spaces, but only three-dimensional ones arise from moduli spaces of sheaves on abelian surfaces.

I will explain a construction that reduces the first gap and eliminates the second gap. First we construct a family of categories over a four-dimensional space that we call "non-commutative abelian surfaces".

By constructing stability conditions, this gives rise to an interpretation of every general Kummer-type Hyperkaehler variety as a moduli space of Bridgeland-stable objects. Finally, I will explain how naturally yields Weil-type abelian fourfolds first studied by O'Grady and Markman, and to a different proof of Markman's result establishing the Hodge conjecture for them.

This is based on joint work with Laura Pertusi, Alex Perry and Xiaolei Zhao.

Yuki Hirano, Length of triangulated categories  

Admissible subcategories of triangulated categories do not satisfy the Jordan-Hölder property. In this talk, we consider similar problems for thick subcategories of triangulated categories. In particular, we discuss length of composition series in derived categories of certain toric surfaces and certain finite dimensional algebras. This is joint work with Martin Kalck and Genki Ouchi.

Soheyla Feyzbakhsh, Hurwitz-Brill-Noether Theory via K3 Surfaces

I will discuss the Brill-Noether theory of a general elliptic K3 surface using wall-crossing with respect to Bridgeland stability conditions. As an application, I will provide an example of a general k-gonal curve from the perspective of Hurwitz-Brill-Noether theory. This is joint work with Gavril Farkas and Andrés Rojas.

WEDNESDAY

Tasuki Kinjo, Intrinsic Donaldson–Thomas theory via component lattices

Donaldson–Thomas theory is a counting theory for coherent sheaves on Calabi–Yau threefolds, or more generally, for objects in an abelian category. In this talk, we introduce a generalization of Donaldson–Thomas theory that extends its scope to non-linear objects, such as G-Higgs bundles on a curve or G-local systems on compact oriented 3-manifolds. A key ingredient of the theory is a combinatorial object called the “component lattice” for algebraic stacks, which generalizes the cocharacter lattice of an algebraic group together with the Weyl group action. If time permits, I will explain several applications of our theory, such as a formulation of the Hodge-theoretic Langlands duality conjecture for the Hitchin system and the computation of the intersection cohomology of quotient varieties. This talk includes several joint works with Chenjing Bu, Ben Davison, Daniel Halpern-Leistner, Andrés Ibáñez Núñez, and Tudor Pădurariu.

Alexander Kuznetsov, Compact degeneration of curves and derived categories 

Given a curve C_0 with two irreducible components C'_0 and C''_0 meeting transversely at a single point and its smoothing {C_t}, I will construct a family of smooth and proper triangulated categories whose general fiber is an augmentation of C_t (i.e., a category with a semiorthogonal decomposition generated by D(C_t) and an extra exceptional object) and whose central fiber an admissible subcategory in a gluing of D(C'_0) and D(C''_0). This family categorifies the family of Jacobians of C_t. 

Emma Lepri, Unitally positive A-infinity algebras and the contraction algebra

By the work of Donovan and Wemyss, the functor of noncommutative deformations of a flopping irreducible rational curve C in a threefold X is representable by an algebra called the contraction algebra. This talk is based on a joint work with J. Karmazyn and M. Wemyss, where we construct a DG-algebra from the data of periodic projective resolution of the simple module on the contraction algebra, and prove that it reconstructs the A-infinity algebra $Ext^*_X(\mathcal{O}_C(-1), \mathcal{O}_C(-1))$, giving an alternative proof of the Donovan-Wemyss conjecture. 

Donatella Iacono, Joint deformations of coherent sheaves and sections

In this talk, we devote our attention to infinitesimal deformations of triples (X,F,s) where F is a coherent sheaf on a smooth variety X and s a section. In particular, we describe a differential graded Lie algebra controlling these deformations over any algebraically closed field of characteristic zero. As an application, we analyse deformations of pairs (variety, divisor). This is based on a joint work in progress with Marco Manetti. 

THURSDAY

Alastair Craw, The Cautis-Logvinenko conjecture 

For a finite subgroup G of SL(3,C), the G-Hilbert scheme is a crepant resolution of the quotient singularity C^3/G, and the universal family determines a derived equivalence between the derived category of G-equivariant coherent sheaves on C^3 on one hand, and the derived category of coherent sheaves on G-Hilb on the other. In their 2009 paper, Cautis and Logvinenko conjectured that this derived equivalence sends the vertex simple G-sheaves, one for each nontrivial irreducible representation of G, to pure sheaves on G-Hilb. I will report on joint work with Ryo Yamagishi towards a proof of a generalisation of this conjecture.  

Will Donovan, Derived symmetries for crepant contractions to hypersurfaces

Given a crepant contraction to a singularity, a symmetry of the derived category of coherent sheaves on the source may often be constructed using the formalism of spherical functors. I will briefly recall this, and discuss work on general constructions of such symmetries for hypersurface singularities (arXiv:2409.19555). This builds on previous results with Segal, and is inspired by work of Bodzenta-Bondal.

Milena Hering, Pushforwards of Frobenius morphisms of seminormal toric varieties and applications

We describe the push forward of a seminormal affine monoid ring under (multiples of) the Frobenius morphism as a direct sum of indecomposable modules. As applications we can compute the F-splitting ratio of such rings, which is a measure of the singularities, and a non-commutative resolution of the Whitney umbrella. This is joint work with Eleonore Faber and Kevin Tucker. 

Yujiro Kawamata, NC Grassmann variety as a moduli space 

I review deformations of sheaves over non-commutative (NC) bases.

NC base is more natural than the commutative base, because the dg algebra governing the deformations is NC.

The deformations are sometimes related to the Hilbert scheme.

I will consider an NC version of the Grassmann variety as an example which becomes an NC scheme defined by pasting NC algebras.

FRIDAY

Francesco Meazzini, Deformations of monomial ideals

We deal with infinitesimal deformations of affine schemes defined by a monomial ideal in a polynomial ring. We discuss few applications: the cotangent cohomology for Stanley-Reisner schemes, some conjectures related to fat points, and some geometric properties of certain Hilbert schemes of points. This is a joint work in progress with Nathan Ilten and Andrea Petracci.

Marco Manetti, Invariant forms on filtered curved Lie algebras and obstructions

Filtered curved Lie algebras are quite common in mathematics; the most basic example is the algebra of smooth differential forms with values in endomorphism of a holomorphic vector bundle equipped with a Chern connection.

A possible way to see  a filtered curved Lie algebra $L$ is as an enrichment of a differential graded Lie algebra $L'$. In this case, to every closed $L$-invariant form $g$ on $L$ we can associate a sequence of semiregularity maps from the cohomology of $L'$ to a graded vector space depending on $g$ (the formal algebraic version of the Buchweitz--Flenner semiregularity maps).

We will prove that these maps are induced by morphisms in the homotopy category of DG-Lie algebras and that they annihilate all the obstructions of the deformation problem controlled by $L'$. This result generalizes a previous theorem by Bandiera--Lepri--Manetti (2021) and  

applies to a wider class of deformation problems.

Amnon Neeman, Vanishing negative K-theory and bounded t-structures 

We will begin with a quick reminder of algebraic K-theory, and a few classical, vanishing results for negative K-theory. The talk will then focus on a striking 2019 article by Antieau, Gepner and Heller - it turns out that there are K-theoretic obstructions to the existence of bounded t-structures.

The result suggests many questions. A few have already been answered, but many remain open. We will concentrate on the many possible directions for future research.