# Applied Mathematics Seminar

Seminars are held at 2:00 pm on Wednesdays in the Access Grid Room ( Carslaw Building, 8th floor), unless otherwise noted.

For more information and to be added to the mailing list, please contact Eduardo G. Altmann .

## Upcoming Seminars

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.

## Previous seminars

### 2017

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 spectral 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.

### 2016

### 2015

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. {\bf 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 {\bf 55}, 014004 (2013) http://dx.doi.org/10.1088/0741-3335/55/1/014004

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. {\bf 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 {\bf 55}, 014004 (2013) http://dx.doi.org/10.1088/0741-3335/55/1/014004