Invited Speakers

Title: The Z-critical equation on surfaces

Abstract: Introduced by Dervan-McCarthy-Sektnan, the Z-critical equations are conjectural complex differential-geometric analogues of Bridgeland stability conditions and seek to strengthen the classical Hitchin-Kobayashi correspondence: a holomorphic vector bundle over a smooth projective variety is slope polystable if and only if it admits a Hermite-Einstein metric. In this talk, I will briefly discuss the motivation and setting of the Z-critical equations, before specialising to the case of line bundles on projective surfaces. In this special case, I will attempt to explain how certain facts from the classical geometry of surfaces give a PDE analogue of a wall-and-chamber decomposition. I will not assume prior knowledge of Kähler geometry. This is joint work with Zakarias Sjöström-Dyrefelt (Aarhus).

Title: The Donovan–Wemyss conjecture on compound Du Val singularities. --An application of the triangulated Auslander–Iyama correspondence--

Abstract: Given an isolated compound Du Val (cDV) singularity with a minimal model, Donovan and Wemyss defined a finite-dimensional algebra called contraction algebra. They conjectured in 2014 that the derived equivalence classes of contraction algebras classify complete local isolated cDV singularities with smooth minimal models. This general form of the conjecture was stated by August in 2018. Strong evidence for the conjecture was provided by these and other authors. In 2018, Hua and Keller proved a derived version of the conjecture. Contraction algebras are isomorphic to endomorphism algebras of 2-cluster tilting objects in the singularity category. These objects classify minimal models (Wemyss 2018). Hua and Keller prove that the derived endomorphism algebras of these objects in the canonical enhancement of the singularity category classify cDV singularities as above. In 2022, in a joint work with Jasso we showed that algebraic triangulated categories with periodic cluster tilting objects have an essentially unique enhancement. This, together with the aforementioned result, implies the original conjecture. The aim of this talk is to explain the main ideas behind this final step.

Title: Birational geometry of moduli of G-constellations

Abstract: For a finite subgroup G of SL_n(C), moduli spaces of G-constellations play an important role in the McKay correspondence since they are natural candidates of crepant resolutions of the quotient singularity C^n/G. In this talk I will explain how the birational geometry of these moduli spaces can be studied and show that every projective crepant resolution of C^3/G is indeed realized as such a moduli space.

Title: Very Functorial resolution of singularities

Abstract: Abstract: The basis of the talk is my proof of the resolution of singularities in characteristic zero, which unlike Hironaka's approach, and all derivatives thereof, constructs the centres in which the algorithm modifies the variety by building up, i.e. looking at increasingly fine infinitesimal data irrespective of the dimension, rather than building down, i.e. cutting with a maximal contact hyperplane and reducing dimension. The talk will aim to complement my GAFA paper,

https://link.springer.com/content/pdf/10.1007/s00039-020-00523-7.pdf

by explaining several simplifications to the already very simple proof (most of the complication in the paper results from the non-geometric case of excellent rings) and how far these simplifications go to removing the hypothesis of characteristic 0.

Title: Boundary algebras arising from uniform Postnikov diagrams on surfaces

Abstract: A Postnikov diagram is an embedding of oriented curves, called strands, in a disk. These diagrams are known to describe the cluster algebra structure of open positroid varieties, with diagrams of uniform type corresponding to a cluster of minors in the Grassmannian Gr(k,n). Each Postnikov diagram can be associated with a dimer algebra, which is the Jacobian algebra of a quiver with potential. Baur-King-Marsh showed that the opposite of the boundary algebra corresponding to such a dimer algebra is isomorphic to a quotient of the preprojective algebra used by Jensen-King-Su to categorify the cluster structure of Gr(k,n). They also determined the boundary algebra for degree two weak Postnikov diagrams arising from general surfaces. This talk will discuss a combinatorial approach to calculating the boundary algebra associated to a uniform Postnikov diagram, and how this can be translated to Postnikov diagrams on other surfaces.

Title: Stability conditions for 3-fold flops

Abstract: For a 3-fold flopping contraction from X to the spectrum Spec(R) of a complete local Gorenstein ring (R,m) with terminal singularity at m, we describe a distinguished connected component of the (normalized) space of Bridgeland stability conditions on certain triangulated categories associated to the flopping contraction. More precisely, we show that the connected component is a regular covering space of the complement of the complexification of a hyperplane arrangement associated to the 3-fold flop. This is joint work with Michael Wemyss.

Title: Global Koszul duality

Abstract: A famous theorem of Lurie and Pridham, tracing its roots back to Deligne, states that in characteristic zero, formal moduli problems are equivalent to dg Lie algebras (dglas). In fact, cocommutative conilpotent dg coalgebras (dgcs) are more fundamental objects than dglas in derived deformation theory: as an example, prorepresentability theorems are often more naturally expressed in terms of dgcs. In the world of noncommutative derived geometry, the noncommutative version of the Lurie-Pridham correspondence has a similar interpretation in terms of the Koszul duality (aka bar-cobar duality) between dgas and conilpotent dgcs. I'll talk about the above ideas, then report on joint work in progress with Andrey Lazarev where we try to remove the word `conilpotent' from the above statement to obtain a more `global' form of Koszul duality, with an eye towards prorepresentability statements for `global' deformation problems.

Title: Enumerative invariants of 3-fold flops: hyperplane arrangements and wall-crossing 

Abstract:  3-fold flopping contractions form a fundamental building block of the higher-dimensional Minimal Model Program. They exhibit extremely rich geometry, which has been investigated by many people over the past half-century. I will present an elegant and visually-pleasing relationship between enumerative invariants of flopping contractions and certain hyperplane arrangements constructed combinatorially from root system data. I will discuss both Gopakumar-Vafa (GV) and Gromov-Witten (GW) invariants, explaining how these are related to one another and how they are encoded in finite and infinite arrangements, respectively. Finally, I will discuss wall-crossing: our combinatorial approach allows us to explicitly construct flops from root system data, leading to a new "direct" proof of the Crepant Transformation Conjecture, with a very explicit formulation. This is joint work with Michael Wemyss. 

Title: The McKay correspondence via VGIT 

Abstract: A Kleinian surface singularity can be `resolved' by a smooth orbifold, and the McKay correspondence famously relates this resolution with the more geometric minimal resolution. In type A it is easy to write down a VGIT problem that interpolates between the orbifold and the minimal resolution, but this has never been done for types D and E. I'll describe some work-in-progress with Tarig Abdelgadir where we fill this gap using a construction inspired by Tannaka duality. 

The speaker will give an introduction to higher homological algebra and the higher analogue of the pair modA , D^b(modA). If time permits, she will explain some results from her PhD thesis.

Title:  Simple-mindedness

Abstract: Module categories have two important types of generators: projective modules and simple modules. Morita theory describes equivalences of module categories in terms of images of projective modules. Tilting theory is the generalisation of Morita theory to derived categories describing equivalences of derived categories in terms of tilting objects. Tilting, silting and cluster-tilting objects, can be thought of as ‘projective-minded objects’.

‘Simple-minded objects’ are generalisations of simple modules. They satisfy Schur’s lemma and a version of the Jordan-Holder theorem, depending on context. Although the theory of simple-minded objects shows many parallels with that of projective-minded objects, it remains relatively undeveloped and is technically more challenging. It remains important to develop this theory because many natural classes of examples, for instance, stable module categories, have no projective-minded objects but do have simple-minded objects. In this talk, I will explain aspects of the theory of simple-minded objects, including mutation and reduction. This talk will be based on joint work with Raquel Coelho Simões and David Ploog.