Applied Mathematics Seminar

Seminars in 2015

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Seminars in 2015, Semester 2

Wednesday November 11

Geoff Vasil (The University of Sydney)

Introducing Dedalus: A new, efficient, accurate, and flexible toolkit for PDEs

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In spite of outward appearances, many of the partial differential equations used in contemporary applied mathematics, and the methods used to solve them, contain enough similarities that one may consider their implementation under a very general framework. In this talk, I describe an equation-agnostic apparatus that incorporates a wide range of possible solving schemes, accurate pseudo-spectral spatial representations, and the expressive python language. Flexibility is a requirement, not an afterthought. From a user perspective, setting up a new science problem entails (i) choosing a spectral basis for the domain; (ii) defining variables and parameters; (iii) symbolically entering equations; (iv) making a choice of solver; (v) defining on-the-fly analysis tasks; (vi) running the code. Dedalus runs efficiently on computing platforms ranging from laptops to large-scale supercomputers. In addition, Dedalus is a community development project. We encourage users to contribute functionality and adaptations. Thus far, Dedalus has primarily been used to study problems arising in astrophysical and geophysical fluid dynamics, but there exist many more potential applications. In the talk, I will describe the basic architecture and algorithms. I will also discuss some of the novel scientific applications that Dedalus is making possible.

Wednesday October 21 (3pm)

CPC Research (The University of Sydney)

CPC Research Projects

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Ed Hancock, Lake-Ee Quek and Alistair Senior will be speaking about their recent research in model reduction, metabolic dynamics and meta-analysis of multi-dimensional nutrition data. Ed, Lake-Ee and Alistair were recently appointed to the CPC as postdocs in Complex Systems and Modelling and their home School is Mathematics and Statistics. This seminar session is an opportunity for them to show us what they were working on and to identify some common ground for collaboration and for building our research portfolio, for the benefit of both the School and the CPC.

Ed Hancock: Dynamical systems and control in systems biology

Lake-Ee Quek: Metabolism, modelling and metabolites

Alistair Senior: Statistical and computational insights in to the evolution of nutritional strategies

Wednesday October 14

Rebecca Chisholm (University of New South Wales)

The evolutionary origins of tuberculosis

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Mycobacterium tuberculosis has an unusual natural history in that the vast majority of its human hosts enter a latent state that is both non-infectious and devoid of any symptoms of disease. From the pathogen perspective, it seems counterproductive to relinquish reproductive opportunities to achieve a dŽtente with the host immune response. However, a small fraction of latent infections regain their transmission potential by reactivating to the disease state. Thus, latency has been argued to provide a safe harbour for future infections which optimises the persistence of M. tuberculosis in human populations. Yet, a fundamental aspect of the evolution of latency in M. tuberculosis and other diseases remains elusive: if a pathogen began interactions with humans as an active disease without latency, how could it begin to evolve latency properties without putting itself at an immediate reproductive disadvantage? Alternatively, if the ancestral form of M. tuberculosis caused only primary progressive disease, it is intriguing that new strains adopting costly latency properties could outcompete the active and transmissible strains. I will discuss how we addressed this paradox with a mathematical model that can reconstruct how latency could have evolved in ancestral mycobacterial infections in early human populations. Results suggest that the emergence of tuberculosis latency may have been enabled by a mechanism akin to cryptic genetic variation in that detrimental latency properties were hidden from natural selection until their expression became evolutionarily favoured.

Wednesday September 23

Philip Maini (Oxford University)

Modelling Collective Cell Motion

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We will review three case studies of mathematical modelling of cell motion: (i) cancer cell invasion, (ii) cranial neural crest cell migration, (iii) intestinal crypt dynamics. In each case, the model takes a different form, ranging from coupled systems of partial differential equations in (i), to a hybrid agent-based model in (ii), to a discrete cell-based simulation model in (iii). It will be shown that in a number of cases these seemingly different approaches are underpinned by a nonlinear diffusion equation. The model results will be critiqued in light of experimental evidence.

Wednesday September 16

Mourad Ismail (University of Central Florida, King Saud University)

A class of 2D orthogonal polynomials

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We survey the literature on orthogonal polynomials in several variables starting from Hermite's work in the late 19th century to the works of Zernike (1920's) and Ito (1950's). We explore combinatorial and analytic properties of the Ito polynomials and offer a general class in 2 dimensions which has interesting structural properties. Connections with certain PDE's will be mentioned.

Wednesday September 9

Michelle Dunbar (University of Wollongong)

Improving Public Transport Accessibility via the Optimisation and Synchronisation of Schedules for Key Transport Modes

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As the population within modern metropolitan cities continues to grow, greater population dispersion means that daily commuters are increasingly faced with longer commute times and journeys consisting of multiple legs; often involving more than one mode of transport. In a bid to discourage the use of the private motor-car and facilitate the uptake of public transport, there is a developing trend towards the construction of centrally-located Transport Hubs, allowing passengers to connect with multiple modes of transport. To assist passengers in connecting with their outbound mode more efficiently, it is desirable to synchronise connecting modal services within the Transport Hub. In this presentation we consider the problem of designing shuttle-bus routes for passengers connecting with one of four different modes of transport at a Transportation Hub. We seek to minimise the average waiting time for passengers, the cost of missed connections at the Hub and the total travel time. Furthermore, we incorporate time-of-day effects and passenger heterogeneity with respect to value-of-time. In addition to commuters, the framework developed is amenable and directly extensible to the perishable good delivery problem for which items possess heterogeneity in delivery priority. Our model is posed as an extension of the vehicle routing problem with time windows, and solved using column generation. We provide a brief outline of our optimisation formulation and results for a number of datasets.

Wednesday August 12 (11am)

Martin Wechselberger (The University of Sydney)

The role of cell volume changes in normal and pathological dynamics of the brain

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I will discuss a recent study of the effect of cell volume on neural dynamics by incorporating cell volume changes together with dynamic ion concentrations and oxygen supply into Hodgkin-Huxley spiking dynamics. I will highlight to role of multiple time-scales structure in these extended conductance based models and show how transitions from normal to pathological states arise.

This is joint work with G. Ullah (Univ South Florida), Y. Wei (UC Riverside), M. Dahlem (Humboldt Univ) and S. Schiff (Penn State).

Wednesday July 29

James Maclaurin (The University of Sydney)

A general framework for stochastic traveling waves and patterns

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In this talk I present a general framework in which to rigorously study the effect of spatio-temporal noise on traveling waves and stationary patterns. In particular the framework can incorporate versions of the stochastic neural field equation that may exhibit traveling fronts, pulses or stationary patterns. To do this, I first formulate a local SDE that describes the position of the stochastic wave up until a discontinuity time, at which point the position of the wave may jump. I then study the local stability of this stochastic front, obtaining a result that recovers a well-known deterministic result in the small-noise limit. I finish with a study of the long-time behavior of the stochastic wave. This was done when I was a member of INRIA Sophia Antipolis, France
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