| CLAN Research Group | Upcoming Meetings | Directions | Past Meetings |
Date: Friday 14th June, 2024
Location: University of Central Lancashire, Preston
Invited Speakers:
Oliver Daisey (Durham University)
Eleonore Faber (University of Graz)
Jan Grabowski (Lancaster University)
| 11:00–11.30 | Registration and Coffee & Tea. |
| Location: Leighton Building, LE003. |
| 11:30–12:30 | Segre products of cluster algebras (slides). |
| Jan Grabowski (Lancaster University) | |
| Location: Leighton Building, LE111. Abstract: Given two cluster algebras, it is natural to ask when and how these can be combined to obtain a new cluster algebra. When the cluster algebras admit gradings, one possibility is the Segre product arising from projective geometry. We will discuss graded cluster algebras and present recent work (https://arxiv.org/abs/2404.14872) with Lauren Hindmarch (Lancaster) on the existence of Segre products of cluster algebras, answering a question of Pressland arising in the context of cluster algebras associated to positroid varieties. |
| 12:30–14:00 | Lunch |
| 14:00–15:00 | Finite type LP cluster structures |
| Oliver Daisey (Durham University) | |
| Location: Leighton Building, LE111. Abstract: Laurent phenomenon (LP) algebras are one possible extension of cluster algebras introduced by Thomas Lam and Pavlo Pylyavskyy. In this framework a seed may have arbitrary irreducible exchange polynomials, providing a substantial course of new examples of cluster structures. However, owing to their complexity, understanding when these cluster structures are of finite type is a very difficult problem in general. In this talk I will introduce these algebras, their basic properties, and some prominent examples of finite type LP cluster structures. In addition, we will discuss some progress on a conjecture of Lam and Pylyavskyy, and demonstrate a computer-assisted classification result. |
| 15:00–15:30 | Coffee Break |
| Location: Leighton Building, LE003. |
| 15:30–16:30 | Friezes and combinatorics of resolutions of curve singularities |
| Eleonore Faber (University of Graz) | |
| Location: Leighton Building, LE111. Abstract: Conway–Coxeter friezes are arrays of positive integers satisfying a determinantal condition. Recently, these combinatorial objects have been of considerable interest in representation theory, since they encode cluster combinatorics of type A. In this talk I will discuss a new connection between Conway–Coxeter friezes and the combinatorics of a resolution of a complex plane curve singularity: via the beautiful relation between friezes and triangulations of polygons one can relate each frieze to the so-called lotus of a curve singularity, which was introduced by Popescu-Pampu. This allows us to interprete some of the entries in the frieze in terms of invariants of the curve singularity, and on the other hand, we can see cluster mutations in terms of the desingularization of the curve. This is joint work with Bernd Schober. |
| 16:30–17:30 | Flash Talks |
| Location: Leighton Building, LE111. Speakers: TBC. |