Past Industrial and Applied Mathematics Seminar

15 May 2014
16:00
Ramon Grima
Abstract
Several experimental studies have shown that the abundance distributions of proteins in a population of isogenic cells may display multiple distinct maxima. Each of these maxima may be associated with a subpopulation of a particular phenotype, the quantification of which is important for understanding cellular decision-making. I will present a novel methodology which allows us to quantify multi-modal gene expression distributions and single cell power spectra in gene regulatory networks. The method is based on an extension of the linear noise approximation; in particular we rigorously show that, in the limit of slow promoter dynamics, these distributions can be systematically approximated as a mixture of Gaussian components. The resulting closed-form approximation provides a practical tool for studying complex nonlinear gene regulatory networks that have thus far been amenable only to stochastic simulation. I will demonstrate the applicability of our approach to several examples and discuss some new dynamical characteristics e.g., how the interplay of transcriptional and translational regulation can be exploited to control the multimodality of gene expression distributions in two-promoter networks and how genetic oscillators can display concerted noise-induced bimodality and noise-induced oscillations.
  • Industrial and Applied Mathematics Seminar
8 May 2014
16:00
Jonathan Ward
Abstract
Adaptive network models, in which node states and network topology coevolve, arise naturally in models of social dynamics that incorporate homophily and social influence. Homophily relates the similarity between pairs of agents' states to their network coupling strength, whilst social influence causes the convergence of coupled agents' states. In this talk, I will describe a deterministic adaptive network model of attitude formation in social groups that incorporates these effects, and in which the attitudinal dynamics are represented by an activator-inhibitor process. I will show that consensus, corresponding to all nodes adopting the same attitudinal state and being fully connected, may destabilise via Turing instability, giving rise to chaotic dynamics. For the case where there are just two agents, I will illustrate, using numerical continuation, how such chaotic dynamics arise.
  • Industrial and Applied Mathematics Seminar
1 May 2014
16:00
Mary Lou Zeeman
Abstract
One of the things sustainability applications have in common with industrial applications is their close connection with decision-making and policy. We will discuss how a decision-support viewpoint may inspire new mathematical questions. For example, the concept of resilience (of ecosystems, food systems, communities, economies, etc) is often described as the capacity of a system to withstand disturbance and retain its functional characteristics. This has several familiar mathematical interpretations, probing the interaction between transient dynamics and noise. How does a focus on resilience change the modeling, dynamical and policy questions we ask? I look forward to your ideas and discussion.
  • Industrial and Applied Mathematics Seminar
13 March 2014
16:00
Marcus Roper
Abstract
Familiar species; humans, mammals, fish, reptiles and plants represent only a razor’s edge of the Earth’s immense biodiversity. Most of the Earth’s multicellular species lie buried in soil, inside of plants, and in the undergrowth, and include millions of unknown species, almost half of which are thought to be fungi. Part of the amazing success of fungi may be the elegant solutions that they have evolved to the problems of dispersing, growing and adapting to changing environments. I will describe how we using both math modeling and experiments to discover some of these solutions. I will focus on (i) how cytoplasmic mixing enables some species to tolerate internal genetic diversity, making them better pathogens and more adaptable, and (ii) how self-organization of these flows into phases of transport and stasis enables cells to function both as transport conduits, and to perform other functions like growth and secretion.
  • Industrial and Applied Mathematics Seminar
6 March 2014
16:00
Beatrice Pelloni
Abstract
In this talk, I will discuss the effect of boundary conditions on the solvability of PDEs that have formally an integrable structure, in the sense of possessing a Lax pair. Many of these PDEs arise in wave propagation phenomena, and boundary value problems for these models are very important in applications. I will discuss the extent to which general approaches that are successful for solving the initial value problem extend to the solution of boundary value problem. I will survey the solution of specific examples of integrable PDE, linear and nonlinear. The linear theory is joint work with David Smith. For the nonlinear case, I will discuss boundary conditions that yield boundary value problems that are fully integrable, in particular recent joint results with Thanasis Fokas and Jonatan Lenells on the solution of boundary value problems for the elliptic sine-Gordon equation.
  • Industrial and Applied Mathematics Seminar
27 February 2014
16:00
Michael Dallaston
Abstract
Two-dimensional viscous fluid flow problems come about either because of a thin gap geometry (Hele-Shaw flow) or plane symmetry (Stokes flow). Such problems can also involve free boundaries between different fluids, and much has been achieved in this area, including by many at Oxford. In this seminar I will discuss some new results in this field. Firstly I will talk about some of the results of my PhD on contracting inviscid bubbles in Hele-Shaw flow, in particular regarding the effects of surface tension and kinetic undercooling on the free boundary. When a bubble contracts to a point, these effects are dominant, and lead to a menagerie of possible extinction shapes. This limiting problem is a generalisation of the curve shortening flow equation from the study of geometric PDEs. We are currently exploring properties of this generalised flow rule. Secondly I will discuss current work on applying a free boundary Stokes flow model to the evolution of subglacial water channels. These channels are maintained by the balance between inward creep of ice and melting due to the flow of water. While these channels are normally modelled as circular or semicircular in cross-section, the inward creep of a viscous fluid is unstable. We look at some simplistic viscous dissipation models and the effect they have on the stability of the channel shape. Ultimately, a more realistic turbulent flow model is needed to understand the morphology of the channel walls.
  • Industrial and Applied Mathematics Seminar
20 February 2014
16:00
Abstract
In Parkinson’s disease, increased power of oscillations in firing rate has been observed throughout the cortico-basal-ganglia circuit. In particular, the excessive oscillations in the beta range (13-30Hz) have been shown to be associated with difficulty of movement initiation. However, on the basis of experimental data alone it is difficult to determine where these oscillations are generated, due to complex and recurrent structure of the cortico-basal-ganglia-thalamic circuit. This talk will describe a mathematical model of a subset of basal-ganglia that is able to reproduce experimentally observed patterns of activity. The analysis of the model suggests where and under which conditions the beta oscillations are produced.
  • Industrial and Applied Mathematics Seminar
13 February 2014
16:00
Abstract
Strongly nonlinear problems, written abstractly in the form N[u]=0, are typically difficult to analyze unless they possess special properties. However, if we are able to find a quasi-solution u_0 in the sense that the residual N[u_0] := R is small, then it is possible to analyze a strongly nonlinear problem with weakly nonlinear analysis in the following manner: We decompose u=u_0 + E; then E satisfies L E = -N_1 [E] - R, where L is the Fre'chet derivative of the operator N and N_1 [E] := N[u_0+E]-N[u_0]-L E contains all the nonlinearity. If L has a suitable inversion property and the nonlinearity N_1 is sufficiently regular in E, then weakly nonlinear analysis of the error E through contraction mapping theorem gives rise to control of the error E. What is described above is quite routine. The only new element is to determine a quasi-solution u_0, which is typically found through a combination of classic orthogonal polynomial representation and exponential asymptotics. This method has been used in a number of nonlinear ODEs arising from reduction of PDEs. We also show how it can be extended to integro-differential equations that arise in study of deep water waves of permanent form. The method is quite general and can in principle be applied to nonlinear PDEs as well. NB. Much of this is joint work with O. Costin and other collaborators.
  • Industrial and Applied Mathematics Seminar
6 February 2014
16:00
Hannah Fry
Abstract
Much of the mathematical modelling of urban systems revolves around the use spatial interaction models, derived from information theory and entropy-maximisation techniques and embedded in dynamic difference equations. When framed in the context of a retail system, the dynamics of centre growth poses an interesting mathematical problem, with bifurcations and phase changes, which may be analysed analytically. In this contribution, we present some analysis of the continuous retail model and corresponding discrete version, which yields insights into the effect of space on the system, and an understanding of why certain retail centers are more successful than others. This class of models turns out to have wide reaching applications: from trade and migration flows to the spread of riots and the prediction of archeological sites of interest, examples of which we explore in more detail during the talk.
  • Industrial and Applied Mathematics Seminar
30 January 2014
16:00
Abstract
Many real-life complex systems arise as a network of simple interconnected individual agents. A central question is to determine how network topology and individual agent dynamics combine to create the global dynamics. In this talk we focus on the case of continuous-time random walks on networks, with a waiting time of the walker on each node assuming arbitrary probability distributions. Such random walks are useful to model diffusion processes over complex temporal networks representing human interactions, often characterized by non-Poissonian contact patterns. We find that the mixing time of the random walker, i.e. the relaxation time for the process to reach stationarity, is determined by a combination of three factors: the spectral gap, associated to bottlenecks in the underlying topology, burstiness, related to the second moment of the waiting time distribution, and the characteristic time of its exponential tail, which is an indicator of the tail `fatness'. We show theoretically that a strong modular structure dampens the importance of burstiness, and empirically that either of the three factors may be dominant in real-life data. These results are available in arXiv:1309.4155
  • Industrial and Applied Mathematics Seminar

Pages