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Nathan Broomhead

Publications and source records attributed to Nathan Broomhead.

13 recordsLinked to original sources
arXiv2026

Homology-based Morphometry of Brain Atrophy: Methods and Applications

Understanding the structure of the brain, and how it changes with time and disease, is a core goal of structural neuroimaging. Contemporary approaches to structural brain analysis are dominated by voxel-wise, mass-univariate methods such as voxel-based morphometry (VBM). However, these techniques require images to be normalized to a standard template, which can obscure subject-specific geometric features. Normalization to a common stereotactic space can also be problematic when comparing groups with substantial brain pathology, lesions, or other anatomical abnormalities. Here, we introduce two complementary pipelines based on persistent homology (PH), a tool from topological data analysis, to quantify multiscale geometric features of structural T1-weighted MRI scans. Pipeline 1 quantifies regional thinning by applying the Euclidean distance transform to tissue masks in a slice-wise manner. Pipeline 2 uses \(\alpha\)-filtrations to measure structural similarity between pairs of scans, capturing sulcal widening and ventricular enlargement. Synthetic experiments with controlled induced lesions showed that Pipeline 1 is best suited to between-subject analyses, whereas Pipeline 2 is better suited to within-subject designs. Applied to real-world data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), Pipeline 1 separated Alzheimer's disease (AD) from cognitively normal (CN) participants using single-modality T1-weighted MRI without nonlinear registration (ROC-AUC = 0.895), with peak effects localized to medial temporal regions. Pipeline 2 captured disease-related longitudinal change, with follow-up scans remaining closest to their own baselines and AD subjects showing greater short-interval change than CN subjects. Together, these pipelines provide interpretable topological biomarkers for cross-sectional group comparisons and longitudinal tracking.

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arXiv2025

An isometry theorem for persistent homology of circle-valued functions

This paper explores persistence modules for circle-valued functions, presenting a new extension of the interleaving and bottleneck distances in this setting. We propose a natural generalisation of barcodes in terms of arcs on a geometric model associated to the derived category of quiver representations. The main result is an isometry theorem that establishes an equivalence between the interleaving distance and the bottleneck distance for circle-valued persistence modules.

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arXiv2024

Simple tilts of length hearts and simple-minded mutation

We characterise when a simple Happel-Reiten-Smalo tilt of a length heart is again a length heart in terms of approximation theory and the existence of a stability condition with a phase gap. We apply simple-minded reduction to provide a sufficient condition for infinite iterability of simple-minded mutation/simple tilting. We use simple-minded mutation pairs to provide a common framework to show that mutation of simple-minded collections (resp. $w$-simple-minded systems, for $w \geq 1$) gives simple-minded collections (resp. $w$-simple-minded systems) under mild conditions, in the process providing a unified proof of results of Alex Dugas and Peter Jorgensen. Finally, we show that under mild conditions, mutation of simple-minded collections is compatible with mutation of $w$-simple-minded systems via a singularity category construction due to Haibo Jin.

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arXiv2023

The heart fan of an abelian category

We apply convex geometry (cones, fans) to homological input (abelian categories, hearts of bounded t-structures) to construct a new invariant of an abelian category, its heart fan. This can be viewed as a `universal phase diagram' for Bridgeland stability conditions with the given heart. When the abelian category is the module category of a finite-dimensional algebra, the heart fan is complete and contains the g-fan as the subfan of full-dimensional cones. The heart fan is also closely related to the wall-and-chamber structure for King semistability.

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arXiv2022

Partial compactification of stability manifolds via massless semistable objects

We introduce two extensions of the space of Bridgeland stability conditions of a triangulated category. First we consider lax stability conditions where semistable objects are allowed to have mass zero but still have a phase. The subcategory of massless objects is thick and there is an induced Bridgeland stability on the quotient category. We study deformations of lax stability conditions. Second we consider the space arising by identifying lax stability conditions which are deformation-equivalent with fixed charge. This second space is stratified by stability spaces of Verdier quotients of the triangulated category by thick subcategories of massless objects. We illustrate our results through examples in which the Grothendieck group has rank $2$. For these, our extended stability spaces can be explicitly described and related to the wall-and-chamber structure of the stability space.

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arXiv2018

Discrete triangulated categories

We introduce and study several homological notions which generalise the discrete derived categories of D. Vossieck. As an application, we show that Vossieck discrete algebras have this property with respect to all bounded t-structures. We give many examples of triangulated categories regarding these notions.

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arXiv2015

Discrete derived categories II: The silting pairs CW complex and the stability manifold

Discrete derived categories were studied initially by Vossieck \cite{Vossieck} and later by Bobiński, Geiß, Skowroński \cite{BGS}. In this article, we define the CW complex of silting pairs for a triangulated category and show that it is contractible in the case of discrete derived categories. We provide an explicit embedding from the silting CW complex into the stability manifold. By work of Qiu and Woolf, there is a deformation retract of the stability manifold onto the silting pairs CW complex. We obtain that the space of stability conditions of discrete derived categories is contractible.

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arXiv2014

Partially ample line bundles on toric varieties

In this note we study properties of partially ample line bundles on simplicial projective toric varieties. We prove that the cone of q-ample line bundles is a union of rational polyhedral cones, and calculate these cones in examples. We prove a restriction theorem for big q-ample line bundles, and deduce that q-ampleness of the anticanonical bundle is not invariant under flips. Finally we prove a Kodaira-type vanishing theorem for q-ample line bundles.

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arXiv2012

Averaging t-structures and extension closure of aisles

We ask when a finite set of t-structures in a triangulated category can be `averaged' into one t-structure or, equivalently, when the extension closure of a finite set of aisles is again an aisle. There is a straightforward, positive answer for a finite set of compactly generated t-structures in a big triangulated category. For piecewise tame hereditary categories, we give a criterion for when averaging is possible, and an algorithm that computes truncation triangles in this case. A finite group action on a triangulated category gives a natural way of producing a finite set of t-structures out of a given one. If averaging is possible, there is an induced t-structure on the equivariant triangulated category.

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arXiv2012

Autoequivalences of toric surfaces

We show that the autoequivalence group of the derived category of any smooth projective toric surface is generated by the standard equivalences and spherical twists obtained from -2-curves. In many cases we give all relations between these generators. We also prove a close link between spherical objects and certain pairs of exceptional objects.

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arXiv2010

Dimer models and Calabi-Yau algebras

In this article we study dimer models, as introduced in string theory, which give a way of writing down a class of non-commutative `superpotential' algebras. Some examples are 3-dimensional Calabi-Yau algebras, as defined by Ginzburg, and some are not. We consider two types of `consistency' condition on dimer models, and show that a `geometrically consistent' model is `algebraically consistent'. We prove that the algebra obtained from an algebraically consistent dimer model is a 3-dimensional Calabi-Yau algebra and finally prove that this gives a non-commutative crepant resolution of the Gorenstein affine toric threefold associated to the dimer model.

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