Tensors are higher dimensional analogues of matrices, used to record data with multiple changing variables. Interpreting tensor data requires finding multi-linear stucture that depends on the application or context. I will describe a tensor-based clustering method for multi-dimensional data. The multi-linear structure is encoded as algebraic constraints in a linear program. I apply the method to a collection of experiments measuring the response of genetically diverse breast cancer cell lines to an array of ligands. In the second part of the talk, I will discuss low-rank decompositions of tensors that arise in statistics, focusing on two graphical models with hidden variables. I describe how the implicit semi-algebraic description of the statistical models can be used to obtain a closed form expression for the maximum likelihood estimate.

The pRED Clinical Pharmacology Disease Modelling Group (CPDMG) aims to better understand the biological basis of inter-patient variability of clinical response to drugs.  Improved understanding of how our drugs drive clinical responses informs which combination dosing regimens (“right drugs”) specific patient populations (“right patients”) are most likely to benefit from. Drug evoked responses are driven by drug-molecular-target interactions that perturb target functions. These direct, "proximal effects" (typically activation and/or inhibition of protein function) propagate across the biological processes these targets participate in via “distal effects” to drive clinical responses. Clinical Systems Pharmacology approaches are used by CPDMG to predict the mechanisms by which drug combinations evoke observed clinical responses. Over the last 5 years, CPDMG has successfully applied these approaches to inform key decisions across clinical development programs. Implementation of these approaches requires: (i) integration of prior relevant biological/clinical knowledge with large clinical and “omics” datasets; (ii) application of supervised machine learning (specifically, Artificial Neural Networks (ANNs)) to transform this knowledge/data into actionable, clinically relevant, mechanistic insights.  In this presentation, key features of these approaches will be discussed by way of clinical examples.  This will provide a framework for outlining the current limitations of these approaches and how we plan to address them in the future.

Single cell RNA-Seq data is challenging to analyse due to problems like dropout and cell type identification. We present a novel clustering 
approach that applies mixture models to learn interpretable clusters from RNA-Seq data, and demonstrate how it can be applied to publicly 
available scRNA-Seq data from the mouse brain. Having inferred groupings of the cells, we can then attempt to learn networks from the data. These 
approaches are widely applicable to single cell RNA-Seq datasets where  there is a need to identify and characterise sub-populations of cells.

Globally, almost 38 million people are living with HIV.  HPTN 071 (PopART) is the largest HIV prevention trial to date, taking place in 21 communities in Zambia and South Africa with a combined population of more than 1 million people.  As part of the trial an individual-based mathematical model was developed to help in planning the trial, to help interpret the results of the trial, and to make projections both into the future and to areas where the trial did not take place. In this talk I will outline the individual-based mathematical model used in the trial, the inference framework, and will discuss examples of how the results from the model have been used to help inform policy decisions.  

 The Steklov eigenvalue problem is an eigenvalue problem whose spectral parameters appear in the boundary condition. On a Riemannian surface with smooth boundary, Steklov eigenvalues have a very sharp asymptotic expansion. Also, a number of interesting sharp bounds for the $k$th Steklov eigenvalues have been known. We extend these results on orbisurfaces and discuss how the structure of orbifold singularities comes into play. This is joint work with Arias-Marco, Dryden, Gordon, Ray and Stanhope.

I will explain what the Willmore Morse Index of unbranched Willmore spheres in Euclidean three-space is and how to compute it. It turns out that several geometric properties at the ends of complete minimal surfaces with embedded planar ends are related to the mentioned Morse index.
One consequence of that computation is that all unbranched Willmore spheres are unstable (except for the round sphere). This talk is based on work with Jonas Hirsch.

 

We build on the literature on financial contagion using models of cross-holdings of equity participations and debt in different seniority classes, and extend them to include bail-ins and contingent convertible debt instruments, two mechanisms of debt-to-equity conversion. We combine these with recently proposed methods of network valuation under stochastic external assets, allowing for the pricing of debt instruments in each seniority layer and the calculation of default probabilities. We show that there exist well-defined valuations for all financial assets cross-held within the system. The full model constitutes an extension of classic asset pricing models that accounts for cross-holdings of debt securities. Our contribution is to add convertible debt to this framework.