D.A.N.C.E. seminar

The following is our current schedule, but see here for previous talks.

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

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.