Many soft materials, such as elastomers and hydrogels, are made of long chain molecules crosslinked to form a three-dimensional network. Their mechanical properties depend on network parameters such as chain density, chain length distribution and the functionality of the crosslinks. Understanding the relationships between the topology of polymer networks and their mechanical properties has been a long-standing challenge in polymer physics.

In this work, we focus on so-called “near-ideal” networks, which are produced by the cross-coupling of star-like macromolecules with well-defined chain length. We developed a computational approach based on random discrete networks, according to which the polymer network is represented by an assembly of non-linear springs connected at junction points representing crosslinks. The positions of the crosslink points are determined from the conditions of mechanical equilibrium. Scaling relations for the elastic modulus and maximum extensibility of the network were obtained. Our scaling relations contradict some predictions of classical estimates of rubber elasticity and have implications for the interpretation of experimental data for near-ideal polymer networks.

Reference: G. Alame, L. Brassart. Relative contributions of chain density and topology to the elasticity of two-dimensional polymer networks. Soft Matter 15, 5703 (2019).

We aim to generate gene coexpression networks from gene expression data. In our networks, nodes represent genes and edges depict high positive correlation in their expression across different samples. Methods based on Pearson correlation are the most commonly used to generate gene coexpression networks. We propose the use of distance correlation as an effective alternative to Pearson correlation when constructing gene expression networks. Our methodology pipeline includes a thresholding step which allows us to discriminate which pairs of genes are coexpressed. We select the value of the threshold parameter by studying the stability of the generated network, rather than relying on exogenous biological information known a priori.

Identifying systemically important countries is crucial for global financial stability. In this work we use (multilayer) network methods to identify systemically important countries. We study the financial system as a multilayer network, where each layer represent a different type of financial investment between countries. To rank countries by their systemic importance, we implement MultiRank, as well a simplistic model of financial contagion. In this first model, we consider that each country has a capital buffer, given by the capital to assets ratio. After the default of an initial country, we model financial contagion with a simple rule: a solvent country defaults when the amount of assets lost, due to the default of other countries, is larger than its capital. Our results show that when we consider that there are various types of assets the ranking of systemically important countries changes. We make all our methods available by introducing a python library. Finally, we propose a more realistic model of financial contagion that merges multilayer network theory and the contingent claims sectoral balance sheet literature. The aim of this framework is to model the banking, private, and sovereign sector of each country and thus study financial contagion within the country and between countries. 

A fundamental problem in the context of Einstein's equations of general relativity is to understand precisely the dynamical evolution of small perturbations of stationary black hole solutions. It is expected that there is a discrete set of characteristic frequencies that play a dominant role at late time intervals and carry information about the nature of the black hole, much like the normal frequencies of a vibrating string. These frequencies are called quasi-normal frequencies or resonances and they are closely related to scattering resonances in the study of Schrödinger-type equations. I will discuss a new method of defining and studying resonances for linear wave equations on asymptotically flat black holes, developed from joint work with Claude Warnick.

Dr Rachel Philip will discuss her experiences working at the interface between academic mathematics and industry. Oxford University Innovation will discuss how they can help academics when interacting with industry. 

Speaker: Daniel Woodhouse (North)
Title: Generalizing Leighton's Graph Covering Theorem
Abstract: Before he ran off and became a multimillionaire, exploiting his knowledge of network optimisation, the computer scientist F. Thomas Leighton proved an innocuous looking result about finite graphs. The result states that any pair of finite graphs with isomorphic universal covers have isomorphic finite covers. I will explain what all this means, and why this should be of tremendous interest to group theorists and topologists.

Speaker: Benjamin Fehrman (South)
Title: Large deviations for particle processes and stochastic PDE
Abstract: In this talk, we will introduce the theory of large deviations through a simple example based on flipping a coin.  We will then define the zero range particle process, and show that its diffusive scaling limit solves a nonlinear diffusion equation.  The large deviations of the particle process about its scaling limit formally coincide with the large deviations of a certain ill-posed, singular stochastic PDE.  We will explain in what sense this relationship has been made mathematically precise.

Speaker: Joseph Keir (North)
Title: Dispersion (or not) in nonlinear wave equations
Abstract: Wave equations are ubiquitous in physics, playing central roles in fields as diverse as fluid dynamics, electromagnetism and general relativity. In many cases of these wave equations are nonlinear, and consequently can exhibit dramatically different behaviour when their solutions become large. Interestingly, they can also exhibit differences when given arbitrarily small initial data: in some cases, the nonlinearities drive solutions to grow larger and even to blow up in a finite time, while in other cases solutions disperse just like the linear case. The precise conditions on the nonlinearity which discriminate between these two cases are unknown, but in this talk I will present a conjecture regarding where this border lies, along with some conditions which are sufficient to guarantee dispersion.

Speaker: Priya Subramanian (South)
Title: What happens when an applied mathematician uses algebraic geometry?
Abstract: A regular situation that an applied mathematician faces is to obtain the equilibria of a set of differential equations that govern a system of interest. A number of techniques can help at this point to simplify the equations, which reduce the problem to that of finding equilibria of coupled polynomial equations. I want to talk about how homotopy methods developed in computational algebraic geometry can solve for all solutions of coupled polynomial equations non-iteratively using an example pattern forming system. Finally, I will end with some thoughts on what other 'nails' we might use this new shiny hammer on.

Persistent homology has been applied to graph classification problems as a way of generating vectorizable features of graphs that can be fed into machine learning algorithms, such as neural networks. A key ingredient of this approach is a filter constructor that assigns vector features to nodes to generate a filtration. In the case where the filter constructor is smoothly tuned by a set of real parameters, we can train a neural network graph classifier on data to learn an optimal set of parameters via the backpropagation of gradients that factor through persistence diagrams [Leygonie et al., arXiv:1910.00960]. We propose a flexible, spectral-based filter constructor that parses standalone graphs, generalizing methods proposed in [Carrière et al., arXiv: 1904.09378]. Our method has an advantage over optimizable filter constructors based on iterative message passing schemes (`graph neural networks’) [Hofer et al., arXiv: 1905.10996] which rely on heuristic user inputs of vertex features to initialise the scheme for datasets where vertex features are absent. We apply our methods to several benchmark datasets and demonstrate results comparable to current state-of-the-art graph classification methods.

Spreading processes on geometric networks are often influenced by a network’s underlying spatial structure, and it is insightful to study the extent to which a spreading process follows that structure. In particular, considering a threshold contagion on a network whose nodes are embedded in a manifold and which has both 'geometric edges' that respect the geometry of the underlying manifold, as well as 'non-geometric edges' that are not constrained by the geometry of the underlying manifold, one can ask whether the contagion propagates as a wave front along the underlying geometry, or jumps via long non-geometric edges to remote areas of the network. 
Taylor et al. developed a methodology aimed at determining the spreading behaviour of threshold contagion models on such 'noisy geometric networks' [1]. This methodology is inspired by nonlinear dimensionality reduction and is centred around a so-called 'contagion map' from the network’s nodes to a point cloud in high dimensional space. The structure of this point cloud reflects the spreading behaviour of the contagion. We apply this methodology to a family of noisy-geometric networks that can be construed as being embedded in a torus, and are able to identify a region in the parameter space where the contagion propagates predominantly via wave front propagation. This consolidates contagion map as both a tool for investigating spreading behaviour on spatial network, as well as a manifold learning technique. 
[1] D. Taylor, F. Klimm, H. A. Harrington, M. Kramar, K. Mischaikow, M. A. Porter, and P. J. Mucha. Topological data analysis of contagion maps for examining spreading processes on networks. Nature Communications, 6(7723) (2015)