Past

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.

We show that two rules are sufficient to automate gradient descent: 1) don't increase the stepsize too fast and 2) don't overstep the local curvature. No need for functional values, no line search, no information about the function except for the gradients. By following these rules, you get a method adaptive to the local geometry, with convergence guarantees depending only on smoothness in a neighborhood of a solution. Given that the problem is convex, our method will converge even if the global smoothness constant is infinity. As an illustration, it can minimize arbitrary continuously twice-differentiable convex function. We examine its performance on a range of convex and nonconvex problems, including matrix factorization and training of ResNet-18.

RF-engineering defines the “perfect” parabolic shape a foldable reflector antenna (e.g. the membrane) should have. In practice it is virtually impossible to design a deployable backing structure that can meet all RF-imposed requirements. Inevitably the shape of the membrane will deviate from its ideal parabolic shape when material properties and pragmatic mechanical design are considered. There is therefore a challenge to model such membranes in order to find the form they take and then use the model as a design tool and perhaps in an optimisation objective function, if tractable. 

The variables we deal with are:
Elasticity of the membrane (anisotropic or orthotropic typ)
Boundary forces (by virtue of the interaction between the membrane and it’s attachment)
Elasticity of the backing structure (e.g. the elasticity properties of the attachment)
Number, location and elasticity of the membrane fixing points

There are also in-orbit environmental effects on such structures for which modelling could also be of value. For example, the structure can undergo thermal shocks and oscillations can occur that are un-dampened by the usual atmospheric interactions at ground level etc. There are many other such points to be considered and allowed for.

Some of the simplest expected cases of Langlands functoriality are the symmetric power liftings Sym^r from automorphic representations of GL(2) to automorphic representations of GL(r+1). I will discuss some joint work with Jack Thorne on the symmetric power lifting for holomorphic modular forms.

Nonlinear generalizations of the Schrödinger equation are of wide applicability to a range of areas including atomic and optical systems, 
plasma physics and water waves.  In this  talk we revisit some principal excitations in atomic and optical systems (such as Bose-Einstein condensates and photo-refractive crystals), namely dark solitonic fronts in single-component systems, and dark-bright waves in multi-component systems. Upon introducing them and explaining their existence and stability properties in one spatial dimension, we will extend them both in the form of stripes and in that rings in two-dimensions, presenting an alternative (adiabatic-invariant based) formulation of their stability and excitations. We will explore their filamentary dynamics, as well as the states that emerge from their transverse (snaking) instability. Then, we will consider these structures even in three dimensions, in the form of planar, as well as spherical shell wave patterns and generalize our adiabatic invariant formulation there. Finally, time permitting, we will give some glimpses of how some of these dynamical features in 1d and 2d generalize in a multi-orbital, time-dependent quantum setting.

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.

In 2013, Bhargava-Shankar proved that (in a suitable sense) the average rank of elliptic curves over Q is bounded above by 1.5, a landmark result which earned Bhargava the Fields medal. Later Bhargava-Gross proved similar results for hyperelliptic curves, and Poonen-Stoll deduced that most hyperelliptic curves of genus g>1 have very few rational points. The goal of my talk is to explain how simple curve singularities and simple Lie algebras come into the picture, via a modified Grothendieck-Brieskorn correspondence.

Moreover, I’ll explain how this viewpoint leads to new results on the arithmetic of curves in families, specifically for certain families of non-hyperelliptic genus 3 curves.

Discussion of https://arxiv.org/abs/1812.04716

Systems of nonlinear polynomial equations arise in a variety of fields in mathematics, science, and engineering.  Many numerical techniques for solving and analyzing solution sets of polynomial equations over the complex numbers, collectively called numerical algebraic geometry, have been developed over the past several decades.  However, since real solutions are the only solutions of interest in many applications, there is a current emphasis on developing new methods for computing and analyzing real solution sets.  This talk will summarize some numerical real algebraic geometric approaches as well as recent successes of these methods for solving a variety of problems in science and engineering.

In recent years it has become abundantly clear that mathematics can do "things" in society; indeed, many more things than in the past. Deep mathematical work now underpins some of the most important aspects of the way society functions. And, as mathematically-trained people, we are constantly promoting the positive impact of mathematics. But if such work is capable of good, then is it not also capable of harm? So how do we begin to identify such potential harm, let alone address it and try and avoid, or at least reduce, it? In this session we will discuss how mathematics is a powerful double-edged sword, and why it must be wielded responsibly.

The second fundamental form of an embedded manifold must satisfy a set of constraint equations known as the Gauß-Codazzi equations. Since work of Chen-Slemrod-Wang, these equations are known to satisfy a particular div-curl structure: under suitable L^p bound on the second fundamental form, the curvatures are weakly continuous. In this talk we explore generalisations of this original result under weaker assumptions. We show how techniques from fluid dynamics can yield interesting insight into the weak continuity properties of isometric embeddings.

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Teichmüller harmonic map flow is a geometric flow designed to evolve combinations of maps and metrics on a surface into minimal surfaces in a Riemannian manifold. I will introduce the flow and describe known existence results, and discuss recent joint work with M. Rupflin that demonstrates how singularities can develop in the metric component in finite time.

 In Classification Theory, Shelah defined several cardinal invariants of a complete theory which detect the presence of certain trees among the definable sets, which in turn quantify the complexity of forking.  In later model-theoretic developments, local versions of these invariants were recognized as marking important dividing lines - e.g. simplicity and NTP2.  Around these dividing lines, a dichotomy theorem of Shelah states that a theory has the tree property if and only if it is witnessed in one of two extremal forms--the tree property of the first or second kind--and it was asked if there is a 'quantitative' analogue of this dichotomy in the form of a certain equation among these invariants.  We will describe these model-theoretic invariants and explain why the quantitative version of the dichotomy fails, via a construction that relies upon some unexpected tools from combinatorial set theory. 

 

On any hyperbolic surface, the number of curves of length at most L is finite. However, it is not immediately clear how quickly this number grows with L. We will discuss Mirzakhani’s breakthrough result regarding the asymptotic behaviour of this number, along with recent efforts to generalise her result using currents.

Let $\mathcal{B}$ be a (unital) commutative Banach algebra and $\Omega$ the set of non-trivial multiplicative linear functionals $\omega : \mathcal{B} \to \mathbb{C}$. Gelfand theory tells us that the kernels of these functionals are exactly the maximal ideals of $\mathcal{B}$ and, as a consequence, an element $b \in \mathcal{B}$ is invertible if and only if $\omega(b) \neq 0$ for all $\omega \in \Omega$. A generalisation to non-commutative Banach algebras is the local principle of Allan and Douglas, also known as central localisation: Let $\mathcal{B}$ be a Banach algebra, $Z$ a closed subalgebra of the center of $\mathcal{B}$ and $\Omega$ the set of maximal ideals of $Z$. For every $\omega \in \Omega$ let $\mathcal{I}_{\omega}$ be the smallest ideal of $\mathcal{B}$ which contains $\omega$. Then $b \in \mathcal{B}$ is invertible if and only if $b + \mathcal{I}_{\omega}$ is invertible in $\mathcal{B} / \mathcal{I}_{\omega}$ for every $\omega \in \Omega$.

From an operator theory point of view, one of the most important features of the local principle is the application to Calkin algebras. In that case the invertible elements are called Fredholm operators and the corresponding spectrum is called the essential spectrum. Therefore, by taking suitable subalgebras, we can obtain a characterisation of Fredholm operators. Many beautiful results in spectral theory, e.g.~formulas for the essential spectrum of Toeplitz operators, can be obtained in this way. However, the central localisation is often not sufficient to provide a satisfactory characterisation for more general operators. In this talk we therefore consider a generalisation where the ideals $\mathcal{I}_{\omega}$ do not originate from the center of the algebra. More precisely, we will start with general $L^p$-spaces and apply limit operator methods to obtain a Fredholm theory that is applicable to many different settings. In particular, we will obtain characterisations of Fredholmness and compactness in many new cases and also rediscover some classical results.

This talk is based on joint work with Christian Seifert.

The study of random matrix moments of moments has connections to number theory, combinatorics, and log-correlated fields. Our results give the leading order of these functions for integer moment parameters by exploiting connections with Gelfand-Tsetlin patterns and counts of lattice points in convex sets. This is joint work with Jon Keating and Theo Assiotis.

Abstract:

Quasicrystals have been observed recently in soft condensed mater, providing new insight into the ongoing quest to understand their formation and thermodynamic stability. I shall explain the stability of certain soft-matter quasicrystals, using surprisingly simple classical field theories, by making an analogy to Faraday waves. This will provide a recipe for designing pair potentials that yield crystals with (almost) any given symmetry.

There are various notions of rank, which measure the complexity of a tensor or polynomial. Cubic surfaces can be viewed as symmetric tensors.  We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The results extend to order three tensors of all sizes, implying the equality of rank and symmetric rank when the symmetric rank is at most seven. We then explore the connection between the rank of a polynomial and the singularities of its vanishing locus, and we find the possible singular loci of a cubic surface of given rank. This talk is based on joint work with Eunice Sukarto.
 

ARC methods are Newton-type solvers for unconstrained, possibly nonconvex, optimisation problems. In this context, a novel variant based on dynamic inexact Hessian information is discussed. The approach preserves the optimal complexity of the basic framework and the main probabilistic results on the complexity and convergence analysis in the finite-sum minimisation setting will be shown. At the end, some numerical tests on machine learning problems, ill-conditioned databases and real-life applications will be given, in order to confirm the theoretical achievements. Joint work with Stefania Bellavia and Benedetta Morini (Università di Firenze). 

Many manufacturing processes require the use of robots to transport parts around a factory line. Some parts, which are very thin (e.g. car doors)
are prone to elastic deformations as they are moved around by a robot. These should be avoided at all cost. A problem that was recently raised by
F.E.E. (Fleischmann Elektrotech Engineering) at the ESGI 158 study group in Barcelona was to ascertain how to determine the stresses of a piece when
undergoing a prescribed motion by a robot. We present a simple model using Kirschoff-Love theory of flat plates and how this can be adapted. We
outline how the solutions of the model can then be used to determine the stresses. 

 I will give an overview of my recent research on the definition of asymptotic charges in asymptotically flat spacetimes, including the definition of subleading and dual BMS charges and the relation to the conserved Newman-Penrose charges at null infinity.

Random graphs were introduced as a convenient example for demonstrating the impossibility of ‘complete disorder’ by Erdos, who also thought that these objects will never become useful in the applied areas outside of pure mathematics. In this talk, I will view random graphs as objects in the field of applied mathematics and discus how the application-driven objectives have set new directions for studying random graphs. I will focus on characterising the sizes of connected components in graphs with a given degree distribution, on the percolation-like processes on such structures, and on generalisations to the coloured graphs. These theoretical questions have interesting implications for studying resilience of networks with nontrivial structures, and for materials science where they explain kinetics-driven phase transitions. Even more surprisingly, the results reveal intricate connections between random graphs and non-linear partial differential equations indicating new possibilities for their analysis.