Matthew Pressland

Matthew Pressland
University of Glasgow | UofG · School of Mathematics and Statistics

PhD

About

16
Publications
584
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53
Citations
Introduction
I am an EPSRC postdoctoral fellow at the University of Glasgow. My research focuses on representation theory, cluster algebras and geometry. I have previously held positions at Universität Stuttgart and the Max Planck Institute for Mathematics in Bonn, and obtained my PhD from the University of Bath.
Additional affiliations
April 2022 - present
University of Glasgow
Position
  • EPSRC Postdoctoral Fellow
January 2022 - April 2022
University of Glasgow
Position
  • Temporary Lecturer
March 2020 - January 2022
University of Leeds
Position
  • EPSRC Postdoctoral Fellow
Education
October 2011 - July 2015
University of Bath
Field of study
  • Mathematical Sciences
October 2007 - July 2011
The University of Warwick
Field of study
  • Mathematics

Publications

Publications (16)
Article
Full-text available
We describe what it means for an algebra to be internally d-Calabi–Yau with respect to an idempotent. This definition abstracts properties of endomorphism algebras of (d−1)-cluster-tilting objects in certain stably (d−1)-Calabi–Yau Frobenius categories, as observed by Keller–Reiten. We show that an internally d-Calabi–Yau algebra satisfying mild ad...
Article
Full-text available
Recently the first author studied multi-gradings for generalised cluster categories, these being 2-Calabi-Yau triangulated categories with a choice of cluster-tilting object. The grading on the category corresponds to a grading on the cluster algebra without coefficients categorified by the cluster category and hence knowledge of one of these struc...
Article
We show that the dimer algebra of a connected Postnikov diagram in the disc is bimodule internally $3$ -Calabi–Yau in the sense of the author’s earlier work [43]. As a consequence, we obtain an additive categorification of the cluster algebra associated to the diagram, which (after inverting frozen variables) is isomorphic to the homogeneous coordi...
Preprint
Full-text available
Cluster algebra structures for Grassmannians and their (open) positroid strata are controlled by a Postnikov diagram D or, equivalently, a dimer model on the disc, as encoded by either a bipartite graph or the dual quiver (with faces). The associated dimer algebra A, determined directly by the quiver with a certain potential, can also be realised a...
Article
Full-text available
We survey results on mutations of Jacobian algebras, while simultaneously extending them to the more general setup of frozen Jacobian algebras, which arise naturally from dimer models with boundary and in the context of the additive categorification of cluster algebras with frozen variables via Frobenius categories. As an application, we show that...
Article
Full-text available
We show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A -module, meaning there are only finitely many indecomposable modules generated by M . Using the canonical tilts of endomorphism al...
Article
Full-text available
We study certain special tilting and cotilting modules for an algebra with positive dominant dimension, each of which is generated or cogenerated (and usually both) by projective-injectives. These modules have various interesting properties, for example, that their endomorphism algebras always have global dimension less than or equal to that of the...
Article
Proposition 5.16 of the author's paper ‘Mutations of frozen Jacobian algebras’ [11] is false. We give several possible remedies with different applications.
Preprint
Full-text available
Motivated by Conway and Coxeter's combinatorial results concerning frieze patterns, we sketch an introduction to the theory of cluster algebras and cluster categories for acyclic quivers. The goal is to show how these more abstract theories provide a conceptual explanation for phenomena concerning friezes, principally integrality and periodicity.
Preprint
Full-text available
We show that the dimer algebra of a connected Postnikov diagram in the disc is bimodule internally 3-Calabi-Yau in the sense of the author's earlier work. As a consequence, we obtain an additive categorification of the cluster algebra associated to the diagram, which (after inverting frozen variables) is isomorphic to the homogeneous coordinate rin...
Preprint
Full-text available
We survey results on mutations of Jacobian algebras, while simultaneously extending them to the more general setup of frozen Jacobian algebras, which arise naturally from dimer models with boundary and in the context of the additive categorification of cluster algebras with frozen variables via Frobenius categories. As an application, we show that...
Preprint
Full-text available
We show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module. The tilted algebra B is related to A by a recollement. We call an A-module M gen-finite if there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras o...
Article
We study the set ${\mathcal{S}}$ of labelled seeds of a cluster algebra of rank n inside a field ${\mathcal{F}}$ as a homogeneous space for the group Mn of (globally defined) mutations and relabellings. Regular equivalence relations on ${\mathcal{S}}$ are associated to subgroups W of Aut Mn ( ${\mathcal{S}}$ ), and we thus obtain groupoids W \ ${\m...
Preprint
Full-text available
We study a set of uniquely determined tilting and cotilting modules for an algebra with positive dominant dimension, with the property that they are generated or cogenerated (and usually both) by projective-injectives. These modules have various interesting properties, for example that their endomorphism algebras always have global dimension at mos...
Preprint
Full-text available
In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables, requiring as input a suitable candidate for the endomorphism algebra of a cluster-tilting object in such a category. In this paper, we construct such candidates in the case of cluster algebras with 'polarised' pr...
Thesis
Full-text available
Cluster categories, introduced by Buan–Marsh–Reineke–Reiten–Todorov and later generalised by Amiot, are certain 2-Calabi–Yau triangulated categories that model the combinatorics of cluster algebras without frozen variables. When frozen variables do occur, it is natural to try to model the cluster combinatorics via a Frobenius category, with the ind...

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