# Applied Mathematics Seminar

For First Semester 2021, seminars will be mostly held online. Zoom links will be provided to the mailing list.

## First Semester 2021

### May

Thursday May 20

Ryan Goh (Boston)

Title: Growing oblique stripes

Schedule: Online at 9am (Zoom link to be sent to mailing list)

Abstract: Spatial growth plays an important role in controlling pattern formation in many different types of physical systems. Specific examples include directional quenching in alloy melts, growing interfaces in biological systems, moving masks in ion milling, eutectic lamellar crystal growth, and traveling reaction fronts, and such growth processes have been shown to select and control spatially periodic patterns, while mediating defects. One of the simplest mechanisms of growth is a directional "quench" which travels across a domain, suppressing patterns in one part of the domain, and exciting them in the other. In this talk we will discuss how a linear phase diffusion equation with a nonlinear boundary condition posed on the half-plane can be used to characterize the formation of stripes oblique to the quenching interface. We use rigorous analysis, formal asymptotics, and numerical continuation to characterize stripe selection for various quenching speeds and stripe angles. Of particular interest, we find that the slow-growth, small-angle regime is governed by the glide-motion of a dislocation defect at the quench interface. Finally, we compare our results numerically to stripe formation in a quenched anisotropic Swift-Hohenberg equation, a prototypical model of pattern formation.

Thursday May 13

Reinout Quispel (La Trobe)

Title: How to discover properties of differential equations, and how to preserve those properties under discretization

Schedule: Carslaw 374 at 10:30am (In-person AND online! Zoom link to be sent to mailing list)

Abstract: This talk will be in two parts. The first part will be introductory, and will address the question: Given an ordinary differential equation (ODE) with certain physical/geometric properties (for example a preserved measure, first and/or second integrals), how can one preserve these properties under discretization? The second part of the talk will cover some more recent work, and address the question: How can one deduce hard to find properties of an ODE from its discretization?

Thursday May 6

Jie Yen Fan (Sydney)

Title: Multi-type age-structured population model

Schedule: Carslaw AGR at 10:30am (In-person AND online! Zoom link to be sent to mailing list)

Abstract: Population process in general setting, where each individual reproduces and dies depending on the state (such as age and type) of the individual as well as the entire population, offers a more realistic framework to population modelling. Formulating the population dynamics as a measure-valued stochastic process allows us to incorporate such dependence. The asymptotics, namely the law of large numbers and the central limit theorem, can be obtained. Some examples, including sexual reproduction and the spread of viral infection will be given.

### April

Thursday April 29

Claire Postlethwaite (Auckland)

Title: A heteroclinic network model of Rock-Paper-Scissors-Lizard-Spock

Schedule: Online at 9am. Zoom link to be sent to mailing list.

Abstract: The well-known game of Rock-Paper-Scissors can be used as a simple model of competition between three species. When modelled in continuous time using differential equations, the resulting system contains a heteroclinic cycle between three equilibrium solutions. The game can be extended in a symmetric fashion by the addition of two further strategies (‘Lizard’ and ‘Spock’): now each strategy is dominant over two of the four other strategies, and is dominated by the remaining two. The differential equation now contains a set of coupled heteroclinic cycles forming a heteroclinic network. In this talk I will discuss how we study the dynamics near this heteroclinic network. In particular, I will show how we are able to identify regions of parameter space in which arbitrarily long periodic sequences of visits are made to the neighbourhoods of the equilibria, and how these regions form a complicated pattern in parameter space.

Thursday April 22

Ryan Goh (Boston)

Title:

Schedule: (POSTPONED TO MAY 20)

Abstract:

Thursday April 15

Yury Stepanyants (USQ)

Title: The asymptotic approach to the description of two-dimensional soliton patterns in the oceans

Schedule: 3:30pm at Quad S227 (In-person AND online! Zoom link to be sent to mailing list)

Abstract: The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe a stationary moving wave patterns consisting of two plane solitary waves moving at an angle to each other. The results obtained within the approximate asymptotic theory is validated by comparison with the exact two-soliton solution of the Kadomtsev–Petviashvili equation. The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin–Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers.

Thursday April 1

Alex Townsend (Cornell)

Title: The art and science of low-rank techniques

Schedule: Online at 9am. Zoom link to be sent to mailing list.

Abstract: Matrices and tensors that appear in computational mathematics are so often well-approximated by low-rank objects. Since random ("average") matrices are almost surely of full rank, mathematics needs to explain the abundance of low-rank structures. We will give various methodologies that allow one to begin to understand the prevalence of compressible matrices and tensors and we hope to reveal underlying links between disparate applications. We will also show how the appearance of low-rank structures can be used in function approximation, fast transforms, and partial differential equation (PDE) learning.

### March

Thursday March 18

Jared Field (Melbourne)

Title: Gamilaraay Kinship Dynamics

Schedule: Carslaw 350 at 3:30pm (In-person AND online! Zoom link to be sent to mailing list)

Abstract: Traditional Indigenous marriage rules have been studied extensively since the mid-1800s. Despite this, they have historically been cast aside as having very little utility. Here, I will walk through some of the interesting mathematics of the Gamilaraay system and show that, instead, they are in fact a very clever construction. Indeed, the Gamilaraay system dynamically trades off kin avoidance -- to minimise incidence of recessive diseases -- against pairwise cooperation, as understood formally through Hamilton's rule.

## Second Semester 2020

### December

Wednesday December 2nd

Samuel Jelbart (Sydney)

Title: Extending the Scope of Geometric Singular Perturbation Theory

Abstract: Multi-scale phenomena in e.g., biology, engineering, and neuroscience are frequently described by singularly perturbed ordinary differential equations with solutions varying over vastly separated timescales, making their analysis a challenging problem. In recent decades, significant progress has been made via the development of Geometric Singular Perturbation Theory (GSPT), which provides a powerful theoretical framework for the analysis of such problems. When combined with a geometric method for the desingularisation of singularities known as blow-up, GSPT can provide a remarkably detailed and geometrically informative understanding of the dynamics.
However, GSPT in its standard form has several limitations which restrict the scope of its applicability. Discontinuity, exponential nonlinearities and 'hidden scales' all present significant challenges to the theory. In this talk, we will explore these limitations in the context of a simple electrical oscillator model - the Le Corbeiller oscillator - and show how they can be overcome using a combination of tools that are adapted from GSPT and the theory of piecewise-smooth (PWS) systems. Our aim will be to understand the onset of multi-scale 'relaxation oscillations' which converge to PWS cycles as a singular perturbation parameter tends to zero. Our main analytical tool is the blow-up method, which must be adapted to resolve degeneracy stemming from (i) the loss of smoothness and (ii) the presence of an essential singularity.
The talk will conclude with a brief discussion on the systematic implementation of these ideas, which have so far been developed only in the context of applications.

### November

Wednesday November 18th

James Meiss (UC Boulder)

Title: Computing Invariant Tori and Resonances using Birkhoff Averages

Abstract: Invariant tori are prominent features of Hamiltonian systems. In particular, integrable systems are foliated by tori with half the dimension of the phase space. KAM theory implies that many of these tori persist under smooth perturbations, but as the perturbations grow, only the robust'’ tori persist in the face of increasing chaos. Perhaps the simplest example of such dynamics is the case of area-preserving maps. John Greene conjectured that the locally most robust rotational circles have rotation numbers that are noble, i.e., have continued fractions with a tail of ones, and that, of these circles, the most robust has golden mean rotation number. For higher dimensional cases, the number theoretic properties of the robust tori are still unknown, though it has been conjectured that those with frequency vectors in cubic irrational field should be most robust.

We develop a method based on a weighted Birkhoff average (an idea due to, Das, Saiki, Sander, and Yorke) to identify chaotic orbits, resonances, and rotational invariant tori. It can quickly compute frequency vectors of the latter to machine precision. Variants of Chirikov’s standard map are used as 2D test cases. As a higher-dimensional test, we study a standard’' family of 3D, volume-preserving maps---to which KAM theory applies, and attempt to identify the most robust two-tori.

(This research is in collaboration with Evelyn Sander of George Mason University)

### October

Wednesday October 28th

Sandro Vaienti (Toulon)

Title: On some recent applications of extreme value theory to dynamical systems

Abstract: We review a few applications of extreme value theory to:
(i) open systems;
(ii) give the distribution of observables defined along the temporal evolution of a dynamical system.
Applications are given for the class of prevalent observables.

Wednesday October 21st

Guo Deng (Macquarie)

Title: Generation, propagation and interaction of solitary waves in integrable versus non-integrable nonlinear lattices

Abstract: The study of lattice dynamics, i.e., the motion of a spatially discrete system governed by a system of differential-difference equations, is a classical subject. Of particular interest are lattices that support the propagation of solitary waves. In this talk, we will compare the properties of two kinds of lattices, one integrable and one non-integrable: the Toda lattice and the Hertzian chain. As is well known, the Toda lattice is an integrable system and has exact soliton solutions. In contrast, the Hertzian chain, which has many physical and engineering applications, is a non-integrable system and no exact solitary-wave solutions are known. Here we will analyze the similarities and differences between the solitary waves in these two systems, we will discuss how each of these systems respond to a velocity perturbation, and we will compare the interaction dynamics of solitary waves. This is a joint work with Dr. Gino Biondini and Dr. Surajit Sen.

Wednesday October 14th

Erika Camacho (Arizona State)

Title: Analyzing two different mathematical models of cone metabolism

Abstract: Cell degeneration, including that resulting in retinal diseases such as retinitis pigmentosa and AMD, is linked to metabolic issues. In the retina, photoreceptor degeneration can result in disturbances of glucose levels and metabolic processes. To identify the key mechanisms in metabolism that may be culprits of this degeneration, we develop and investigate two mathematical models of the metabolic pathway of aerobic glycolysis in a healthy cone photoreceptor. We develop two different nonlinear systems of enzymatic functions and differential equations. In one case, we mathematically model cone molecular and photoreceptor cellular interactions. In the other case, within a single cone cell, we consolidate some of the metabolic processes in the glycolytic pathway and focus on the glucose, lactate, and pyruvate levels. We perform numerical simulations, use available metabolic data to estimate parameters and fit the models to this data. We conduct uncertainty and sensitivity analysis to identify the processes that have the largest impact on each system.

In the molecular and cellular level model, we consider the case of a healthy cone, a cone with low levels of glucose, and a cone with low and no rod-derived cone viability factor (RdCVF). The three key processes identified are metabolism of fructose-1,6-bisphosphate, production of glycerol-3-phosphate and competition that rods exert on cone resources. The first two processes are proportional to the partition of the carbon flux between glycolysis and the pentose phosphate pathway or the Kennedy pathway, respectively. The last process is the rods’ competition for glucose, which may explain why rods also provide the RdCVF signal to compensate.

In the other model, we use bifurcation techniques and identify a bistable regime, biologically corresponding to a healthy versus a pathological state. The system exhibits a saddle node bifurcation and hysteresis. Model simulations reveal the modulating effect of external lactate in bringing the system to steady state; the bigger the difference between external lactate and initial internal lactate concentrations, the longer the system takes to achieve steady state. Sensitivity analysis reveals that the rate of b-oxidation of ingested outer segment fatty acids consistently plays an important role in the concentration of glucose, G3P, and pyruvate, whereas the extracellular lactate level consistently plays an important role in the concentration of lactate and acetyl coenzyme A.

The ability of these mechanisms to affect key metabolites’ levels (as revealed in our analyses) signifies the importance of inter-dependent and inter-connected feedback processes modulated by and affecting cone’s metabolism.

### September

Wednesday , Sept 30 at 9am

Sarah Iams (Harvard)

Title: Patterned vegetation in drylands: satellite imagery and models

Abstract: Banded vegetation patterns are surprisingly common in drylands. In these ecosystems, water is a limiting nutrient, and ecohydrological processes are considered to be relevant to the formation and maintenance of patterns. Modeling challenges for capturing and predicting the evolution of these patterns include a lack of first principles mechanisms around which to build a model, and the range of timescales that are relevant to the system. Timescales include the rapid timescales of rainfall and runoff (minutes to hours), the seasonal timescales of plant growth and death (days to months), and the longer time scales associated with pattern evolution (years to decades). In this talk, I will discuss results from studies based on satellite data, I will provide a brief overview of reaction-advection-diffusion modeling approaches, and I will present models where we have worked to incorporate topographic variation and fast-slow switching to capture the dynamics of the system.

## First Semester 2020

Wednesday Mar 11, 2pm in Carslaw 373

Ian Melbourne (Warwick)

Title: Deterministic homogenization for fast-slow dynamical systems (DIFFERENT LOCATION: CARSLAW 175)

Abstract:  We consider deterministic fast-slow systems where the fast dynamics is assumed to be (non)uniform hyperbolic. The aim is to prove that the slow dynamics converges to a stochastic differential equation (with the correct interpretation of the stochastic integrals). Various results in this direction will be described.

Wednesday Mar 4, 2pm in Carslaw 373

Bob Rink (VU Amsterdam)

Title: Bifurcations in networks - is synchrony just symmetry?

Abstract:  Networks of coupled nonlinear dynamical systems arise as models throughout the sciences. Such network systems may display unexpected collective behaviour even if the individual dynamical systems that it is made of are quite simple. This behaviour includes (partial) synchronisation, where (some of) the agents of the network evolve in unison (think of the simultaneous firing of neurons). It has also been observed that synchrony in networks often emerges and breaks through unusual bifurcation scenarios. This talk is motivated by the question how we can predict and compute these synchrony breaking bifurcation scenarios from intrinsic geometric properties of the network. One of these geometric properties is an algebraic structure that we called "hidden network symmetry". This structure lets us invoke representation theory to determine the generic local behaviour of network dynamical systems.  This is joint work with Jan Sanders and Eddie Nijholt.

## Second Semester 2019

Wednesday Nov 20, 2pm in Carslaw 373

Ivan Graham (Bath)

Title: Uncertainty quantification for PDEs

Abstract: In this talk I'll give an overview of work in the uncertainty quantification of PDEs with random input data, where the main objective is to compute expected values of quantities of interest derived from the solutions of the PDEs. I'll give some practical examples and then I'll explain how, via parametrization, the random PDE can be written as a parametrized family of deterministic PDEs with parameter lying in a possibly (very) high dimensional space. Such problems can then be solved by sampling the PDE (often many times over) and then averaging, to obtain expected values.  A successful algorithm then consists of (a) making good choices of points in high-dimensional parameter space at which to sample the data, (b) computing the samples of the data, and (c) fast computation of samples of the PDE, very many of which may be needed.   In recent years there are many successful algorithms combining (a), (b) and (c) for some classes of PDEs, particularly the diffusion equation, and I'll describe a method which uses quasi-Monte Carlo for (a), circulant embedding for (b) and algebraic multigrid for (c). Recently I've been working on the frequency domain wave equation, which arises in the study of waves in random media. There the problems which arise are much more difficult particularly because there is no nice method to achieve (c), and so there are many open problems. I'll present some recent progress in this area.

Wednesday Nov 13, 2pm in Carslaw 373

Andrea Bertozzi (UCLA)

Title: A theory for undercompressive shocks in tears of wine

Abstract: We revisit the tears of wine problem for thin films in water-ethanol mixtures and present a new model for the climbing dynamics. The new formulation includes a Marangoni stress balanced by both the normal and tangential components of gravity as well as surface tension which lead to distinctly different behavior. The combined physics can be modeled mathematically by a scalar conservation law with a nonconvex flux and a fourth order regularization due to the bulk surface tension. Without the fourth order term, shock solutions must sastify an entropy condition - in which characteristics impinge on the shock from both sides. However, in the case of a nonconvex flux, the fourth order term is a singular perturbation that allows for the possibility of undercompressive shocks in which characteristics travel through the shock. We present computational and experimental evidence that such shocks can happen in the tears of wine problem, with a protocol for how to observe this in a real life setting.

Wednesday Oct 30, 2pm in Carslaw 373

Jan Obloj (Oxford)

Title: Optimal Transport with a Martingale Constraint: theory, applications and numerics

Abstract: Optimal transportation is a very rich and well-established field in mathematics. I consider here its variant where the transport has a direction and an additional martingale, or barycentre preservation, constraint. I will explain how this problem, called the Martingale Optimal Transport (MOT), arises naturally in (robust) financial mathematics and how it links with the classical Skorokhod embedding problem in probability. I will then discuss some recent results on structure of martingale transports. Finally, I will present recent advances on numerical methods for such problems via Linear Programming and/or deep Neural Networks methods. Based on joint works with Pietro Siorpaes and with Gaoyue Guo.

Wednesday Oct 25, 2pm in Carslaw 373

Jitesh Gajjar (Manchester)

Title: Problems with the numerical solution of initial value problems in triple-deck theory

Abstract: There are a number of example problems arising in triple-deck theory which we have attempted to solve recently where the solution technique seems to generate unexplained behaviour. For example in the classic supersonic compression ramp problem discussed in Logue, Gajjar & Ruban (2014) wavepackets appear in the results which do not appear to be related to anything physical. A similar situation arises in trying to solve the classic initial value vibrator problem of Terentev (1981). In this talk we discuss a closely related problem of boundary layer flow past localised heating elements. The linearised initial-value problem is solved analytically as well as numerically, and we will discuss the strange wavepacket behaviour which is also present.

Wednesday Oct 23, 2pm in the AGR room

Matthew Holden (Queensland)

Title: The Dynamics of Wildlife Crime

Abstract: Illegal harvest of wildlife (poaching) is one of the greatest threats to biodiversity. Most countries try to reduce poaching by increasing law enforcement to catch and punish poachers. But despite best efforts from police, poaching is more frequent now than ever. In this talk, we present simple ordinary differential equation models of poachers and wildlife, to explore why law-enforcement has failed to stem the poaching problem. We then use these models to project the performance of controversial, alternative, management actions, such as, campaigns to reduce consumer demand for illegal wildlife products, and legalising trade of these products.

Wednesday Oct 16, 2pm in the AGR room

Nathan Duignan (Sydney)

Title: On the Simultaneous Binary Collision

Abstract: We explore the work of my recent PhD thesis, namely, theory surrounding the singularity at a simultaneous binary collision in the 4-body problem. It is known that any attempt to remove the singularity via block regularisation will result in a regularised flow that is no more than  differentiable with respect to initial conditions. Through an example based analysis of planar systems, this concept of block regularisation is defined. Then, this curious loss of differentiability is investigated through a blow-up procedure and a new proof of the  regularity in the collinear problem is provided. In the process, it is revealed that the critical manifold from the blow-up consists of two manifolds of normally hyperbolic saddle singularities which are connected by a manifold of heteroclinics. By utilising recent work on transitions near such objects and their normal forms, an asymptotic series of the transition past the singularity is explicitly computed. It becomes remarkably apparent that the finite differentiability at  is due to the inability to construct a set of integrals local to the simultaneous binary collision.

Wednesday Oct 9, 2pm in the AGR room

Christopher Lustri (Macquarie)

Title: The Role of Stokes lines in Physical Systems

Abstract: Systems with small parameters are often studied using asymptotic techniques. Despite the ubiquity of these techniques, many classical asymptotic methods are unable to capture behaviour that occurs on an exponentially small scale, which lies "beyond all orders" of power series in the small parameter. Typically this does not cause any issues; this behavior is too small to have a measurable impact on the overall behaviour of the system. I will showcase two systems in which exponentially small contributions have a significant effect on the overall system behaviour.   The first system, which I will discuss in detail, will be nonlinear waves propagating through particle chains with periodic masses. I will show that it is typically possible for Toda and FPUT lattices for certain combinations of parameters - determined by the exponentially small system behaviour - to produce solitary waves that propagate indefinitely. The second system, which I will discuss more briefly, will be the shape of bubbles in a steadily translating Hele-Shaw cell. By studying exponentially small effects, it is possible to construct exotic bubble shapes which correspond to recent laboratory experiments.

Wednesday Sept 25, 2pm in the AGR room

Theo Vo (Monash)

Title: French ducks roam free across the brain!

Abstract: Rhythms in the brain are vital for all aspects of physiological development and function. At the cellular level, neural rhythms typically manifest as electrical signals known as bursts, consisting of long periods of inactivity interspersed with rapid trains of closely spaced action potentials. Bursts are the basic units of neural information and have been proposed to support numerous functional roles, such as synchronization between neuronal populations, attention, synaptic plasticity, and memory and awareness. In this seminar, we examine the dynamics of bursting from the viewpoint of canard (French: duck) theory. We explain the origins and properties of bursting in various contexts, such as in hormone and neurotransmitter secretion in the pituitary gland. We also discuss how the predictions from canard theory can be tested in vitro.

Wednesday Sept 18, 2pm in the AGR room

Title: Paradoxes across the scales: model reduction from fine-scale to coarse-scale dynamics

Abstract: Let's first prove that negative probabilities are OK! The question is when? and how? Answer: when modelling the dynamics of a high-D system by a low-D system---such model reduction is the theme of this talk.   Then let's discuss how averaging is unsound despite many claiming it is exact! Of course, such averaging underpins many conservation PDEs in space, and hence these PDEs may mislead!  Lastly, let's see how non-autonomous/stochastic systems have to be modelled by uncertain variables! Examples discussed include quasi-stationary probability, thin fluid films, shear dispersion, spatial birth and death, lattice systems, Brownian motion, and population models. Remember these paradoxes and their resolution whenever you consider dynamics at multiple levels of model resolution.

Wednesday Sept 4, 2pm in the AGR room

Martin Wechselberger (Sydney)

Title: Regularisation of shock waves in reaction-nonlinear diffusion models: a geometric singular perturbation theory approach

Abstract: Reaction-nonlinear diffusion models arising in the context of cell migration and population dynamics can exhibit the property of aggregation – or backward diffusion. While this is physically relevant, mathematically it causes such models to break down. The aggregation causes shocks to form, and the solutions are no longer computable.  To account for shocks, modellers have employed the technique of regularisation – adding additional small higher order terms to these models to smooth out the shocks. These regularisation techniques have been widely employed in models of chemical phase-separation, though they have gone relatively unnoticed in biological models until very recently.  We have developed techniques from the field of geometric singular perturbation theory to resolve similar issues of shock formation in a different class of models, so-called advection-reaction models (hyperbolic balance laws). In this presentation, we will tackle the question of existence and formation of shocks in regularised reaction-nonlinear diffusion models using geometric singular perturbation theory.

Wednesday August 21, 2pm in the AGR room

Imene Khames (INSA Rouen)

Title: Nonlinear Network Wave Equation: Periodic Solutions and Graph Characterizations

Abstract: We study the discrete nonlinear wave equation in arbitrary finite networks. This is a general model, where the usual continuum Laplacian is replaced by the graph Laplacian. We consider such a wave equation with a cubic on-site nonlinearity which is the discrete \Phi^4 model, describing a mechanical network of coupled nonlinear oscillators or an electrical network where the components are diodes or Josephson junctions.  In the first part, we investigate the extension of the linear normal modes of the graph Laplacian into nonlinear periodic orbits. Normal modes -whose Laplacian eigenvectors are composed uniquely of {1}, {-1,1} or {-1,0,1}- give rise to nonlinear periodic orbits for the discrete \Phi^4 model. We perform a systematic linear stability (Floquet) analysis of these orbits and show the modes coupling when the orbit is unstable. Then, we characterize graphs having Laplacian eigenvectors in {-1,1} and {-1,0,1} using graph spectral theory.  In the second part, we investigate periodic solutions that are exponentially (spatially) localized. Assuming a large amplitude localized initial condition on one node of the graph, we approximate its evolution by the Duffing equation. The rest of the network satisfies a linear system forced by the excited node. This approximation is validated by reducing the discrete \Phi^4 equation to the discrete nonlinear Schrodinger equation and by Fourier analysis. These results relate nonlinear dynamics to graph spectral theory.

Wednesday August 7, 2pm in the AGR room

Vera Roshchina (UNSW)

Title: Faces of convex sets: dimensions and regularity

Abstract: The facial structure of convex sets can be surprisingly complex, and unexpected irregularities of the arrangements of faces give rise to badly behaved sets and various counterexamples. In this talk I will focus on specific properties of facial structure that capture irregularities in the facial structure of the set (dimensions of faces, singularity degree, facial exposure and facial dual completeness). I will also talk about some classic results related to faces of convex sets, mention some new results and counterexamples and will relate this to several open problems in convex algebraic geometry and the geometry of polytopes.

## Previous seminars

### 2019

#### First Semester

Wednesday March 27, 2pm in the AGR room

Dr. Peter Cudmore (Systems Biology Laboratory, the University of Melbourne)

Title: On Emergence in Complex Physical Systems

Abstract: Many problems in biology, physics and engineering involve predicting and controlling complex systems, loosely defined as interconnected system-of-systems. Such systems can exhibit a variety of interesting non-equilibrium features such as emergence and phase transitions, which result from mutual interactions between nonlinear subsystems. Modelling these systems is a task in-and-of itself, as systems can span many physical domains and evolve of multiple time scales. Nonetheless, one wishes to analyse the geometry of these models and relate both qualitative and quantitative insights back to the physical system. Beginning with the modelling and analysis of a coupled optomechanical systems, this talk presents some recent results concerning the existence and stability of emergent oscillations. This forms the basis for a discussion of new directions in symbolic computational techniques for complex physical systems as a means to discuss emergence more generally.

Wednesday May 8, 2pm in the AGR room

Dr. Heather McCreadie (Aberystwyth University, UK)

Title: Autonomous Curve Fitting of the Dst index during Geomagnetic Storms

Abstract: A technique has been developed to fit all types of geomagnetic storms identified in the Dst. A lognormal fitting procedure may be used to describe any storm by setting the lognormal standard deviation to greater than 0.9. The fit needs to be constrained around the peak time and the scaling factor determined. This will enable an autonomous method for fitting any type of storm within the Dst. The unique factor identifying the relationship between the main and recovery phase of a storm is the lognormal mean.

Wednesday May 22, 2pm in the AGR room

Dr. Paul Griffiths (Coventry University, UK)

Title: Temperature dependent viscosity flows - analysis and applications

Abstract: In this talk we will consider two and three dimensional boundary layer flows. The stability of both the flat plate and rotating disk boundary layers will be discussed in the context of fluids that exhibit a variational viscosity. In particular, we will present linear stability results for fluids with viscosity that varies as a function of temperature. Numerical results (neutral curves, growth rates and energy analyses) will be supported by asymptotic predictions at large Reynolds numbers. The influence of an enforced axial flow will also be discussed in the context of Chemical Vapour Deposition (CVD).

Wednesday June 5, 2pm in the AGR room

Prof. Vladimir Dragovic (UT Dallas, USA)

Title: Triangular Schlesinger systems, Painleve VI equations, and superelliptic curves

Abstract: We study the Schlesinger system in the case when the unknown matrices of arbitrary size (p×p) are triangular and the eigenvalues of each matrix form an arithmetic progression with a rational difference q, the same for all matrices. We show that such a system possesses a family of solutions expressed via periods of meromorphic differentials on the Riemann surfaces of superelliptic curves. We determine the values of the difference q, for which our solutions lead to explicit polynomial or rational solutions of the Schlesinger system. As an application of the (2 × 2)-case, we obtain explicit sequences of rational solutions and one-parameter families of rational solutions of Painleve VI equations. This is a joint work with Renat Gontsov and Vasilisa Shramchenko.

Wednesday 12 of June 2019, 2pm in the AGR room

Dr. Vijay Rajagopal (Dept. of Biomedical Engineering, University of Melbourne)

Title: Dissecting the role of the internal architecture of cardiac cells on calcium signaling in the heart using computational models.

Abstract: Calcium plays a central role in how our hearts beat. Each heartbeat is governed by the cyclic rise and fall of calcium in the cell cytoplasm through various co-ordinated and tightly regulated electrical and chemical processes. Calcium also plays a crucial role in determining when our heart muscle will grow in order to increase the force with which the heart beats as long-term demand for blood supply is increased. This process of cell and heart muscle growth is termed hypertrophy and is analogous to how our skeletal muscles grow through weight training. Exactly how calcium can regulate beat-to-beat muscle contraction and also send a signal to the cell nucleus for long-term growth is unclear. In this talk I will present our research into how the spatial organisation of ion-channels that govern calcium concentration in the cytoplasm affect calcium dynamics in the cell for beat-to-beat contraction and hypertrophic growth.

### 2018

#### Second Semester

Wednesday October 10, 2pm in the AGR room

Dr. Minh-Ngoc Tran (Business School, University of Sydney)

Title: Bayesian Deep Net GLM and GLMM

Abstract: Deep feedforward neural networks (DFNNs) are a powerful tool for functional approximation. We describe flexible versions of generalized linear and generalized linear mixed models incorporating basis functions formed by a DFNN. Efficient computational methods for high-dimensional Bayesian inference are developed using Gaussian variational approximation, with a parsimonious but flexible factor parametrization of the covariance matrix. We implement natural gradient methods for the optimization, exploiting the factor structure of the variational covariance matrix in computation of the natural gradient. Our flexible DFNN models and Bayesian inference approach lead to a regression and classification method that has a high prediction accuracy, and is able to quantify the prediction uncertainty in a principled and convenient way. We also describe how to perform variable selection in our deep learning method. The proposed methods are illustrated in a wide range of simulated and real-data examples, and the results compare favourably to a state of the art flexible regression and classification method in the statistical literature, the Bayesian additive regression trees (BART) method. User-friendly software packages in Matlab and R implementing the proposed methods are available at https://github.com/VBayesLab.

Wednesday September 19, 2pm in the AGR room

Dr. Lachlan Smith (University of Sydney)

Title: Chaos and the flow capture problem: Polluting is easy, cleaning is hard

Abstract: Where do you place pollutant capture units? When objects move through heterogeneous flow environments, such as oceanic micro-plastics, the answer is not obvious. We formulate flow capture problems, involving flows and sinks, and, using dynamical systems techniques, show that blindly positioning capture units carries high risk of failure. Capture efficiency depends on capture rate: long-term efficiency decreases as the number of capture units increases, whereas short-term efficiency increases. Doubling numbers of capture units can more than double the capture rate. The formal description of flow capture problems will impact engineering solutions ranging from atmospheric CO2 capture to oceanic micro-plastic pollution.

Wednesday August 29, 1pm (ONE HOUR EARLIER THAN USUAL!) in the AGR room

Dr. Robyn Araujo (Queensland Univ. of Tech.)

Title: Robust Perfect Adaptation in Complex Bionetworks

Abstract: Robustness, and the ability to function and thrive amid changing and unfavourable environments, is a fundamental requirement for all living systems. Moreover, it has been a long-standing mystery how the extraordinarily complex communication networks inside living cells, comprising thousands of different interacting molecules, are able to exhibit such remarkable robustness since complexity is generally associated with fragility. In this talk I will give an overview of our recent research on robustness in cellular signalling networks, with an emphasis on the robust functionality known as Robust Perfect Adaptation (RPA). This work is now published in Nature Communications, and is available here: https://rdcu.be/M46K. This work has suggested a resolution to the complexity-robustness paradox through the discovery that robust adaptive signalling networks must be decomposable into topological basis modules of just two possible types. This newly-discovered modularisation of complex bionetworks has important implications for evolutionary biology, embryology and development, cancer research and drug development.

Wednesday September 5, 2pm in the AGR room

Dr. Justin Tzou (Macquarie University)

Title: Stability analysis of localised patterns in two and three spatial dimensions

Abstract: We present a matched asymptotics framework for constructing and analysing the stability of localised patterns that arise in singularly perturbed activator-inhibitor reaction-diffusion systems. In two spatial dimensions, by way of analyses of nonlocal eigenvalue problems, we resolve two long-standing problems regarding 1) the stability of spot patterns to oscillatory instabilities, and 2) the stability of stripe patterns to break-up instabilities, the latter motivated by the persistence of striped vegetation patterns on steep hillsides. In three spatial dimensions, we calculate explicit stability thresholds for self-replication and annihilation of spots, and derive a gradient flow that governs their slow dynamics. Joint work with Theodore Kolokolnikov, Michael J. Ward, and Shuangquan Xie.

#### First semester

Monday (!) March 26, 2pm in the AGR room

Prof. Gunther Uhlmann (University of Washington)

Title: Journey to the Center of the Earth

Abstract: We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also several applications in optics and medical imaging among others.

The problem can be recast as a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.

We will also describe some recent results, join with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary. No previous knowledge of Riemannian geometry will be assumed.

Wednesday February 21, 2pm in the AGR room

Prof. Herbert Huppert (University of Cambridge)

Title: How to frack into and out of trouble.

Abstract: After a short introduction to the mechanism and politics of fracking, the talk will concentrate on the fluid mechanics and elastodynamics of driving fluid into cracks and the quite different response when the pressure is released and the fluid flows back out. Development of the governing equations will be presented along with their numerical solution and asymptotic analysis in certain useful limits. Videos of laboratory experiments will be shown and the results compared with the theoretical predictions.

Wednesday March 7, 2pm in the AGR room

Prof. Martin Wechselberger (Applied Maths, University of Sydney)

Title: Two-stroke relaxation oscillators

Abstract: In classic van der Pol-type oscillator theory, a relaxation cycle consists of two slow and two fast orbit segments per period (slow-fast-slow-fast). A possible alternative relaxation oscillator type consists of one slow and one fast segment only. In electrical circuit theory, Le Corbeiller (published in IEEE 1960) termed this type a two-stroke oscillator (compared to the four-stroke vdP oscillator). I will provide examples of two-stroke relaxation oscillators and discuss these problems from a geometric singular perturbation theory point of view "beyond the standard form". It is worth mentioning that Fenichel's seminal work on geometric singular perturbation theory (published in JDE 1979) discusses this more general setting, but it has not received much attention in the literature.

Wednesday March 14, 2pm in the AGR room

Prof. Dmitry Pelinovsky (McMaster University, Canada)

Title: Rogue periodic waves in the focusing MKDV and NLS equations

Abstract: Rogue periodic waves stand for gigantic waves on a periodic background. The nonlinear Schrodinger equation in the focusing case admits two families of periodic wave solutions expressed by the Jacobian elliptic functions dn and cn. Both periodic waves are modulationally unstable with respect to long-wave perturbations. Exact solutions for the rogue periodic waves are constructed by using the explicit expressions for the periodic eigenfunctions of the Zakharov–Shabat spectral problem and the Darboux transformations. These exact solutions generalize the classical rogue wave (the so-called Peregrine’s breather). Computations of rogue periodic waves rely on properties of the nonlinear Schrodinger equation due to its integrability.

### 2017

#### Second semester

Monday December 11, 2pm in the AGR room

Dr. Yulia Peet (Arizona State University)

Title: Overlapping and Moving Grid Approaches with Spectral-Element Methods: Concepts and Applications

Abstract: In this talk, we present our recent development of overlapping and moving grid methodology for high-fidelity computations of fluid flow problems with spectral element methods. Spectral element methods belong to a class of high-order methods that combine exponential convergence of global spectral methods with geometrical flexibility of finite-element methods. High-order methods possess low dissipation and low dispersion errors and are well suited for high-accuracy simulations of turbulent flows. The current development of overlapping and moving grid approaches enables the application of spectral element methods to a larger class of problems that involve moving bodies and complex geometries. In this talk, the fundamental concepts of both the spectral element methodology and the overlapping grid approach will be discussed, followed by a description and analysis of the methods that we have developed, paying a special attention to the concepts of stability and accuracy of the proposed methodology. We will proceed by discussing an application of the developed method to Direct Numerical Simulations of airfoil dynamic stall in the presence of upstream disturbances. We conclude by showing further potential extensions of the current methodology and list new applications that can be successfully tackled with this method.

Wednesday July 19

Prof. Boris Khesin (Department of Mathematics, University of Toronto, Canada)

Title: Hamiltonian dynamics of vortex membranes

Abstract: We show that an approximation of the hydrodynamical Euler equation describes the skew-mean-curvature flow on vortex membranes in any dimension. This generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for dynamics of higher-dimensional vortex filaments and vortex sheets as singular 2-forms (Green currents) with support of codimensions 2 and 1, respectively.

Wednesday July 26

Dr. Marianito Rodrigo (School of Mathematics and Applied Statistics University of Wollongong)

Title: On a fractional matrix exponential and an explicit method for its calculation

Abstract: The matrix exponential arises in many applications, particularly in the solution of linear systems of ordinary differential equations. The nth derivative of the matrix exponential is equal to the nth power of the matrix multiplied by the matrix exponential. What is the analogue of this when the ordinary derivative is replaced by a fractional derivative? In this talk I will define a fractional matrix exponential and then give an explicit method for calculating the fractional matrixexponential. An overview of the fractional calculus will be given.

Wednesday August 2

A/Prof Zhi-An Wang (Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong)

Title: Boundary layers arising from chemotaxis models

Abstract: The original well-known Keller-Segel system describing the chemotactic wave propagation remains poorly understood in many aspects due to the logarithmic singularity. As the chemical assumption rate is linear, the singular Keller-Segel model can be converted, via a Cole-Hopf type transformation, into a system of viscous conservation laws without singularity. In this talk, we first consider the dynamics of the transformed Keller-Segel system in a bounded interval with time-dependent Dirichlet boundary conditions. By imposing some conditions on the boundary data, we show that boundary layer profiles are present as chemical diffusion tends to zero and large-time profile of solutions will be determined by the boundary data (i.e. boundary stabilization). We employ the refined (weighted) energy estimates with the "effective viscous flux" technique to show the emergence of boundary layer profiles. For asymptotic dynamics of solutions, we develop a new idea by exploring the convexity of an entropy expansion to get the basic $L^1$-estimate, on which our results are built up by the method of energy estimates. Finally we gain the results for the original singular Keller-Segel system by reversing the Cole-Hopf transformation. Numerical simulations are performed to interpret our analytical results and their implications.

Wednesday August 9

Prof Kenji Kajiwara (Institute of Mathematics for Industry, Kyushu University, Japan)

Title: Construction and simulation of discrete integrable model for soil water infiltration problem

Abstract: In this talk, we propose an integrable model and its discretization describing one-dimensional soil water infiltration problem. The model is formulated as the nonlinear boundary value problem for a nonlinear diffusion-convection equation, which is transformed to the Burgers equation by a certain independent variable transformation incorporating the dependent variable, called the hodograph transformation or the reciprocal transformation. We construct the discrete model preserving the underlying integrability nature and formulate it as the self-adaptive moving mesh scheme. If we require the numerical stability and high-precision coincidence with the special case where the exact solution is obtained, we need some investigation and modification of the discrete model from the point of view different from integrability. We discuss this point and show some numerical results.

This talk is based on the paper arXiv:1705.03129 by D. Triadis (Kyushu/La Trobe) , P. Broadbridge (La Trobe), K. Maruno (Waseda) and myself.

Wednesday September 6

Dr. Sophie Calabretto (Department of Mathematics,Macquarie University)

Title: Flow external to a rotating torus (or a sphere)

Abstract: The unsteady flow generated due to the impulsive motion of a torus or sphere is a paradigm for the study of many temporally developing boundary layers. The boundary layer is known to exhibit a finite-time singularity at the equator. We present results of a study that focuses upon the behaviour of the flow after the onset of this singularity. Our computational results demonstrate that the singularity in the boundary layer manifests as the ejection of a radial jet. This radial jet is preceded by a toroidal starting vortex pair, which detaches and propagates away from the sphere. The radial jet subsequently develops an absolute instability, which propagates upstream towards the sphere surface.

Wednesday September 13

Dr. Ananta K. Majee (Mathematisches Institut, University of Tuebingen, Germany)

Title: On stochastic optimal control in ferromagnetism

Abstract: In this presentation, we study an optimal control problem for the stochastic Landau-Lifshitz-Gilbert equation on a bounded domain in R^d (d = 1, 2, 3). We first establish existence of a relaxed optimal control for relaxed version of the problem. As the control acts in the equation linearly, we then establish existence of an optimal control for the underlying problem. Furthermore, convergence of a structure presrving finite element approximation for d = 1 and physically relevant computational studies will be discussed.

Wednesday October 4

Dr. Lewis Mitchell (School of Mathematical Sciences, University of Adelaide)

Title: Information flows in online social networks

Abstract: The flow of information online is a significant factor in social contagion, rumour and “fake news” propagation, and protest organisation. Further, online social platforms provide a unique opportunity for computational social scientists to observe individuals’ spontaneous interactions over social ties, often through structural or temporal proxies for information. However, such approaches do not leverage the full extent of information available, namely the time-ordered textual content of messages. Here we apply information-theoretic techniques to social media data to identify the extent to which predictive information is encoded in social ties, and that in principle one can profile an individual from their contacts even if the individual is hidden'' within the network. Analysis of Twitter users shows that 95% of the potential predictive accuracy attainable for an individual is embedded within their social ties, and numerical simulations on a paradigmatic model of information flow shows that these techniques are robust.

Wednesday October 11

Dr. David A. Smith (Science (Mathematics), Yale-NUS, Singapore)

Title: Nonlocal Problems for Linear Evolution Equations

Abstract: Linear evolution equations, such as the heat and linearized KdV equations, are commonly studied on finite spatial domains via initial-boundary value problems. Typically, the boundary conditions specify information about the solution and its derivatives at the edges of the spatial domain. Alternatively, in place of the boundary conditions, consider "multipoint conditions", where one specifies some linear combination of the solution and its derivatives evaluated at internal points of the spatial domain. A further generalization is the "nonlocal" specification of the integral over space of the solution against some continuous weight. We describe a general framework for studying such problems, and provide solution representations for 2nd and 3rd order examples.

#### First semester

Wednesday January 25

Dr. Paul Griffiths (Oxford Brookes University, UK)

Title: Shear-thinning: A stabilising effect? Yes, no, maybe?

Abstract: In this talk we will investigate how viscosity effects the stability of a fluid flow. By assuming a shear-thinning viscosity relationship, where an increase in shear-rate results in a decrease in fluid viscosity, we show that flows can be both stabilised or destabilised, depending on (i) the fluid model in question and (ii) the ‘amount’ of shear-thinning the fluid exhibits. Using a two-dimensional boundary-layer flow as our ‘toy model’ we are able to show equivalence between different shear-thinning models. The effect shear-thinning has on important parameters such as the critical Reynolds number, and the maximum frequency of the disturbances will be discussed and interpreted in the wider context.

Wednesday February 22

Dr. Maria Vlassiou (Eindhoven University of Technology, Netherlands)

Title: Heavy-traffic limits for layered queueing networks

Abstract: Heavy-traffic limits for queueing networks are a topic of continuing interest. Presently, the class of networks for which these limits have been rigorously derived is restricted. An important ingredient in such work is the demonstration of state space collapse (SSC), which loosely speaking shows that in diffusion scale the queuing process for the stochastic model can be approximately recovered as a continuous lifting of the workload process. This often results in a reduction of the dimensions of the original system in the limit, leading to improved tractability. In this talk, we discuss diffusion approximations of layered queuing networks through two examples.

In the first example, we establish a heavy-traffic limit through SSC for a computer network model. For this model, SSC is related to an intriguing separation of time scales in heavy traffic. The main source of randomness occurs at the top layer; the interactions at the other layer are shown to converge to a fixed point at a faster time scale.

The second example focuses on a network of parallel single-server queues, where the speeds of the servers are varying over time and governed by a single continuous-time Markov chain. We obtain heavy-traffic limits for the distributions of the joint workload, waiting-time and queue length processes. We do so by using a functional central limit theorem approach, which requires the interchange of steady-state and heavy-traffic limits. For this model, we show that the SSC property does not hold.

Wednesday March 1

Dr. Daniel Lecoanet (Princeton University, Princeton, USA)

Title: Measuring Core Stellar Magnetic Field using Wave Conversion

Abstract: By studying oscillation modes at the surface of stars, astrophysicists are able to infer characteristics of their deep interior structure. This was first applied to observations of the Sun, but recently space-based telescopes have measured oscillations in many other stars, leading to many new mysteries in stellar structure and evolution. Recent work has suggested that low dipole oscillation amplitudes in evolved red giant branch stars may indicate strong core magnetic fields. Here we present both numerical simulations and analytic calculations of the interactions of waves with a strong magnetic field. We can solve the problem very accurately by using the WKB approximation to reduce the 2D PDE into a series of ODEs for different heights. We find that magnetic fields convert the buoyancy-driven waves observable at the surface of the star to magnetic waves, which are not present at the surface, in agreement with observations.

Wednesday March 22

Sheehan Olver (School of Mathematics and Statistics, University of Sydney)

Title: Solving PDEs on triangles using multivariate orthogonal polynomials

Abstract: Univariate orthogonal polynomials have long history in applied and computational mathematics, playing a fundamental role in quadrature, spectral theory and solving differential equations with spectra methods. Unfortunately, while numerous theoretical results concerning multivariate orthogonal polynomials exist, they have an unfair reputation of being unwieldy on non-tensor product domains. In reality, many of the powerful computational aspects of univariate orthogonal polynomials translate naturally to multivariate orthogonal polynomials, including the existence of Jacobi operators and the ability to construct sparse partial differential operators, a la the ultrapsherical spectral method [Olver & Townsend 2012]. We demonstrate these computational aspects using multivariate orthogonal polynomials on a triangle, including the fast solution of general partial differential equations.

Wednesday April 5

Professor Shige Peng (Shandong University, Jinan, China)

Title: Backward Stochastic Differential Equations Driven by G-Brownian Motion in Finance

Abstract: We present some recent developments in the theory of Backward Stochastic Differential Equations (BSDEs) driven by a new type of a Brownian motion under a nonlinear expectation space and we discuss applications of this new class of BSDEs to financial models in which the uncertainty of volatility is taken into account.

Wednesday April 12

Professor Holger Dullin (School of Mathematics, University of Sydney)

Title: A new twisting somersault - 513XD

Abstract: Abstract: Modelling an athlete as a system of coupled rigid body we derive a time-dependent reduced Euler equation for the dynamics of shape changing bodies. Reconstruction allows to recover the full dynamics in SO(3), and the number of somersaults is decomposed into a geometric phase and a dynamics phase. A kick model is used to approximate the dynamics, and using the insight gained from this we propose a new 10 meter platform twisting somersault dive (FINA code 513XD) that incorporates 5 full twists.

Wednesday April 19 Different Location! Carslaw room 535

Prof. Nihat Ay (Max-Planck-Institute for the Mathematics in the Sciences, Leipzig, Germany)

Title:Information Geometry and its Application to Complexity Theory

Abstract: In the first part of my talk, I will review information-geometric structures and highlight the important role of divergences. I will present a novel approach to canonical divergences which extends the classical definition and recovers, in particular, the well-known Kullback-Leibler divergence and its relation to the Fisher-Rao metric and the Amari-Chentsov tensor.

Divergences also play an important role within a geometric approach to complexity. This approach is based on the general understanding that the complexity of a system can be quantified as the extent to which it is more than the sum of its parts. In the second part of my talk, I will motivate this approach and review corresponding work.

References:

1. N. Ay, S.I. Amari. A Novel Approach to Canonical Divergences within Information Geometry. Entropy (2015) 17: 8111-8129.
2. N. Ay, J. Jost, H. V. Le, L. Schwachhöfer. Information geometry and sufficient statistics. Probability Theory and Related Fields (2015) 162: 327-364.
3. N. Ay, J. Jost, H. V. Le, L. Schwachhöfer. Parametrized measure models. Bernoulli (2016) accepted. arXiv:1510.07305.
4. N. Ay, J. Jost, H. V. Le, L. Schwachhöfer. Information geometry. Ergebnisse der Mathematik und Ihrer Grenzgebiete/A Series of Modern Surveys in Mathematics, Springer 2017, forthcoming book.
5. N. Ay. Information Geometry on Complexity and Stochastic Interaction. Entropy (2015) 17(4): 2432-2458.

Wednesday April 26

Professor Robert Dewar (Research School of Physics & Eng., Australian National Univ., Canberra)

Title:Variational constructions of almost-invariant tori for 1 1/2-D Hamiltonian systems

Abstract: Action-angle variables are normally defined only for integrable systems, but in order to describe 3D magnetic field systems a generalization of this concept was proposed recently [1,2] that unified the concepts of ghost surfaces and quadratic-flux-minimizing (QFMin) surfaces (two strategies for minimizing action gradient). This was based on a simple canonical transformation generated by a change of variable, $\theta = \theta(\Theta ,\zeta)$, where $\theta$ and $\zeta$ (a time-like variable) are poloidal and toroidal angles, respectively, with $\Theta$ a new poloidal angle chosen to give pseudo-orbits that are (a) straight when plotted in the $\zeta,\Theta$ plane and (b) QFMin pseudo-orbits in the transformed coordinate. These two requirements ensure that the pseudo-orbits are also (c) ghost pseudo-orbits, but they do not uniquely specify the transformation owing to a relabelling symmetry. Variational methods of solution that remove this lack of uniqueness are discussed.

[1] R.L. Dewar and S.R. Hudson and A.M. Gibson, Commun. Nonlinear Sci. Numer. Simulat. 17, 2062 (2012) http://dx.doi.org/10.1016/j.cnsns.2011.04.022
[2] R.L. Dewar and S.R. Hudson and A.M. Gibson, Plasma Phys. Control. Fusion 55, 014004 (2013) http://dx.doi.org/10.1088/0741-3335/55/1/014004

Wednesday May 3

Prof. Michael Small (The University of Western Australia)

Title: Communities Within Networks

Abstract: Many complex systems are naturally represented as networks which lack an underlying geodesic space. That is, elements of the network are naturally represented by their interconnection and not by their position in any real space. A favourite problem in complex systems is then how best to infer sensible communities from the network adjacency matrix. To be able to better frame this question, we first need to more precisely say something about what we mean by "sensible" communities. The usual way to do this is to define a statistical measure that quantifies the relative number of inter- to intra- community links - which we call "modularity". With this in mind, there are several methods one can apply to choose suitable sets of communities which achieve local optimality of this measure. I will describe some standard methods and some of our own approaches to this problem. Most recently we have developed methods that embed the network in a suitable geodesic space and then borrow ideas from computational clustering algorithms to detect communities (joint work with Arif Mahmood, formerly of UWA now with Qatar University). If I get time, I hope to finish by spending a few minutes talking about generative algorithms for networks with communities - the problem here is that while we have algorithms to generate networks with specific "nice" properties (preferential attachment, for example), and we have algorithms to generate communities, the algorithms to generate "nice" networks with communities are rather clunky.

Wednesday May 17

Dr. Milena Radnovic (The University of Sydney)

Title:Geometry, billiards, integrability.

Abstract: Starting from the celebrated Poncelet porism, we will present classical and modern results concerning integrable billiards.

Wednesday May 24

Dr. David Galloway (The University of Sydney)

Title:Slow-burning instabilities of Dufort-Frankel finite differencing.

Abstract: Du Fort-Frankel is a tactic to stabilise Richardson's unstable 3-level leapfrog time-stepping scheme. By including the next time level in the right hand side evaluation, it is implicit, but it can be re-arranged to give an explicit updating formula, thus apparently giving the best of both worlds. Textbooks prove unconditional stability for the heat equation, and extensive use on a variety of advection-diffusion equations has produced many useful results. Nonetheless, for some problems the scheme can fail in an interesting and surprising way, leading to instability at very long times. An analysis for a simple teaching problem involving a pair of evolution equations that describe the spread of a rabies epidemic gives insight into how this occurs. An even simpler modified diffusion equation suffers from the same instability. Attempts to fix the rabies problem by additional averaging are described. One method works for a limited parameter range but beyond that, instability can take a very long time to appear and its analysis displays interesting subtleties. This is joint work with David Ivers.