Cluster Structures in the North 2023
June 28, 2023
University of Central Lancashire, Preston
University of Central Lancashire, Preston
| Home | Registration | Schedule | Directions | Participants |
| 10:00-11.00 | Registration and Coffee & Tea. |
| 11:00-12:00 | Talk 1: A structural view of maximal green sequences. |
| Nicholas Williams (Lancaster University) | |
| Abstract: Maximal green sequences are maximal paths of finite length in an oriented cluster exchange graph. They were introduced by Keller in the context of Donaldson--Thomas invariants, although they had already appeared independently in the physics literature in the work of Cecotti, Cordova, and Vafa. In this talk, I will explain recent joint work with Mikhail Gorsky, where we study the structure of the set of all maximal green sequences of a finite-dimensional algebra. Indeed, we show that there is a natural equivalence relation on this set which can be interpreted in several different ways, underscoring its significance. There are then three natural partial orders on the set of equivalence classes, analogous to the partial orders on silting complexes and generalising the higher Stasheff--Tamari orders on triangulations of three-dimensional cyclic polytopes. Analogous to how the partial orders on silting complexes have the same Hasse diagram, we conjecture these orders to be equal. We prove this conjecture in the case of Nakayama algebras. |
| 12:00-13:00 | Talk 2: Simple-mindedness in negative Calabi-Yau cluster categories of the hereditary type. |
| Raquel Coelho Guardado Simoes (Lancaster University) | |
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Abstract: Simple-minded systems were introduced by Koenig-Liu as an abstraction of non-projective simple modules in stable module categories. In this talk, we will consider simple-minded systems in certain orbit categories of the bounded derived category of a hereditary algebra, which can be considered to be negative Calabi-Yau cluster categories. Simple-minded systems in this setting turn out to play the role of cluster-tilting objects in the positive Calabi-Yau setup. In this talk, I will explain the connection between simple-minded systems in negative Calabi-Yau cluster categories of hereditary algebras, simple-minded collections in their bounded derived category, and positive noncrossing partitions, generalising results by Buan-Reiten-Thomas and Iyama-Jin. This is based on joint work with David Pauksztello and David Ploog. |
| 13:00-14:30 | Lunch |
| 14:30-15:30 | Talk 3: Expansion formulae via good matchings of loop graphs. |
| Jon Wilson (University of Central Lancashire) | |
| Abstract: In 2011 Musiker, Schiffler, and Williams obtained expansion formulae for surface cluster algebras via perfect matchings of snake graphs. For singly and doubly notched arcs their formulae required the notion of γ-symmetric perfect matchings and γ-compatible pairs of γ-symmetric perfect matchings, respectively. Due to technicalities, in the case of doubly notched arcs, their proof had to exclude twice-punctured closed surfaces. In this talk, I will describe how their approach may be unified by considering good matchings of loop graphs. Furthermore, building on joint work with Christof Geiss and Daniel Labardini-Fragoso, I will outline a new proof of this result, which verifies the formulae in full generality. |
| 15:30-16:30 | Talk 4: Growth of frieze patterns. |
| Emine Yıldırım (University of Leeds) | |
| Abstract: This talk is about an ongoing Winart project with Karin Baur, Emily Gunawan, Léa Bittmann and Gordana Todorov. We investigate a certain property, called "growth coefficient", for friezes coming from affine type D. |
| 16:30- | Wine and Cheese |
| 19:00- | Conference Dinner |