Titles and Abstracts

14:00-14:50 - Catherine Drysdale

Title: Computing Spectra of Infinite Dimensional Operators - What do we need?  

Abstract: In this talk, I look at three examples regarding the numerical computation of spectra for infinite-dimensional non-self-adjoint operators. Two conditions are needed to compute the spectrum of operators in one limit. Namely, knowledge of how the operator acts on vectors and also how the resolvent norm behaves near the spectrum. We will explore examples that illustrate the two conditions and how we can overcome them in certain contexts. These include the non-self-adjoint Ginzburg-Landau equation and a convection-diffusion equation associated with fluid rotating in a cylinder. 

16:00-16:50 - Sandeep Shirgill

Title: Using deep learning to compare spatial organisation of proteins within cells imaged using single-molecule localisation microscopy.

Abstract: Unlike traditional microscopy images, single-molecule localisation microscopy (SMLM) data is pointillist, consisting of xyz coordinates of labelled molecules. Currently, no published method characterises the spatial organisation of proteins in a cell and how it varies between different cells or within the same cell under different conditions, such as before and after drug treatment. In this study, we employ a machine-learning approach to embed SMLM regions of interest (ROIs) into a two-dimensional latent space, facilitating the identification of similar ROIs. Initially, ROIs are processed through a contrastive-learning neural network trained to recognize biologically relevant differences, such as different levels of clustering within the cells and the shape and size of clusters, while ignoring irrelevant variations, such as cluster positions and ROI rotations. The neural network embeds these ROIs into a high-dimensional latent space, where closer ROIs indicate greater similarity in their spatial organisations. To visualise the data, we use the dimensionality reduction technique Uniform Manifold Approximation and Projection (UMAP) to project the high-dimensional latent space into two dimensions. Using this technique, the diversity that exists for nanoscale organisational similarity between cells, proteins, species etc. could be revealed. This would provide the first information on the diversity of nanoscale organisation.

17:00-17:50 - Bradly Deeley

Title: Transient propagation of invasive plants in the heterogeneous landscape 

Abstract: Understanding the propagation of invasive plants at the beginning of invasive spread is important as it can help practitioners eradicate harmful species more efficiently. In this talk it will be shown how invasive species propagate at the short-time scale in the homogeneous environment [1], and investigate the question of whether the transient dynamics become different when the homogeneous landscape is transformed into the heterogeneous one by introducing a road in the spatial domain. The integro-difference framework is employed in the model to simulate spatio-temporal dynamics of a stage-structured invasive plant in the heterogeneous landscape where the road is considered as a “hostile environment”. We will demonstrate how the propagation of the invasive species at short times is changed by the presence of a road in comparison with transient propagation in the homogeneous landscape. Among other parameters, we are interested in the impact the road width makes on the propagation regime at short times, and it will be argued in the talk that invasion slows down in the presence of a wide road. 

[1] Deeley, B., Petrovskaya, N. Transient Propagation of the Invasion Front in the Homogeneous Landscape and in the Presence of a Road. Bull Math Biol 86, 78 (2024). https://doi.org/10.1007/s11538-024-01302-3