Past Industrial and Applied Mathematics Seminar

It is thought that the hairy legs of water walking arthropods are able to remain clean and dry because the flexibility of the hairs spontaneously moves drops off the hairs. We present a mathematical model of this bending-induced motion, or bendotaxis, and study how it performs for wetting and non-wetting drops. Crucially, we show that both wetting and non-wetting droplets move in the same direction (using physical arguments and numerical solutions). This suggests that a surface covered in elastic filaments (such as the hairy leg of insects) may be able to universally self-clean. To quantify the efficiency of this effect, we explore the conditions under which drops leave the structure by ‘spreading’ rather than translating and also how long it takes to do so.

Many species of insects adhere to vertical and inverted surfaces using footpads that secrete thin films of a mediating fluid. The fluid bridges the gap between the foot and the target surface. The precise role of this liquid is still subject to debate, but it is thought that the contribution of surface tension to the adhesive force may be significant. It is also known that the footpad is soft, suggesting that capillary forces might deform its surface. Inspired by these physical ingredients, we study a model problem in which a thin, deformable membrane under tension is adhered to a flat, rigid surface by a liquid droplet. We find that there can be multiple possible equilibrium states, with the number depending on the applied tension and aspect ratio of the system. The presence of elastic deformation  ignificantly enhances the adhesion force compared to a rigid footpad. A mathematical model shows that the equilibria of the system can be controlled via two key parameters depending on the imposed separation of the foot and target surface, and the tension applied to the membrane. We confirm this finding experimentally and show that the system may transition rapidly between two states as the two parameters are varied. This suggests that different strategies may be used to adhere strongly and then detach quickly.

We present a framework for the design, analysis and application of computational multiscale methods for slow-fast high-dimensional stochastic processes. We call these processes "microscopic'', and assume existence of an approximate "macroscopic'' model that captures the slow behaviour of a selected set of macroscopic state variables. The methodology combines short bursts of microscopic simulation with extrapolation at the macroscopic level. The methodology requires the careful study of a few key algorithmic ingredients. First, we need to properly initialise the microscopic system, based on a given macroscopic state and (possibly) a prior microscopic state that contains additional information about the system. Second, we need to control the variance of the noise that originates from the microscopic Monte Carlo simulation. Third, we need to analyse stability of the extrapolation step. We will discuss these aspects on two types of model problems -- scale-separated SDEs and kinetic equations -- and show the efficacity of the resulting methods in diverse applications, ranging from tumor growth to fusion energy.

The statistical physics governing phase-ordering dynamics following a symmetry breaking rst-order phase transition is an area of active research. The Coarsening/Ageing of the ensemble of phase domains, wherein irreversible annihilation or joining of domains yields a growing characteristic domain length, is an omniprescent feature whose universal characteristics one would wish to understand. Driven kinetic Ising models and growing nano-faceted crystals are theoretically important examples of such Coarsening (Ageing) Dynamical Systems (CDS), since they additionally break thermodynamic uctuation-dissipation relations. Power-laws for the growth in time of the characteristic size of domains, and a concomitant scale-invariance of associated length distributions, have so frequently been empirically observed that their presence has acquired the status of a principle; the so-called Dynamic-Scaling Hypothesis. But the dynamical symmetries of a given CDS- its Coarsening Group G - may include more than the global spatio-temporal scalings underlying the Dynamic Scaling Hypothesis. In this talk, I will present a recently developed theoretical framework (Ref.[1]) that shows how the symmetry group G of a Coarsening (ageing) Dynamical System necessarily yields G-equivariance (covariance) of its universal statistical observables. We exhibit this theory for a variety of model systems, of both thermodynamic and driven type, with symmetries that may also be Emergent (Ref. [2,3]) and/or Hidden. We will close with a magical theoretical coarsening law that combines Lorentzian and Parabolic symmetries!

References
[1] Lorentzian symmetry predicts universality beyond scaling laws, SJ Watson, EPL 118 (5), 56001, (Aug.2, 2017) Editor's Choice
[2] Emergent parabolic scaling of nano-faceting crystal growth Stephen J. Watson, Proc. R. Soc. A 471: 20140560 (2015)
[3] Scaling Theory and Morphometrics for a Coarsening Multiscale Surface, via a Principle of Maximal Dissipation", Stephen

Phytoplankton moving in the ocean, spermatozoa making their way  through the female reproductive tract and harmful bacteria that form biofilms on implanted medical devices interact with a surrounding fluid. Their length scales are small enough so that viscous effects dominate inertial effects allowing the resulting fluid dynamics to be described by the linear Stokes equations. However,  nonlinear behavior can occur because these structures are flexible and their form evolves with the flow. In addition, the fluid environment may also  be complex because of embedded microstructures that further complicate the dynamics.  We will discuss recent successes and challenges in describing these elastohydrodynamic systems.

Brain convolutions are specificity of mammals. Varying in intensity according to the animal species, it is measured by an index called the gyrification index, ratio between the effective surface of the cortex compared to its apparent surface. Its value is closed to 1 for rodents (smooth brain), 2.6 for new-borns and 5 for dolphins.  For humans, any significant deviation is a signature of a pathology occurring in fetal life, which can be detected now by magnetic resonance imaging (MRI). We propose a simple model of growth for a bilayer made of the grey and white matter, the grey matter being in cortical position. We analytically solved the Neo-Hookean approximation in the short and large wavelength limits. When the upper layer is softer than the bottom layer (possibly, the case of the human brain), the selection mechanism is dominated by the physical properties of the upper layer. When the anisotropy favours the growth tangentially as for the human brain, it decreases the threshold value for gyri formation. The gyrification index is predicted by a post-buckling analysis and compared with experimental data. We also discuss some pathologies in the model framework.

Discontinuous solutions, such as cracks or cavities, can suddenly appear in elastic solids when a limiting condition is reached. Similarly, self-contacting folds can nucleate at a free surface of a soft material subjected to a critical compression. Unlike other elastic instabilities, such as buckling and wrinkling, creasing is still poorly understood. Being invisible to linearization techniques, crease nucleation is a problem of high mathematical complexity.

In this talk, I will discuss some recent theoretical insights solving the quest for both the nucleation threshold and the emerging crease morphology.  The analytic predictions are in  agreement with experimental and numerical data. They prove a fundamental insight either for understanding the creasing onset in living matter, e.g. brain convolutions, or for guiding engineering applications, e.g. morphable meta-materials.

In this talk we consider a system of interacting Brownian particles. When diffusing particles interact with each other their motions are correlated, and the configuration space is of very high dimension. Often an equation for the one-particle density function (the concentration) is sought by integrating out the positions of all the others. This leads to the classic problem of closure, since the equation for the concentration so derived depends on the two-particle correlation function. We discuss two  common closures, the mean-field (MFA) and the Kirkwood-superposition approximations, as well as an alternative approach, which is entirely systematic, using matched asymptotic expansions (MAE). We compare the resulting (nonlinear) diffusion models with Monte Carlo simulations of the stochastic particle system, and discuss for which types of interactions (short- or long-range) each model works best. 

Understanding the evolution of a solidification front is important in the study of solidification processes. Mathematically, self-similar solutions exist to the Stefan problem when the liquid domain is assumed semi-infinite, and such solutions have been extensively studied in the literature. However, in the case where the liquid region is finite and sufficiently small, such of solutions no longer hold, as in this case two solidification fronts will move toward each other and interact. We present an asymptotic analysis for the two-front Stefan problem with a small amount of constitutional supercooling and compare the asymptotic results with numerical simulations. We finally discuss ongoing work on the same problem near the time when the two fronts are close to colliding.
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Silicon is produced from quartz rock in electrode-heated furnaces by using carbon as a reduction agent. We present a model of the heat and mass transfer in an experimental pilot furnace and perform an asymptotic analysis of this model. First, by prescribing a steady state temperature profile in the furnace we explore the leading order reactions in different spatial regions. We next utilise the dominant behaviour when temperature is prescribed to reduce the full model to two coupled partial differential equations for the time-variable temperature profile within the furnace and the concentration of solid quartz. These equations account for diffusion, an endothermic reaction, and the external heating input to the system. A moving boundary is found and the behaviour on either side of this boundary explored in the asymptotic limit of small diffusion. We note how the simplifications derived may be useful for industrial furnace operation.

Understanding the spatial distribution of organisms throughout an environment is an important topic in population ecology. We briefly review ecological questions underpinning certain mathematical work that has been done in this area, before presenting a few examples of spatially structured population models. As a first example, we consider a model of two species aggregation and clustering in two-dimensional domains in the presence of heterogeneity, and demonstrate novel aggregation mechanisms in this setting. We next consider a second example consisting of a predator-prey-subsidy model in a spatially continuous domain where the spatial distribution of the subsidy influences the stability and spatial structure of steady states of the system. Finally, we discuss ongoing work on extending such results to network-structured domains, and discuss how and when the presence of a subsidy can stabilize predator-prey dynamics."

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Compaction is a primary process in the evolution of a sedimentary basin. Various 1D models exist to model a basin compacting due to overburden load. We explore a multi-dimensional model for a basin undergoing mechanical and chemical compaction. We discuss some properties of our model. Some test cases in the presence of geological features are considered, with appropriate numerical techniques presented.

Soft active materials are largely employed to realize devices (actuators), where deformations and displacements are triggered by a wide range of external stimuli such as electric field, pH, temperature, and solvent absorption. The effectiveness of these actuators critically depends on the capability of achieving prescribed changes in their shape and size and on the rate of changes. In particular, in gel–based actuators, the shape of the structures can be related to the spatial distribution of the solvent inside the gel, to the magnitude and the rate of solvent uptake.

In the talk, I am going to discuss some results obtained by my group regarding surface patterns arising in the transient dynamics of swelling gels [1,2], based on the stress diffusion model we presented a few years ago [3]. I am also going to show our extended stress diffusion model suited for investigating swelling processes in fiber gels, and to discuss shape formation issues in presence of fiber gels [4-6].

[1]   A. Lucantonio, M. Rochè, PN, H.A. Stone. Buckling dynamics of a solvent-stimulated stretched elastomeric sheet. Soft Matter 10, 2014.

[2]   M. Curatolo, PN, E. Puntel, L. Teresi. Full computational analysis of transient surface patterns in swelling hydrogels. Submitted, 2017.

[3]   A. Lucantonio, PN, L. Teresi. Transient analysis of swelling-induced large deformations in polymer gels. JMPS 61, 2013.

[4]   PN, M. Pezzulla, L. Teresi. Anisotropic swelling of thin gel sheets. Soft Matter 11, 2015.

[5]   PN, M. Pezzulla, L. Teresi. Steady and transient analysis of anisotropic swelling in fibered gels. JAP 118, 2015.

[6]   PN, L. Teresi. Actuation performances of anisotropic gels. JAP 120, 2016.

Understanding the outcome of a collision between liquid drops (merge or bounce?) as well their impact and spreading over solid surfaces (splash or spread?) is key for a host of processes ranging from 3d printing to cloud formation. Accurate experimental observation of these phenomena is complex due to the small spatio-temporal scales or interest and, consequently, mathematical modelling and computational simulation become key tools with which to probe such flows.

Experiments show that the gas surrounding the drops can have a key role in the dynamics of impact and wetting, despite the small gas-to-liquid density and viscosity ratios. This is due to the formation of gas microfilms which exert their influence on drops through strong lubrication forces.  In this talk, I will describe how these microfilms cannot be described by the Navier-Stokes equations and instead require the development of a model based on the kinetic theory of gases.  Simulation results obtained using this model will then be discussed and compared to experimental data.

Considerable attention has been recently focused on the study of muscle tissues viewed as prototypes of new materials that can actively generate stresses. The intriguing mechanical properties of such systems can be linked to hierarchical internal architecture. To complicate matters further, they are driven internally by endogenous mechanisms supplying energy and maintaining non-equilibrium.  In this talk we review the principal mechanisms of force generation in muscles and discuss the adequacy of the available mathematical models.

What can maths tell us about this topic? Do mathematicians even have a seat at the table, and should we? What do we know about directed networks and dynamical systems that can contribute to this?

We consider the implications of the mathematical modelling and analysis of neurone-to-neurone dynamical complex networks. We explain how the dynamical behaviour of relatively small scale strongly connected networks lead naturally to non-binary information processing and thus to multiple hypothesis decision making, even at the very lowest level of the brain’s architecture. This all looks a like a a loose  coupled array of  k-dimensional clocks. There are lots of challenges for maths here. We build on these ideas to address the "hard problem" of consciousness - which other disciplines say is beyond any mathematical explanation for ever! 

We discuss how a proposed “dual hierarchy model”, made up from both externally perceived, physical, elements of increasing complexity, and internally experienced, mental elements (which we argue are equivalent to feelings), may support a leaning and evolving consciousness. We introduce the idea that a human brain ought to be able to re-conjure subjective mental feelings at will. An immediate consequence of this model  is that finite human brains must always be learning and forgetting and that any possible subjective internal feeling that might be fully idealised only with a countable infinity of facets, could never be learned completely a priori by zombies or automata: it may be experienced more and more fully by an evolving human brain (yet never in totality, not even in a lifetime). 

Networks of interacting oscillators give rise to collective dynamics such as localized frequency synchrony. In networks of neuronal oscillators, for example, the location of frequency synchrony could encode information. We discuss some recent persistence results for certain dynamically invariant sets called weak chimeras, which show localized frequency synchrony of oscillators. We then explore how the network structure and interaction allows for dynamic switching of the spatial location of frequency synchrony: these dynamics are induced by stable heteroclinic connections between weak chimeras. Part of this work is joined with Peter Ashwin (Exeter).

Many systems in nature consist of stochastically interacting agents or particles. Stochastic processes have been widely used to model such systems, yet they are notoriously difficult to analyse. In this talk I will show how ideas from statistics can be used to tackle some challenging problems in the field of stochastic processes.

In the first part, I will consider the problem of inference from experimental data for stochastic reaction-diffusion processes. I will show that multi-time distributions of such processes can be approximated by spatio-temporal Cox processes, a well-studied class of models from computational statistics. The resulting approximation allows us to naturally define an approximate likelihood, which can be efficiently optimised with respect to the kinetic parameters of the model. 

In the second part, we consider more general path properties of a certain class of stochastic processes. Specifically, we consider the problem of computing first-passage times for Markov jump processes, which are used to describe systems where the spatial locations of particles can be ignored.  I will show that this important class of generally intractable problems can be exactly recast in terms of a Bayesian inference problem by introducing auxiliary observations. This leads us to derive an efficient approximation scheme to compute first-passage time distributions by solving a small, closed set of ordinary differential equations.

In Soft Matter, octupolar order is not just an exotic mathematical curio. Liquid crystals have already provided a noticeable case of soft ordered materials for which a (second-rank) quadrupolar order tensor may not suffice to capture the complexity of the condensed phases they can exhibit. This lecture will discuss the properties of a third-rank order tensor capable of describing these more complex phases. In particular, it will be shown that octupolar order tensors come in two separate, equally abundant variants. This fact, which will be given a simple geometric interpretation, anticipates the possible existence of two distinct octupolar sub-phases. 

Bubble Dynamics

We shall discuss certain generalisations of the Rayleigh Plesset equation for bubble dynamics

Self-assembly of a filament by curvature-inducing proteins

We explore a simplified macroscopic model of membrane shaping by means of curvature-sensing proteins. Equations describing the interplay between the shape of a freely floating filament in a fluid and the adhesion kinetics of proteins are derived from mechanical principles. The constant curvature solutions that arise from this system are studied using weakly nonlinear analysis. We show that the stability of the filament’s shape is completely characterized by the parameters associated with protein recruitment and establish that in the bistable regime, proteins aggregate on the filament forming regions of high and low curvatures. This pattern formation is then followed by phase-coarsening that resolves on a time-scale dependent on protein diffusion and drift across the filament, which contend to smooth and maintain the pattern respectively. The model is generalized for multiple species of proteins and we show that the stability of the assembled shape is determined by a competition between proteins attaching on opposing sides.

In recent years, ideas from the world of partial differential equations (PDEs) have found their way into the arena of graph and network problems. In this talk I will discuss how techniques based on nonlinear PDE models, such as the Allen-Cahn equation and the Merriman-Bence-Osher threshold dynamics scheme can be used to (approximately) detect particular structures in graphs, such as densely connected subgraphs (clustering and classification, minimum cuts) and bipartite subgraphs (maximum cuts). Such techniques not only often lead to fast algorithms that can be applied to large networks, but also pose interesting theoretical questions about the relationships between the graph models and their continuum counterparts, and about connections between the different graph models.

The immune system finds very rare amounts of pathogens and responds against them in a timely and efficient manner. The time to find and respond against pathogens does not vary appreciably with the size of the host animal (scale invariant search and response). This is surprising since the search and response against pathogens is harder in larger animals.

The first part of the talk will focus on using techniques from computer science to solve problems in immunology, specifically how the immune system achieves scale invariant search and response. I use machine learning techniques, ordinary differential equation models and spatially explicit agent based models to understand the dynamics of the immune system. I will talk about Hierarchical Bayesian non-linear mixed effects models to simulate immune response in different species.

The second part of the talk will focus on taking inspiration from the immune system to solve problems in computer science. I will talk about a model that describes the optimal architecture of the immune system and then show how architectures and strategies inspired by the immune system can be used to create distributed systems with faster search and response characteristics.

I argue that techniques from computer science can be applied to the immune system and that the immune system can provide valuable inspiration for robust computing in human engineered distributed systems.

What makes cities successful? A complex systems approach to modelling urban economies

Urban centres draw a diverse range of people, attracted by opportunity, amenities, and the energy of crowds. Yet, while benefiting from density and proximity of people, cities also suffer from issues surrounding crime, congestion and density. Seeking to uncover the mechanisms behind the success of cities using novel tools from the mathematical and data sciences, this work uses network techniques to model the opportunity landscape of cities. Under the theory that cities move into new economic activities that share inputs with existing capabilities, path dependent industrial diversification can be described using a network of industries. Edges represent shared necessary capabilities, and are empirically estimated via flows of workers moving between industries. The position of a city in this network (i.e., the subnetwork of its current industries) will determine its future diversification potential. A city located in a central well-connected region has many options, but one with only few peripheral industries has limited opportunities.

We develop this framework to explain the large variation in labour formality rates across cities in the developing world, using data from Colombia. We show that, as cities become larger, they move into increasingly complex industries as firms combine complementary capabilities derived from a more diverse pool of workers. We further show that a level of agglomeration equivalent to between 45 and 75 minutes of commuting time maximizes the ability of cities to generate formal employment using the variety of skills available. Our results suggest that rather than discouraging the expansion of metropolitan areas, cities should invest in transportation to enable firms to take advantage of urban diversity.

This talk will be based on joint work with Eduardo Lora and Andres Gomez at Harvard University.

Hamilton-Jacobi-Bellman equations for dynamic pricing

I will discuss the Hamilton-Jacobi-Bellman (HJB) equation, which is a nonlinear, second-order, terminal value PDE problem. The equation arises in optimal control theory as an optimality condition.

Consider a dynamic pricing problem: over a given period, what is the best strategy to maximise revenues and minimise the cost of unsold items?

This is formulated as a stochastic control problem in continuous time, where we try to find a function that controls a stochastic differential equation based on the current state of the system.

The optimal control function can be found by solving the corresponding HJB equation.

I will present the solution of the HJB equation using a toy problem, for a risk-neutral and a risk-averse decision maker.

Cells adapt their metabolic state in response to changes in the environment.  I will present a systematic framework for the construction of flux graphs to represent organism-wide metabolic networks.  These graphs encode the directionality of metabolic fluxes via links that represent the flow of metabolites from source to target reactions.  The weights of the links have a precise interpretation in terms of probabilities or metabolite flow per unit time. The methodology can be applied both in the absence of a specific biological context, or tailored to different environmental conditions by incorporating flux distributions computed from constraint-based modelling (e.g., Flux-Balance Analysis). I will illustrate the approach on the central carbon metabolism of Escherichia coli, revealing drastic changes in the topological and community structure of the metabolic graphs, which capture the re-routing of metabolic fluxes under each growth condition.

By integrating Flux Balance Analysis and tools from network science, our framework allows for the interrogation of environment-specific metabolic responses beyond fixed, standard pathway descriptions.

Averaging, either spatial or temporal, is a powerful technique in complex multi-scale systems.

However, in some situations it can be difficult to justify.

For example, many real-world networks in technology, engineering and biology have a function and exhibit dynamics that cannot always be adequately reproduced using network models given by the smooth dynamical systems and fixed network topology that typically result from averaging. Motivated by examples from neuroscience and engineering, we describe a model for what we call a "functional asynchronous network". The model allows for changes in network topology through decoupling of nodes and stopping and restarting of nodes, local times, adaptivity and control. Our long-term goal is to obtain an understanding of structure (why the network works) and how function is optimized (through bifurcation).

We describe a prototypical theorem that yields a functional decomposition for a large class of functional asynchronous networks. The result allows us to express the function of a dynamical network in terms of individual nodes and constituent subnetworks.

The stability of structures continues to be scientifically fascinating and technically important.  Shell buckling emerged as one of the most challenging nonlinear problems in mechanics more than fifty years ago when it was intensively studied.  It has returned to life with new challenges motivated not only by structural applications but also by developments in the life sciences and in soft materials.  It is not at all uncommon for slightly imperfect thin cylindrical shells under axial compression or spherical shells under external pressure to buckle at 20% of the buckling load of the perfect shell.  A historical overview of shell buckling will be presented followed by a discussion of recent work by the speaker and his collaborators on the buckling of spherical shells.  Experimental and theoretical work will be described with a focus on imperfection-sensitivity and on viewing the phenomena within the larger context of nonlinear stability. 

A computational and asymptotic analysis of the solutions of Carrier's problem  is presented. The computations reveal a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurcations as the bifurcation parameter tends to zero. The method of Kuzmak is then applied to construct asymptotic solutions to the problem. This asymptotic approach explains the bifurcation structure identified numerically, and its predictions of the bifurcation points are in excellent agreement with the numerical results. The analysis yields a novel and complete taxonomy of the solutions to the problem, and demonstrates that a claim of Bender & Orszag is incorrect.