Forthcoming events in this series
The transition to turbulent fluid flow
Abstract
It is well known that the Navier-Stokes equations of viscous fluid flow do not give good predictions of when a viscous flow is likely to become unstable. When classical linearized theory is used to explore the stability of a viscous flow, the Navier-Stokes equations predict that instability will occur at fluid speeds (Reynolds numbers) far in excess of those actually measured in experiments. In response to this discrepancy, theories have arisen that suggest the eigenvalues computed in classical stability analysis do not give a full account of the behaviour, while others have suggested that fluid instability is a fundamentally non-linear process which is not accessible to linearized stability analyses.
In this talk, an alternative account of fluid instability and turbulence will be explored. It is suggested that the Navier-Stokes equations themselves might not be entirely appropriate to describe the transition to turbulent flow. A slightly more general model allows the possibility that the classical viscous fluid flows predicted by Navier-Stokes theory may become unstable at Reynolds numbers much closer to those seen in experiments, and so might perhaps give an account of the physics underlying turbulent behaviour.
Visualizing Multi-dimensional Persistent Homology
Abstract
Persistent homology is a tool for identifying topological features of (often high-dimensional) data. Typically, the data is indexed by a one-dimensional parameter space, and persistent homology is easily visualized via a persistence diagram or "barcode." Multi-dimensional persistent homology identifies topological features for data that is indexed by a multi-dimensional index space, and visualization is challenging for both practical and algebraic reasons. In this talk, I will give an introduction to persistent homology in both the single- and multi-dimensional settings. I will then describe an approach to visualizing multi-dimensional persistence, and the algebraic and computational challenges involved. Lastly, I will demonstrate an interactive visualization tool, the result of recent work to efficiently compute and visualize multi-dimensional persistent homology. This work is in collaboration with Michael Lesnick of the Institute for Mathematics and its Applications.
Mathematical modelling and numerical simulation of LiFePO4 cathodes
Abstract
LiFePO4 is a commercially available battery material with good theoretical discharge capacity, excellent cycle life and increased safety compared with competing Li-ion chemistries. During discharge, LiFePO4 material can undergo phase separation, between a highly and lowly lithiated form. Discharge of LiFePO4 crystals has traditionally been modelled by one-phase Stefan problems, which assume that phase separation occurs.
Recent work has been using phase-field models based on the Cahn-Hilliard equation, which only phase-separates when thermodynamically favourable. In the past year or two, this work has been having considerable impact in both theoretical and experimental electrochemistry.
Unfortunately, these models are very difficult to solve numerically and involve large, coupled systems of nonlinear PDEs across several different size scales that include a range of different physics and cannot be homogenised effectively.
This talk will give an overview of recent developments in modelling LiFePO4 and the sort of strategies used to solve these systems numerically.
Computational Modeling of the Eukaryotic Cytoskeleton
Abstract
Acto-myosin network growth and remodeling in vivo is based on a large number of chemical and mechanical processes, which are mutually coupled and spatially and temporally resolved. To investigate the fundamental principles behind the self-organization of these networks, we have developed detailed physico-chemical, stochastic models of actin filament growth dynamics, where the mechanical rigidity of filaments and their corresponding deformations under internally and externally generated forces are taken into account. Our work sheds light on the interplay between the chemical and mechanical processes, and also will highlights the importance of diffusional and active transport phenomena. For example, we showed that molecular transport plays an important role in determining the shapes of the commonly observed force-velocity curves. We also investigated the nonlinear mechano-chemical couplings between an acto-myosin network and an external deformable substrate.
16:00
Collective dynamics and self-organization
Abstract
We are interested in large systems of agents collectively looking for a
consensus (about e.g. their direction of motion, like in bird flocks). In
spite of the local character of the interactions (only a few neighbours are
involved), these systems often exhibit large scale coordinated structures.
The understanding of how this self-organization emerges at the large scale
is still poorly understood and offer fascinating challenges to the modelling
science. We will discuss a few of these issues on a selection of specific
examples.
16:00
Landing or take-off of fluids and bodies
Abstract
The talk is on impacts, penetrations and lift-offs involving bodies and fluids, with applications that range from aircraft and ship safety and our tiny everyday scales of splashing and washing, up to surface movements on Mars. Several studies over recent years have addressed different aspects of air-water effects and fluid-body interplay theoretically. Nonlinear interactions and evolutions are key here and these are to be considered in the presentation. Connections with experiments will also be described.
16:00
Capillary multipoles, shape anisotropy, and flocculation in 2D: the case of anisotropic colloids at fluid interfaces
Abstract
The synthesis of complex-shaped colloids and nanoparticles has recently undergone unprecedented advancements. It is now possible to manufacture particles shaped as dumbbells, cubes, stars, triangles, and cylinders, with exquisite control over the particle shape. How can particle geometry be exploited in the context of capillarity and surface-tension phenomena? This talk examines this question by exploring the case of complex-shaped particles adsorbed at the interface between two immiscible fluids, in the small Bond number limit in which gravity is not important. In this limit, the "Cheerio's effect" is unimportant, but interface deformations do emerge. This drives configuration dependent capillary forces that can be exploited in a variety of contexts, from emulsion stabilisation to the manufacturing of new materials. It is an opportunity for the mathematics community to get involved in this field, which offers ample opportunities for careful mathematical analysis. For instance, we find that the mathematical toolbox provided by 2D potential theory lead to remarkably good predictions of the forces and torques measured experimentally by tracking particle pairs of cylinders and ellipsoids. New research directions will also be mentioned during the talk, including elasto-capillary interactions and the simulation of multiphase composites.
16:00
Stochastic-Dynamical Methods for Molecular Modelling
Abstract
Molecular modelling has become a valuable tool and is increasingly part of the standard methodology of chemistry, physics, engineering and biology. The power of molecular modelling lies in its versatility: as potential energy functions improve, a broader range of more complex phenomena become accessible to simulation, but much of the underlying methodology can be re-used. For example, the Verlet method is still the most popular molecular dynamics scheme for constant energy molecular dynamics simulations despite it being one of the first to be proposed for the purpose.
One of the most important challenges in molecular modelling remains the computation of averages with respect to the canonical Gibbs (constant temperature) distribution, for which the Verlet method is not appropriate. Whereas constant energy molecular dynamics prescribes a set of equations (Newton's equations), there are many alternatives for canonical sampling with quite different properties. The challenge is therefore to identify formulations and numerical methods that are robust and maximally efficient in the computational setting.
One of the simplest and most effective methods for sampling is based on Langevin dynamics which mimics coupling to a heat bath by the incorporation of random forces and an associated dissipative term. Schemes for Langevin dynamics simulation may be developed based on the familiar principle of splitting. I will show that the invariant measure ('long term') approximation may be strongly affected by a simple re-ordering of the terms of the splitting. I will describe a transition in weak numerical order of accuracy that occurs (in one case) in the t->infty limit.
I will also entertain some more radical suggestions for canonical sampling, including stochastic isokinetic methods that enable the use of greatly enlarged timesteps for expensive but slowly-varying force field components.
16:00
Theory and experiments are strongly connected in nonlinear mechanics
Abstract
A perturbative method is introduced to analyze shear bands formation and
development in ductile solids subject to large strain.
Experiments on discrete systems made up of highly-deformable elements [1]
confirm the validity of the method and suggest that an elastic structure
can be realized buckling for dead, tensile loads. This structure has been
calculated, realized and tested and provides the first example of an
elastic structure buckling without elements subject to compression [2].
The perturbative method introduced for the analysis of shear bands can be
successfuly employed to investigate other material instabilities, such as
for instance flutter in a frictional, continuum medium [3]. In this
context, an experiment on an elastic structure subject to a frictional
contact shows for the first time that a follower load can be generated
using dry friction and that this load can induce flutter instability [4].
The perturbative approach may be used to investigate the strain state near
a dislocation nucleated in a metal subject to a high stress level [5].
Eshelby forces, similar to those driving dislocations in solids, are
analyzed on elastic structures designed to produce an energy release and
therefore to evidence configurational forces. These structures have been
realized and they have shown unexpected behaviours, which opens new
perspectives in the design of flexible mechanisms, like for instance, the
realization of an elastic deformable scale [6].
[1] D. Bigoni, Nonlinear Solid Mechanics Bifurcation Theory and Material
Instability. Cambridge Univ. Press, 2012, ISBN:9781107025417.
[2] D. Zaccaria, D. Bigoni, G. Noselli and D. Misseroni Structures
buckling under tensile dead load. Proc. Roy. Soc. A, 2011, 467, 1686.
[3] A. Piccolroaz, D. Bigoni, and J.R. Willis, A dynamical interpretation
of flutter instability in a continuous medium. J. Mech. Phys. Solids,
2006, 54, 2391.
[4] D. Bigoni and G. Noselli Experimental evidence of flutter and
divergence instabilities induced by dry friction. J. Mech. Phys.
Solids,2011,59,2208.
[5] L. Argani, D. Bigoni, G. Mishuris Dislocations and inclusions in
prestressed metals. Proc. Roy. Soc. A, 2013, 469, 2154 20120752.
[6] D. Bigoni, F. Bosi, F. Dal Corso and D. Misseroni, Instability of a
penetrating blade. J. Mech. Phys. Solids, 2014, in press.
16:00
Quantifying multimodality in gene regulatory networks
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.
16:00
Chaotic dynamics in a deterministic adaptive network model of attitude formation in social groups
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.
Mathematical questions in sustainability and resilience
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.
"Myco-fluidics": physical modeling of fungal growth and dispersal
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.
The effect of boundary conditions on linear and nonlinear waves
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.
Problems in free boundary Hele-Shaw and Stokes flows
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.
Mathematical modelling of abnormal beta oscillations in Parkinson’s disease
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.
Quasi-solution approach towards nonlinear problems
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.
Urban growth and decay
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.
Bottlenecks, burstiness and fat tails regulate mixing times of diffusion over temporal networks
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
Classifier ensembles: Does the combination rule matter?
Abstract
Combining classifiers into an ensemble aims at a more accurate and robust classification decision compared to that of a single classifier. For a successful ensemble, the individual classifiers must be as diverse and as accurate as possible. Achieving both simultaneously is impossible, hence compromises have been sought by a variety of ingenious ensemble creating methods. While diversity has been in the focus of the classifier ensemble research for a long time now, the importance of the combination rule has been often marginalised. Indeed, if the ensemble members are diverse, a simple majority (plurality) vote will suffice. However, engineering diversity is not a trivial problem. A bespoke (trainable) combination rule may compensate for the flaws in preparing the individual ensemble members. This talk will introduce classifier ensembles along with some combination rules, and will demonstrate the merit of choosing a suitable combination rule.
Random matrices and the asymptotic behavior of the zeros of the Taylor approximants of the exponential function
Abstract
The plan: start with an introduction to several random matrix ensembles and discuss asymptotic properties of the eigenvalues of the matrices, the last one being the so-called "Normal Matrix Model", and the connection described in the title will be explained. If all goes well I will end with an explanation of asymptotic computations for a new normal matrix model example, which demonstrates a form of universality.
(NOTE CHANGE OF VENUE TO L2)
Network dynamics and meso-scale structures
Abstract
The dynamics of networks of interacting systems depend intricately on the interaction topology. Dynamical implications of local topological properties such as the nodes' degrees and global topological properties such as the degree distribution have intensively been studied. Mesoscale properties, by contrast, have only recently come into the sharp focus of network science but have
rapidly developed into one of the hot topics in the field. Current questions are: can considering a mesoscale structure such as a single subgraph already allow conclusions on dynamical properties of the network as a whole? And: Can we extract implications that are independent of the embedding network? In this talk I will show that certain mesoscale subgraphs have precise and distinct
consequences for the system-level dynamics. In particular, they induce characteristic dynamical instabilities that are independent of the structure of the embedding network.
Leftovers are just fine
Abstract
After an MISG there is time to reflect. I will report briefly on the follow up to two problems that we have worked on.
Crack Repair:
It has been found that thin elastically weak spray on liners stabilise walls and reduce rock blast in mining tunnels. Why? The explanation seems to be that the stress field singularity at a crack tip is strongly altered by a weak elastic filler, so cracks in the walls are less likely to extend.
Boundary Tracing:
Using known exact solutions to partial differential equations new domains can be constructed along which prescribed boundary conditions are satisfied. Most notably this technique has been used to extract a large class of new exact solutions to the non-linear Laplace Young equation (of importance in capillarity) including domains with corners and rough boundaries. The technique has also been used on Poisson's, Helmholtz, and constant curvature equation examples. The technique is one that may be useful for handling modelling problems with awkward/interesting geometry.