D.A.N.C.E. seminar

The following is a list of previous talks.

Speaker: Andrew Hanlon
Date: 10 February 2025, 4pm CET
Title: Birational King’s conjecture
slides | video

Abstract King's conjecture proposed that there is a full strong exceptional collection of line bundles on any smooth projective toric variety. Although the conjecture turned out to be false, it continues to motivate research on the derived categories of toric varieties. I will explain that King's conjecture in fact holds for a natural category glued from the derived categories of birational toric varieties. The talk is based on joint work with Ballard, Berkesch, Brown, Cranton Heller, Erman, Favero, Ganatra, and Huang.

Speaker: Matt Booth
Date: 17 February 2025, 4pm CET
Title: Nonsmooth Calabi—Yau structures for algebras and coalgebras
slides | video

Abstract I'll talk about a generalised notion of Calabi-Yau structure for dg (co)algebras, before giving a brief review of algebra-coalgebra Koszul duality, and explaining why this `nonsmooth Calabi—Yau' condition is dual to a symmetric Frobenius condition. There is also an analogous one-sided version: Gorenstein (co)algebras are Koszul dual to Frobenius (co)algebras. This leads to a surprising example: the ring k[[x]] of formal power series, equipped with its natural topology, is a pseudocompact Frobenius algebra. As an application of the above theory, we obtain a new characterisation of Poincaré duality spaces, which for simply connected spaces recovers Félix-Halperin-Thomas's notion of Gorenstein space. This talk is based on forthcoming work joint with Joe Chuang and Andrey Lazarev.

Speaker: Hannah Dell
Date: 3 March 2025, 4pm CET
Title: Categorical Torelli for cyclic covers
slides | video

Abstract Since any Fano variety can be recovered from its derived category up to isomorphism, we ask whether less information determines the variety - this is called a categorical Torelli question. In this talk, we consider an n-fold cover X → Y ramified in a divisor Z. The cyclic group of order n acts on X. We study how a certain subcategory of Db(X) (the Kuznetsov component) behaves under this group action. We combine this with techniques from topological K-theory and Hodge theory to prove that this subcategory determines X for two new classes of Fano threefolds which arise as double covers of (weighted) projective spaces. This is joint work with Augustinas Jacovskis and Franco Rota (arXiv:2310.13651).

Speaker: Thilo Baumann
Date: 17 March 2025, 4pm CET
Title: Noncommutative plane curves
slides | video

Abstract A noncommutative plane curve is defined by a central homogeneous element in a 3-dimensional Artin—Schelter regular algebra. If the algebra is finite over its center, we show how a noncommutative plane curve can be understood as the restriction of an order on the projective plane along a (commutative) curve. This observation provides a general method for studying noncommutative plane curves of arbitrary degree. As an application, we explain how our approach connects to the existing results in degrees two and three. Furthermore, it extends the dictionary between orders and stacks in dimension one. This is joint work with Pieter Belmans and Okke van Garderen.

Speaker: Sridhar Venkatesh
Date: 31 March 2025, 4pm CEST
Title: Derived characterizations of singularities
slides | video

Abstract We characterize well known classes of singularities of complex projective varieties in terms of statements about generation in the derived category of coherent sheaves. This includes classical cohomological singularities such as rational singularities, Du Bois singularities, and rational pairs (both in the sense of Schwede-Takagi and Kollár-Kovács). This is based on joint work with Pat Lank and joint work with Pat Lank and Peter McDonald.

Speaker: Isambard Goodbody
Date: 28 April 2025, 4pm CEST
Title: Immaculate complexes and strong generation

Abstract Strong generation of the derived category of a (non-commutative) scheme can be used to extract properties of the scheme. Many results of Neeman, Orlov, Rouquier and others support this. For example, Neeman showed the perfect complexes over a Noetherian separated scheme admit a strong generator if and only if it is regular of finite dimension. We define the immaculate complexes over a Noetherian scheme as the bounded complexes of finite sums of indecomposable injectives. We show that a Noetherian scheme is regular and of finite dimension if and only if the immaculates are strongly generated by the indecomposable injectives.

Speaker: Dylan Spence
Date: 12 May 2025, 4pm CEST
Title: The derived McKay correspondence for rank 2 reflection groups and semiorthogonal decompositions of equivariant derived categories
slides | video

Abstract I will speak about some recent work of the author and collaborators which extend the two-dimensional derived McKay correspondence to certain reflection groups. Along the way we’ll talk about the history of the McKay correspondence, equivariant derived categories, and future directions.

Speaker: Marina Godinho
Date: 26 May 2025, 4pm CEST
Title: New derived symmetries for varieties with (relative) tilting bundles
video

Abstract A ring morphism p: A ⟶B satisfying certain mild assumptions induces a derived endomorphism of A and a derived endomorphism of B, which are closely related. In fact, the derived endomorphism of A is the twist around the restriction of scalars functor, and the derived endomorphism of B is the corresponding cotwist. I will discuss settings in which these endomorphisms are derived equivalences and use this technology to construct new derived autoequivalences of varieties with (relative) tilting bundles. We will construct some interesting examples of these new autoequivalence for quotient singularities.

Speaker: Parth Shimpi
Date: 9 June 2025, 4pm CEST
Title: Drifting towards rational curves
video

Abstract Rational curves and their neighbourhoods in surfaces and 3-folds provide a playground for rich interplay of algebra and geometry, the first instance of such interaction being Beilinson’s observation that the derived category of a projective line admits infinitely many algebraic t-structures. We can walk between these, one step at a time, using various techniques of tilting, mutation, and application of the Picard group action. Then upon iterating these operations, a `fixed—point theorem’ reveals itself: the geometric category of coherent sheaves is naturally a limit of algebraic hearts. I will describe the convex—geometric and combinatorial tools used to study the result, and how it is used to classify t-structures and spherical objects in the local derived category of a flopping curve in a 3-fold.

Speaker: Alekos Robotis
Date: 23 June 2025, 4pm CEST
Title: The space of augmented stability conditions
slides | video

Abstract I will motivate and give an overview of recent joint work with Daniel Halpern-Leistner which constructs an enlargement of the space of Bridgeland stability conditions of a triangulated category D. The new points parametrise new categorical structures called augmented stability conditions, which combine properties of semiorthogonal decompositions and stability conditions.

Speaker: Sebastian Opper
Date: 20 October 2025, 4pm CET
Title: Derived Picard groups and Lie theory

Abstract We discuss how to compute derived Picard groups (enhanced autoequivalence groups) of Fukaya categories of surfaces after Bocklandt and Haiden-Katzarkov-Kontsevich, and the related nodal stacky curves. I will also explain how solving this problem naturally leads to a very general analogy between Lie groups, Lie algebras, derived Picard groups and Hochschild cohomology.

Speaker: Calum Crossley
Date: 3 November 2025, 4pm CET
Title: Categorical resolutions from geometry
slides | video

Abstract This will be a survey on categorical resolutions, in the context of conjectures relating derived categories to birational geometry. Examples using matrix factorizations give a geometric perspective, shedding light on various aspects of these resolutions: crepancy and minimality, null categories, and relations to singular equivalences.

Speaker: Céline Fietz
Date: 17 November 2025, 4pm CET
Title: Categorical resolutions of cuspidal singularities

Abstract In this talk I will explain that there exists a particularly small (“crepant”) categorical resolution of the derived category of a projective variety with an isolated A_2/cuspidal singularity. More importantly, one can describe generators of its kernel in a very explicit way: In the case of an even dimensional variety with an isolated A_2 singularity, the kernel can be generated by two 2-spherical objects, which are related to spinor sheaves on a nodal quadric and induce autoequivalences on the categorical resolution.

Speaker: Prashanth Sridhar
Date: 8 December 2025, 4pm CET
Title: Differential Graded Noncommutative Geometry
slides

Abstract Pioneering work of Artin-Tate-Van den Bergh-Zhang extends important aspects of projective geometry to the noncommutative (nc) setting. In particular, the derived category of such a nc scheme shares many features with the derived category of a classical one. In this talk, I'll discuss extensions of some classical and modern results in the theory of nc projective geometry to nc spaces associated to dg-algebras. The focus will be on applications to projective varieties: for instance, this approach results in an analog of a landmark theorem of Orlov concerning the derived category of a complete intersection for any projective variety.

Speaker: Aporva Varshney
Date: 15 December 2025, 4pm CET
Title: Stringy Kähler moduli of flops using GIT
slides

Abstract According to predictions coming from physicists and mirror symmetry, the fundamental group of the "stringy Kahler moduli space" of a variety acts on the derived category by autoequivalences. However, it is unclear how to compute or even define this space in full generality. I will discuss two approaches given in the literature, one using Bridgeland stability manifolds and the other based in GIT and window subcategories. We will see that these two approaches agree for single curve threefold flops of lengths one and two. This is based on work in arxiv:2508.05285.

Speaker: Andrei Konovalov
Date: 16 March 2026, 4pm CET
Title: Algebraic K-theory of singular hypersurfaces
video

Abstract Considering short exact sequences of enhanced triangulated categories naturally leads to the notion of algebraic K-theory. Although recent years have seen substantial progress in the area, there are still few complete computations of (higher) K-theory of singular varieties: in positive characteristic there are essentially none in dimension >1 in the literature, and the situation in characteristic 0 is not much better. In this talk, I will show how recent developments in algebraic K-theory can be used to understand the K-theory of singular varieties, focusing on cubic surfaces and threefolds, where complete computations can be obtained in many cases.

Speaker: Amal Mattoo
Date: 30 March 2026, 4pm CEST
Title: Phantoms on Rational Surfaces

Abstract Phantom categories, or admissible subcategories on which all additive invariants vanish, were once considered pathological phenomena unlikely to occur on simple enough varieties. So when Krah constructed a phantom on a blowup of $\mathbf{P}^{2}$, it came as a surprise and disproved several conjectures! Since then, there has been work on extending this construction to other rational surfaces. In this talk, I will explain these constructions and how they are proved using Kuznetsov's machinery of heights. I will also describe some tools used to study these phantoms, like Hochschild cohomology and a spectral sequence for computing Hom's between objects.

Speaker: Dave Murphy
Date: 13 April 2026, 4pm CEST
Title: Piano Algebras and Dissections of Marked Surfaces

Abstract Paquette and Yıldırım constructed a triangulated (cluster) category that is modelled by a combinatorial completion of a discrete infinity-gon with finitely many so-called accumulation points, building upon cluster categories introduced by Igusa and Todorov as the stable Frobenius category of matrix factorisations over a cyclic poset. It turns out that in the case of a single accumulation point, the Paquette--Yıldırım category is triangle equivalent to the perfect derived category of k[x], seen as a dga with trivial differential and x placed in degree -1. This leads us to ask, for any given number of accumulation points, can we construct a dga A such that the perfect derived category of A is triangle equivalent to the Paquette-Yıldırım category? In this talk we shall show how to construct these algebras using methods introduced to study gentle algebras, and show how our methods are far more general and may lead to triangulated categories modelled by arbitrary surfaces with infinitely many marked points on each boundary component. This talk is based on joint work with Marina Godinho.

Speaker: Gopinath Sahoo
Date: 27 April 2026, 4pm CEST
Title: Tensor t-structures and perversity functions
video

Abstract For a Noetherian scheme X admitting a dualizing complex, Bezrukavnikov-Deligne and independently Gabber and Kashiwara showed that any monotone comonotone perversity function on X gives rise to a t-structure on the bounded derived category of X. In a recent preprint (arXiv:2412.18009), we introduced the notion of tensor t-structures via the action of perfect complexes on the bounded derived category, and proved that these coincide exactly with those coming from perversity functions. This work builds on our earlier results on t-structures for unbounded derived categories of Noetherian schemes. In this talk, I will explain how these results are related and briefly review their history.

Speaker: Giovanna Le Gros
Date: 4 May 2026, 4pm CEST
Title: Semi-Bousfield classes in the derived category of a commutative noetherian ring

Abstract For a rigidly-compactly generated triangulated category with a fixed t-structure, we introduce semi-Bousfield classes. Examples of semi-Bousfield classes include both Bousfield classes and coaisles of compactly generated tensor t-structures. We will mainly consider the case of the unbounded derived category of a commutative noetherian ring with the standard t-structure. In this case, we can describe the semi-Bousfield classes which come from perversities, which are integer valued functions on the prime spectrum of the ring. Moreover, this assignment is compatible with the classification of compactly generated t-structures by sp-filtrations due to Alonso-Jeremías-Saorín, and localising subcategories by subsets of the spectrum due to Neeman. This is based on joint work with Dolors Herbera and Michal Hrbek.

Speaker: Lukas Bertsch
Date: 11 May 2026, 4pm CEST
Title: Noncommutative resolutions of Kleinian singularities
slides | video

Abstract The McKay correspondence establishes a derived equivalence between the classical minimal resolution and a noncommutative crepant resolution of a Kleinian surface singularity. Based on joint work with Ruth Wye, I will explain how the McKay correspondence extends to a larger class of noncommutative crepant resolutions of the singularity, and how their Hilbert schemes of points are related through variation of GIT quotients (VGIT). Time permitting, I will also sketch some recent ideas from work in progress with Austin Hubbard on how to relate the resolutions themselves via VGIT by taking into account the variation of monoidal structures on their mutual derived category.

Speaker: Daigo Ito
Date: 25 May 2026, 4pm CEST
Title: Tensor-generation on toric varieties
slides | video

Abstract In the study of derived categories of coherent sheaves, ample line bundles play a fundamental role: their tensor powers generate the derived category. We call a line bundle with this property tensor-generating. For toric varieties, tensor-generation admits a purely combinatorial criterion, and there exist many tensor-generating line bundles that are neither ample nor anti-ample, including examples on complete non-projective toric varieties. In this talk, I will further explore the relationship between tensor-generation and nonstandard autoequivalences of the derived category. This is joint work in progress with Michael Zeng and Xiangru Zeng.

Speaker: Fei Peng
Date: 8 June 2026, 4pm CEST
Title: Generation and approximation in derived categories of algebraic stacks
slides

Abstract In this talk, I will discuss recent results on generation and compact approximation in derived categories of quasi-coherent sheaves on algebraic stacks. I will explain how Lipman–Neeman-type approximation methods can be extended to the stacky setting using descent techniques. Time permitting, I will mention applications and related questions concerning the Rouquier dimension. This is based on joint work with Jack Hall, Alicia Lamarche, and Pat Lank.