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

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

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Speaker: Hannah Dell
Date: 3 March 2025, 4pm CET
Title: Categorical Torelli for cyclic covers
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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).

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

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

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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.

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Speaker: Dylan Spence
Date: 12 May 2025, 4pm CEST
Title: TBA

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Speaker: Marina Godinho
Date: 26 May 2025, 4pm CEST
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Speaker: Uttaran Dutta
Date: 9 June 2025, 4pm CEST
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Speaker: Alekos Robotis
Date: 23 June 2025, 4pm CEST
Title: TBA

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