D.A.N.C.E. seminar

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: TBA

Abstract TBA