## Mathematics Colloquium

## Mathematics Colloquium

The Departmental Colloquia are held on Friday afternoons, from 3:00 to 4:00pm, during the teaching semesters and are followed by refreshments.

#### 2018 Series...

###### Week 10 - 04 May

**Date:** Friday, 04 May 2018**Speaker: **Prof Anthony Henderson (University of Sydney)

**Venue: **E7B 146 ACE room

**Title: **Enhancing the Jordan canonical form

**Abstract:**

In undergraduate linear algebra, the Jordan canonical form theorem tells us that each similarity class of n x n complex matrices is similar to one in a very simple block-diagonal form. From the Lie-theoretic viewpoint, this theorem is about classifying the orbits of the general linear group in its adjoint representation. This suggests natural follow-up questions about the closures of the orbits, and raises the hope of finding analogous orbit classifications for other representations of algebraic groups. After explaining some of this general context, I will focus on a case which, despite its close proximity to the Jordan canonical form theorem, was worked out only ten years ago by Pramod Achar (Louisiana State University) and myself.

###### Week 7 - 13 April

**Date:** Friday, 13 April 2018**Speaker: **Dr Bishnu Lamichhane (University of Newcastle)

**Venue: **E7B 146 ACE room

**Title: **Computing and visualising the solutions of partial differential equations

**Abstract:**

The finite element method has become the most powerful approach in solving partial differential equations arising in modern engineering and physical applications. We present computation and visualisation of the solutions of some applied partial differential equations using the finite element method for most of our examples. Our examples come from solid and fluid mechanics, heat conduction in sliding meshes and image processing.

###### Week 7 - 12 April

**Date:** Thursday, 12 April 2018**Speaker: **Prof Christopher Howls (University of Southampton, UK)

**Venue: **E7B 146 ACE room

**Title: **Complex rays, cones of silence and catastrophes, or when to turn an asymptotic blind eye

**Abstract:**

We develop a complex-ray approach to the calculation of high-frequency flow-interaction effects for 3D jet engine noise in a moving media obeying the Lilley equation. The method copes with caustics, complex rays and the multiplicity of ray solutions. Excellent agreement with modal and numerical solutions is achieved. For off-axis point sources, the solution field is rich in the canonical, but easily calculable, caustic structures of catastrophe theory. This reveals not only previously unknown additional upstream beaming of engine noise, but also some novel, vanishing caustic behaviour. The result is not only new engineering but also a striking novel phenomenon of direct relevance to asymptotic beyond all orders.

###### Week 6 - 06 April

**Date:** Friday, 06 April 2018**Speaker: **Prof Dmitry Pelinovsky (McMaster University Canada)

**Venue: **E7B 146 ACE room

**Title: **Reversing interfaces in the slow diffusion equation with strong absorption

**Abstract:**

Bifurcations of self-similar solutions for reversing interfaces are studied in the slow diffusion equation with strong absorption. We show that such bifurcations occur at points where the confluent hypergeometric functions satisfying Kummer's differential equation are truncated to finite polynomials. A two-scale asymptotic method is employed to obtain the local dependencies of the self-similar reversing interfaces near the bifurcation points. Numerical shooting techniques are developed to confirm the asymptotic results.

###### Week 4 - 23 March

**Date:** Friday, 23 March 2018**Speaker: **Prof Norman Dancer (University of Sydney)

**Venue: **E7B 146 ACE room

**Title: **The Fucik spectrum

**Abstract:**

We discuss the Fucik spectrum, which is {(α,β) ∈R² : −∆u = αu^{+}+βu‾ in Ω,u = 0 on ∂Ω,has a non-trivial solution}.

Here Ω is a ﬁxed bounded open set in R². We discuss positive results, counterexamples, and conjectures and connections with more classical problems. The proofs of results tend to involve linear and nonlinear analysis.

###### Week 3 - 16 March

**Date:** Friday, 16 March 2018**Speaker: **Prof Holger Dullin (University of Sydney)

**Venue: **E7B 146 ACE room

**Title: **Quantum monodromy in the Kepler problem

**Abstract:**

The motion of the Earth around the Sun is well described by Kepler’s problem, which is a beautiful example of a super-integrable system. Its full (sometimes called “hidden”) symmetry group is SO(4), which makes every orbit periodic. The quantum analogue of the Kepler problem is the Hydrogen atom, which can be separated in spherical coordinates and leads to the well known spherical harmonics. We found that when the system is separated in spheroidal coordinates instead, the lattice of the joint spectrum of the corresponding commuting operators lacks global basis. This phenomenon is called quantum monodromy, and I will trace its origin back to the classical dynamics of the Kepler problem and show that it can be found generally in spheroidal harmonics.

###### Week 2 - 09 March

**Date:** Friday, 09 March 2018**Speaker: **Dr Kevin Coulembier (University of Sydney)

**Venue: **E7B 146 ACE room

**Title: **Ideals in monoidal categories

**Abstract:**

I will discuss some topics in the study of “tensor ideals” in monoidal categories (categories equipped with a tensor product). One version of such ideals are studied in tensor triangulated geometry, where the Balmer spectrum of a tensor triangulated category mimics the notion of a scheme from algebraic geometry. After reviewing some generalities and surprising connections, I will mainly focus on a different version of tensor ideals, which I recently classified in Deligne’s universal monoidal categories. This led to some new applications in the representation theory of Lie supergroups.

###### Week 2 - 08 March

**Date:** Thursday, 08 March 2018**Speaker: **Professor John Sader (University of Melbourne)

**Venue: **E7B 146 ACE room

**Title: **Why does an inverted-flag flap in a uniform steady flow?

**Abstract:**

The dynamics of a cantilevered elastic sheet, with a uniform steady flow impinging on its clamped-end, have been studied widely and provide insight into the stability of flags and biological phenomena. Recent measurements show that reversing the sheet's orientation, with the flow impinging on its free-edge, dramatically alters its dynamics. In contrast to the conventional flag, which exhibits (small-amplitude) flutter above a critical flow speed, the inverted-flag displays large-amplitude flapping over a finite band of flow speeds. In this talk, a combination of mathematical theory, scaling analysis and measurement is used to investigate the origin of this large-amplitude flapping motion. Flapping is found to be a vortex-induced vibration and is periodic predominantly, with a transition to chaos as flow speed increases. These findings have implications to leaf motion and other biological processes, such as the dynamics of hairs, because they also can present an inverted-flag configuration.

###### Session 1, Week 1 - 02 March

**Date:** Friday, 02 March 2018**Speaker: **Professor Nalini Joshi (University of Sydney)

**Venue: **E7B 146 ACE room

**Title:** Symmetry through geometry

**Abstract:** Symmetry is an essential part of our description of the world. The quality of being made up of exactly similar parts facing each other is all around us: one day reflects another and the days fill out the year in the same way that similar hexagonal compartments fill out a honeycomb. The mathematical description of symmetries is built from only two operations: reflections and translations. In two dimensions, these give rise to triangular, hexagonal and square tilings of the plane. But in higher dimensions, many more tiling patterns are available. One of the many questions that arise is how to go from higher dimensional tilings to two-dimensional ones. I will show how to use these ideas to link two major theories that arise in mathematical physics.

###### 09 February

**Date:** Friday 09 February, 2018**Speaker:** Professor Herbert Huppert (University of Cambridge, UK)**Venue: **E7B 146 ACE room

**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.the HF radar community.

### Past Colloquia

#### 2017 Series...

###### Session 1 - Week 1 - 03 Mar 2017

**Date:** Friday 3 March 2017 **Speaker:** Emeritus Professor Ross Street (Macquarie)**Venue: **TBA**Title:** The Natural Transformation in Mathematics**Abstract:** The goal is to give some idea of what category theory is about: some history, some examples, some concepts, and an application to physics. The subject officially began in 1945 with papers focussing on examples and applications to

group theory. The authors were prepared to look at the collection of all groups as a mathematical object; this was quite controversial at the time. By now category theory has become a vital language for expressing much of mathematics and has

found many significant applications. A feature of the subject is the use of diagrams made of arrows. The arrows are an abstraction of functions f from one set A to another set B. More recently, a dual viewpoint, where f is depicted as a node with

input string A and output string B, has led to deep connections with knot theory, invariants for low dimensional manifolds, and the branch of theoretical physics called quantum field theory. The talk should be accessible to senior mathematics majors.

###### Week 2 - 10 Mar 2017 - Cancelled

###### Week 3 - 17 Mar 2017

**Date:** Friday 17 March 2017 **Speaker:** Dr Cecilia González Tokman (University of Queensland)**Venue: **E6B 149**Title:** Non-autonomous dynamical systems and multiplicative ergodic theorems**Abstract:** Non-autonomous dynamical systems yield very flexible models for the study of time-dependent systems, with driving mechanisms allowed to range from deterministic forcing to stationary noise. Multiplicative ergodic theorems (METs) encompass fundamental information for the study of transport phenomena in such systems, including Lyapunov exponents, invariant measures and coherent structures.

In this talk we will discuss recent developments on METs, motivated by applications in the geophysical sciences. We will then address related stability questions, which arise naturally in the context of non-autonomous systems from the use of numerical approximation schemes, as well as from the presence of modelling errors and noise. (This talk is based on joint work with Gary Froyland and Anthony Quas)

###### Week 4 - 24 Mar 2017

**Date:** Friday 24 March 2017 **Speaker:** Dr Pierre Portal (ANU)**Venue: **TBA**Title:** Harmonic Analysis in Rough Contexts**Abstract:** In recent years, perspectives on what constitutes the ``natural" framework within which to conduct various forms of mathematical analysis have shifted substantially. The common theme of these shifts can be described as a move towards roughness, i.e. the elimination of smoothness assumptions that had previously been considered fundamental. Examples include partial differential equations on domains with a boundary that is merely Lipschitz continuous, geometric analysis on metric measure spaces that do not have a smooth structure, and stochastic analysis of dynamical systems that have nowhere differentiable trajectories.

In this talk, aimed at a general mathematical audience, I describe some of these shifts towards roughness, placing an emphasis on harmonic analysis, and on my own contributions. This includes the development of heat kernel methods in situations where such a kernel is merely a distribution, and applications to deterministic and stochastic partial differential equations.

###### Week 5 - 31 Mar 2017

**Date:** Friday 31 March 2017 **Speaker:** Dr. Emily Riehl (Johns Hopkins University)**Venue: **E7A 801-803**Title:** Functoriality in algebra and topology**Abstract:** This talk will survey mathematical innovations involving functoriality. In the first examples - drawn from algebraic topology, algebraic geometry, and topological data analysis - the functors in question are “large” objects, bridging two large categories. In the second half of the talk, we will turn to “small” examples, functors indexed by small categories, and illustrate how such diagrams provide efficient combinatorial models of algebraic or topological data. This connects to recent work in progress, developed in conversation with John Bourke, Richard Garner, and Dominic Verity, to develop a unified framework for inductive arguments for those functors indexed by a certain family of categories.

###### Week 6 - 7 Apr 2017

**Date:** Friday 7 Apr 2017 **Speaker:** Dr Norman Do (Monash University)**Venue: **E7A 801-803**Title:** Counting surfaces: A mixed bag of combinatorics, geometry, and physics**Abstract:** Given some polygons, how many ways can you glue their edges together to create a particular surface? This enumeration is governed by two simple objects - a "spectral curve" and a "quantum curve" - that are related by a mysterious process called "quantisation". We will discuss exactly what this means and why it is mysterious, before observing the same structure in seemingly unrelated problems that involve permutations, knots and more. The talk will be G-rated, in the sense that almost no prerequisites are required!

###### Week 7 - 14 April 2017 - No talk due to Good Friday Public Holiday

###### Week 8 - 5 May 2017

**Date:** Friday 5 May 2017 **Speaker:** Dr Robert Marangell (Sydney University)**Venue: **E7A 801 (12 Wally's Walk roof top)**Title:** Travelling Wave Dynamics in Mathematical Biology**Abstract:** In their independent seminal works in 1937, Fisher, and Kolmogorov, Pescunov and Petrovskii (F-KPP) were among the first apply the analysis of travelling waves to a problem in mathematical biology. Subsequently, travelling waves have appeared in a host of mathematical biological problems, including chemototaxis, nerve impluse propagaion, intestinal crypt dynamics, tumour growth, wound healing, and population migrations, to name just a few. The first part of my talk will focus on some examples: specifically, a (classic) chemotactic model, and a Wolbachia infection model. The second half of my talk will discuss how a dynamical systems approach can shed light on the evolution of travelling waves. Using the F-KPP equation as a motivating example, I will show how much of the dynamic behaviour of a travelling wave is encoded in the spectrum of an associated linear equation.

###### Week 9 - 12 May 2017

**Date:** Friday 12 May 2017 **Speaker:** Dr Joshua Ross (University of Adelaide)**Venue: **E7A 801 (12 Wally's Walk roof top)**Title:** Mathematical problems in pandemic influenza response.**Abstract:** The emergence of a novel strain of influenza poses an ever-present threat to our health and well-being. Whilst a vaccine is available that typically provides protection against seasonal influenza, the development and production of a vaccine for a novel strain will take at least five months. Furthermore, the characteristics of the strain, pertinent to its threat and method of control, are obviously largely unknown. Mathematics and statistics are key to tackling this problem. I will present some of the contributions I have made to this topic and some insights they have provided.

###### Week 10 - 19 May 2017

**Date:** Friday 19 May 2017 **Speaker:** Dr Tanya Evans (Auckland, New Zealand)**Venue: **E7A 801 (12 Wally's Walk roof top)**Title:** An intra-departmental professional development model which is fun. . .but actually works**Abstract:** In this presentation we will give an overview of a model of professional development and highlight my personal experience in this project which led to a remarkable transformation of my lecturing practice. We will also talk more generally about transferable mechanisms for examining and improving our overall teaching practices. The model of professional development grew out of an inter-departmental initiative at the University of Auckland in which a group of lecturers meets regularly during the year as part of an ongoing professional development programme. The group is very diverse in the nature of the courses they teach and their mathematical research interests, including algebra, analysis, applied mathematics and mathematics education. At these meetings, we view a short excerpt from a video-recording of one of the lecturers from the group, which might be at either the undergraduate or graduate level. We then discuss aspects of the excerpt, with the discussions guided by the ROG (Resources, Orientations & Goals) framework formulated by Schoenfeld (2010).

References: Paterson, J., & Evans. T. (2013). Audience insights: Feed forward in professional development. In D. King, B. Loch & L. Rylands (Eds.), Proceedings of Lighthouse Delta, the 9th Delta conference of teaching and learning of undergraduate mathematics and statistics Through the Fog (pp.132-140). Kiama, Australia: Delta. Leong, Y. H., Ho, W. K. & Evans, T. (2016). Videos in teacher professional development, Discussion Group, Proceedings of the 13th International Congress on Mathematical Education (ICME), Hamburg, 24-31 July 2016: ICME. Barton, B., Oates, G., Paterson, P., & Thomas, M. O. J. (2015). A marriage of continuance: professional development for mathematics lecturers. Mathematics Education Research Journal, 27(2), 147-164. Schoenfeld, A. H. (2010). How we think. A theory of goal-oriented decision making and its educational applications. Routledge: New York.

###### Week 11 - 26 May 2017

**Date:** Friday 26 May 2017 **Speaker:** Dr Christopher Lustri (Macquarie University)**Venue: **E7A 801 (12 Wally's Walk roof top)**Title:** Applications of Exponential Asymptotic Methods**Abstract:** The utility of asymptotic series expansions has been long-established within applied mathematics for providing approximations to exact solutions in some asymptotic limit. However, these methods are typically unable to capture behaviour that is exponentially-small in the limit, irrespective of how many terms of the series one chooses to take. Behaviour on this scale is described as lying "beyond-all-orders".

This talk will be divided into two parts. In the first part, I will discuss how exponential asymptotic methods may be used to obtain information about behaviour that occurs on an exponentially-small scale, and in particular, how such methods uncover behaviour known as the Stokes Phenomenon. In the second part of the talk, I will discuss applications of these methods to problems arising in fluid dynamics and particle lattices.

###### Week 12 - 2 June 2017

**Date:** Friday 2 June 2017 **Speaker:** A/Prof. Zihua Guo (Monash University)**Venue: **E7A 801 (12 Wally's Walk roof top)**Title:** Recent development on the long time behaviour to some quadratic dispersive systems**Abstract:** In this talk I will survey some recent results on the study of the long time behaviour to some quadratic dispersive systems such as Zakharov system and Gross-Pitaevskii equation. These equations have quadratic nonlinear terms which usually cause considerable difficulties to study the long-time behaviour in low dimensions.

###### Week 13 - 9 June 2017

**Date:** Friday 2 June 2017 **Speaker:** Prof. Martin Weschelberger (University of Sydney)**Venue: **E7A 801 (12 Wally's Walk roof top)**Title:** Stellar Winds: The Force Awakens Through Ducks**Abstract:** Looking at the gas dynamics of stars under the assumption of spherical symmetry, I will show that transonic events in such systems are canard phenomena — peculiar solution structures identified in geometric singular perturbation problems. Consequently, stellar winds are carried by `supersonic ducks', and canard theory provides a mathematical framework for this astrophysical phenomenon. This is collaborative work with Paul Carter (University of Arizona) and Edgar Knobloch (UC Berkeley) published in Nonlinearity 30 (2017), 1006-1033.

###### Session 2 - Week 1 - 04 August 2017

**Date:** Friday 4 August 2017 **Speaker:** Professor Stuart Anderson (Adelaide)**Venue: **E7B 146 ACE**Title:** Mathematical problems arising in HF radar investigations of the geophysical environment**Abstract:** It is a common feature among remote sensing technologies that very considerable progress can be made with relatively simple physical and mathematical models. This is often fortuitous as, without getting `runs on the board' at an early stage, support for further development may evaporate and consign a technology to the graveyard of good-in-principle ideas. But, having survived to puberty, it is equally common for progress to slow or even stall because the detail and fidelity of measurements fail to meet the rising expectations of the users. At this point there is no option but to identify the factors that are limiting performance and to develop techniques to mitigate them. Typically the research problems which arise at this point demand a marriage, or even a menage-a-trois, between the disciplines of physics, engineering and mathematics. In the case of HF radar in its diverse roles, several topics within the realm of applied mathematics are of special interest at present, and it so happens that these are strongly represented in the research activities of the Department of Mathematics at Macquarie University, specifically in the areas of (i) fluid dynamics, oceanic and atmospheric sciences, (ii) nonlinear phenomena, (iii) inverse problems, (iv) optimisation, and (v) computational mathematics. In this talk I shall illustrate the application of these branches of mathematics to the problems currently of concern to the HF radar community.

###### Week 2 - 11 August 2017

**Date:** Friday 11 August 2017 **Speaker:** Dr Alessandro Ottazzi (UNSW)**Venue: **E7B 146 ACE**Title:** An introduction to sub-Riemannian geometry and its applications**Abstract:** Sub-Riemannian geometry is a flourishing subject of research that appears in many different areas of pure and applied mathematics. It is a generalisation of Riemannian geometry that provides models in robotics, aerospace engineering, medical imaging, and neurobiology. Applications in pure mathematics include metric geometry, PDEs, harmonic analysis, and geometric control theory. In this presentation I will give an introduction "by examples" to sub-Riemannian geometry and I will briefly describe a couple of models: a special car and the primary visual cortex. If there is some time left, I will also discuss in one example how simple calculus notions differ from the Euclidean case.

The talk is tailored for a general audience.

###### Week 3 - 18 August 2017

**Date:** Friday 18 August 2017 **Speaker:** Prof Gary Froyland (UNSW)**Venue: **E7B 146 ACE**Title:** Dynamics, Mixing and Coherence**Abstract:** Coherent structures in geophysical flows play fundamental roles by organising fluid flow and obstructing transport. For example, in the ocean, coherence manifests itself at global scales down to scales of at least tens of kilometres, and strongly influences the transportation of heat, salt, nutrients, phytoplankton, pollution, and garbage. I will describe some recent mathematical constructions, ranging across dynamical systems, probability, and geometry, which enable the accurate identication and tracking of such structures, and the quantication of associated mixing and transport properties. I will present case studies from a variety of geophysical settings.

###### Week 4 - 23 August 2017

**Date:** Friday 23 August 2017 **Speaker:** Dr Yoni Nazarathy (UQ)**Venue: **E7B 146 ACE**Title:** Reward Observing Restless Multi Armed Bandits**Abstract:** Much of operations research has to do with constraints. You want to do your best, but need to operate within certain bounds. A key question in the field is: what choice to make so as to maximise reward? You often try to answer this question in a dynamic manner over time, occasionally dealing with uncertainty and randomness. One such class of problems is the case where you have D assets that evolve in some random manner over time. An asset can be in "good" state or \bad" state. You can only choose K out of D such assets at every time slot. Which do you choose? Clearly giving priority to those in "good" state makes sense. But perhaps you don't have full information about the state of the assets. What do you do then? In this talk we'll show different approaches and variants of this problem and discuss some solution and performance analysis methods. Some concepts to be encountered are Markov Chains, Partially Observable Markov Decision Processes and Restless Bandits.

###### Week 5 - 1 September 2017

**Date:** Friday 01 September 2017 **Speaker:** A/Prof Ngamta (Natalie) Thamwattana (UoW)**Venue: **E7B 146 ACE**Title: **Modelling carbon nanostructures: mathematics, mechanics and molecular dynamics**Abstract: ** The talk will consist of three parts. First I will look at modelling mathematically the interaction between carbon nanostructures, including graphene, nanotubes and fullerenes. Two particular problems will be considered: interactions between graphene oxide and water and the spiral motion of molecules inside nanotubes. Molecular dynamics simulations for these problems will also be presented. Secondly I will show how to use calculus of variations to predict shapes of carbon nanostructures, such as graphene folds and wrinkles. Finally, I will touch on a recent result on modelling electromaterials in dye-sensitized solar cells.

###### Week 6 - 8 September 2017

**Date:** Friday 08 September 2017 **Speaker:** Dr Shane Keating (UNSW)**Venue: **E7B 146 ACE**Title:** Estimating turbulent mixing using stochastic filtering and superresolution of satellite imagery**Abstract: **The era of earth-observing satellites has revolutionised our understanding of our planet and the dynamical processes that shape it. In many real-world geophysical systems, however, estimates of turbulent mixing and transport are limited by the resolution of available observations. In this talk, I will describe a suite of stochastic filtering strategies for estimating mixing in turbulent geophysical flows from "superresolved" satellite imagery obtained by combining coarse observations with an efficient stochastic parameterization for the unresolved scales.

The method enhances the effective resolution of satellite observations by exploiting the effect of spatial aliasing and generates an optimal estimate of small scales using standard Bayesian inference. The technique is tested in quasigeostrophic simulations driven by realistic climatological shear and stratication profiles. Two applications are considered: calculating poleward ocean eddy heat flux from satellite altimetry, and estimating the three-dimensional upper ocean velocity field from superresolved sea-surface temperature imagery. In each case, the superresolved satellite observations result in a considerable improvement in estimates of turbulent fluxes compared with the raw observations.

###### Week 7 - 15 September 2017 - No talk this week

###### Week 8 - 6 October 2017

**Date:** Friday 6 October 2017 **Speaker: **Prof Scott McCue **Venue: ** E7B 146 (ACE room)**Title:** Linear and nonlinear ship waves: wake angles and time-frequency analysis**Abstract:** It is commonly believed that the half-angle which encloses a Kelvin ship wave pattern is simply arcsin(1/3) (roughly 19.5 degrees), provided the fluid is deep and the disturbance is small. However, observations and calculations for sufficiently fast-moving ships suggest that the apparent wake angle decreases with ship speed. We explore this phenomenon with a toy model, considering linear and nonlinear versions, in both infinite depth and finite depth. It turns out that the apparent wake angle also decreases for sufficiently slow-moving ship, which provides another interesting mathematical limit that involves exponential asymptotics and novel wave patterns. Finally, if time permits, the final topic of this walk will concern time-frequency analysis of ship wakes. Here, we analyse wave signals with short-time Fourier transforms and spectrograms, and describe some attempts to match our models with experimental data.

###### Week 9 - 13 October 2017

**Date:** Friday 13 October 2017 **Speaker: **Dr Justin Tzou (Macquarie) **Venue: ** E7B 146 (ACE room) **Title:** Mean first passage time problems and localised pattern formation - analysis, results, and surprising connections**Abstract: **Mean first passage time (MFPT) problems, a classic example of which is the gambler's ruin problem, generally ask the question - how long on average does it take for a random walker to first encounter a set of targets? Analysis of these problems in the past has been restricted to the scenario where targets are stationary. In this talk, we discuss how to derive and analyse the boundary-value problem associated with moving targets, and report some counterintuitive results. For localised spot patterns in reaction-diffusion systems, we will demonstrate a hybrid asymptotic-numerical method for obtaining key analytic results for their stability and dynamics. Finally, we draw some surprising links between these two seemingly very different problems.

###### Week 10 - 20 October 2017

**Date:** Friday 20 October 2017 **Speaker: **Prof. Michael Cowling (UNSW) **Venue: **E7B 146 (ACE room)**Title: ** The Brascamp-Lieb inequalities**Abstract:** About 20 years ago, the mathematical physicists Herm Brascamp and Eliott Lieb discovered a family of

inequalities, which now bear their names. These inequalities generalise several of the classical inequalities of analysis,

including Hölder's inequality, Young's convolution inequality, and the Loomis-Whitney inequality. The Brascamp-Lieb inequalities depend on linear maps *Lj* with a common domain *V* and different ranges *Vj* and a family of indices *θj* in [0,1]. The inequalities hold for some but not all of the possible indices. For applications in partial differential equations it is important to understand for which *Lj* and *θj* the inequalities hold, and whether the constants that appear in the inequalities depend continuously on the parameters. In this talk, I review the inequalities, and discuss some recent progress on these questions.

###### Week 11 - 27 October 2017

**Date:** Friday 27 October 2017 **Speaker:** Daniel Hauer (USyd) **Venue: ** E7B 146 (ACE room)**Title:** Non-concavity of Robin eigenfunctions

###### Week 12 - 3 November 2017

**Date:** Friday 3 November 2017 **Speaker: **Dr. Philipp Braun from University of Newcastle **Venue: **E7B 146 (ACE room)**Title: **Lyapunov and Control Lyapunov Functions: Stability of and Feedback design for Nonlinear Systems**Abstract: **Lyapunov’s second method is one of the most successful tools for analyzing stability properties of dynamical systems. To illustrate the idea behind Lyapunov’s second method we review existing results on Lyapunov functions and (nonsmooth) control Lyapunov functions in the context of stability and stabilization of nonlinear dynamical systems. Moreover, we highlight open problems and results on the ongoing research topic of control Lyapunov functions for destabilization of nonlinear systems. The talk concludes with ideas combining the concepts of stabilizing and destabilizing controllers based on the knowledge of appropriate control Lyapunov functions. The results presented in the talk are illustrated and motivated on the examples of an inverted pendulum, a nonholonomic integrator and Artstein’s circles.

###### Week 13 - 10 November 2017

**Date:** Friday 10 November 2017 **Speaker: **Dr Jiakun Liu from University of Wollongong **Venue: **E7B 146 (ACE room)**Title: **Optimal transportation on the hemisphere**Abstract: **In this talk, we first give a brief introduction to the optimal transportation problem. Then we study the optimal transportation on the hemisphere with the cost function $c(x,y)=d^2(x,y)/2$, where $d$ is the Riemannian distance of the round sphere. The potential function satisfies a Monge-Ampere type equation with a natural boundary condition. In this critical case, the hemisphere does not satisfy the $c$-convexity assumption. We obtain the a priori oblique derivative estimate, and in the special case of dimension two, we obtain the boundary $C^2$ estimate. Our proof does not require the smoothness of densities. This is a joint work with S.-Y. Alice Chang and Paul Yang.

#### 2016 Series

**Date:** 11 November 2016**Speaker:** Dr Richard Garner (Macquarie University)**Venue:** AHH Lecture Theatre 1.200**Title: ***Homotopy type theory***Abstract: ** Homotopy type theory is a new area of mathematics which, over the past ten or so years, has successfully combined aspects of the highly constructive disciplines of type theory and functional programming, and the highly non-constructive disciplines of algebraic topology and homotopy theory. The sheer unlikeliness of the pairing has been the source of both fascination and suspicion among workers in both fields. We attempt to give an introduction to the area comprehensible to a general mathematical audience.

**Date:** 4 November 2016**Speaker:** Associate Professor Catherine Greenhill (University of New South Wales)**Venue:** C5C Collaborative Forum**Title: ***Colouring random graphs and hypergraphs***Abstract:** A colouring of a graph (or hypergraph) is a map which assigns a colour to each vertex such that no edge is monochromatic. If there are k available colours then this map is called a k-colouring, and the minimum value of k such that a k-colouring exists is called the chromatic number of the graph. Graph colourings are fundamental objects of study, with applications in many areas including statistical physics and radio frequency assignment. The chromatic number of random graphs has been studied since the pioneering work of Erdős and Renyi (1960). We will take a tour through some of the major results in this area, and the methods used to prove them, including the probabilistic method and martingale arguments. I will also discuss some results on the chromatic number of hypergraphs with a linear number of edges (joint work with Colin Cooper and Martin Dyer and, subsequently, Peter Ayre and Amin Coja-Oghlan.) This work uses a more analytic approach, inspired by ideas from statistical mechanics.

**Date:** 28 October 2016**Speaker:** Professor Andrew Francis (University of Western Sydney)**Venue:** C5C Collaborative Forum**Title: ***Bacterial genome rearrangements and phylogeny in the Cayley graph***Abstract: **Modelling bacterial genome rearrangement operations as group actions on the space of all possible genomes provides a one-to-one correspondence between genome space and the group that acts. This means that a subset of genomes defines a set of points on the Cayley graph of the group, and a phylogeny on those genomes is represented by a Steiner tree on those points. In this talk I will describe this viewpoint and several related results. First, I will show how group theory can be used to calculate the "minimal distance" between genomes. Then I will describe a more nuanced view of the distance between genomes through a maximum likelihood estimate, and finally, I will describe some algorithmic results relating to the median problem for three genomes on the Cayley graph.

**Date: **21 October 2016**Speaker:** Dr Melissa Tacy (Australian National University)**Venue:** AHH Lecture Theatre 1.200**Title:**** **Semiclassical analysis in PDE**Abstract: **Semiclassical analysis arose as a set of techniques for studying the high energy (or semiclassical) limit of quantum mechanics. These techniques however can be used for a wide range of problems in PDE that feature a large (or small) parameter. In this talk I will discuss some of the applications of semiclassical analysis and the intuitions that drive this theory.

**Date: **14 October 2016**Speaker: **Professor Mary Myerscough (University of Sydney)**Venue:** AHH Lecture Theatre 1.200**Title: ***Why do hives die? Using mathematics to solve the problem of honey bee colony collapse.***Abstract: **Honey bees are vital to the production of many foods which need to be pollinated by insects. Yet in many parts of the world honey bee colonies are in decline. A crucial contributor to hive well-being is the health, productivity and longevity of its foragers. When forager numbers are depleted due to stressors in the colony (such as disease or malnutrition) or in the environment (such as pesticides) there are significant effects. These include a reduction in the amount of food (nectar and pollen) that can be collected and a reduction of the colony's capacity to raise brood (eggs, larvae and pupae) to produce new adult bees to replace lost or old bees.

We use a set of differential equation models to explore the effect on the hive of high forager death rates. We track the population of brood, hive bees (who work inside the hive) and foragers (who bring back food to the hive) and we track stored food. Using data from experimental research we devised functions that described the effect of the age that bees first become foragers on their success and lifespan as foragers. In particular we examine what happens when bees become foragers at a comparatively young age and how this can lead to a sudden rapid decline of adult bees and the death of the colony.

**Date:** 7 October 2016**Speaker: **Dr Julie Clutterbuck (Monash University)**Venue: **AHH Lecture Theatre 1.200**Title: ***Extreme eigenvalues***Abstract: **Each bounded domain has a sequence of eigenvalues associated to it. These are *determined* by the geometry of the domain, but do not* completely encode* the geometry. A natural question is to ask: which domains optimise the eigenvalues? For example, which domains have the smallest or largest first eigenvalue, or have the largest gap between eigenvalues? This is a rather old problem, with connections to the isoperimetric problem. I will describe some old and new results.

**Date:** 16 September 2016**Speaker:** Associate Professor Lesley Ward (University of South Australia)**Venue:** AHH Lecture Theatre 1.200**Title:** *Harmonic Analysis on Spaces of Homogeneous Type***Abstract: **The Calder\'on-Zygmund theory in harmonic analysis deals with singular integral operators and the function spaces on which they act. Early impetus came from problems in partial differential equations and Fourier theory. Much effort has been devoted to generalising the Calder\'on-Zygmund theory in several directions. Here we focus on the generalisation from functions defined on Euclidean spaces to functions defined on spaces of homogeneous type. The underlying space $\mathbb R^n$, equipped with the Euclidean metric and Lebesgue measure, is replaced by a general set X equipped only with a metric or quasi-metric and a doubling measure. In particular, the group structure and the Fourier transform are missing. Varied examples of spaces of homogeneous type arise in Riemannian geometry, several complex variables, and Lie theory. The goal is to build on this widely applicable foundation a Calder\'on-Zygmund theory which is as complete as it can be, recovering the classical results where possible and finding appropriate replacements or analogues where needed. I will survey some current progress towards this goal.

**Date:** 9 September 2016**Speaker:** Dr Georgy Sofronov (Department of Statistics, Macquarie University)**Venue**: AHH Lecture Theatre 1.200**Title:** *The theory of multiple optimal stopping rules and its applications***Abstract: **We observe a sequence of random variables and have to decide when we must stop, given that there is no recall allowed, that is, a random variable once rejected cannot be chosen later on. Our decision to stop depends on the observations already made, but does not depend on the future which is not yet known. The objective is to nd an optimal procedure that maximizes an expected gain. We consider problems when at least two stops are required, for example, a sequential problem of selling several identical assets over a nite time horizon.

**Date: **2 September 2016**Speaker:** Dr Luke Bennetts (University of Adelaide)**Venue:** AHH Lecture Theatre 1.200**Title: ***Water wave interactions with line arrays of vertical cylinders***Abstract: **In a highly cited paper, Maniar & Newman (J Fluid Mech, 1997) considered the impact of surface water waves on supports for bridges or other oshore structures, modelled by line arrays of vertical cylinders. They showed that the cylinders experience extreme resonant loads (i.e. hydrodynamic forces) at certain wave frequencies. Over the following decade, a sequence of papers by Evans, Porter, Linton and others, showed that the resonances are caused by excitation of so-called Rayleigh-Bloch waves "trapped" modes propagating along the array and decaying away from it. I'll summarise this previous work, then show how random perturbations in cylinder locations damp the resonances, and connect this with the phenomenon of Anderson localisation.

**Date:** 26 August 2016**Speaker:** Dr Peter Kim (University of Sydney)**Venue:** AHH Lecture Theatre 1.200**Title: ***Modelling evolution of post-menopausal human longevity: The Grandmother Hypothesis***Abstract: **Human post-menopausal longevity makes us unique among primates, but how did it evolve? One explanation, the Grandmother Hypothesis, proposes that as grasslands spread in ancient Africa displacing foods ancestral youngsters could eectively exploit, older females whose fertility was declining left more descendants by subsidizing grandchildren and allowing mothers to have new ospring sooner. As more robust elders could help more descendants, selection favoured increased longevity while maintaining the ancestral end of female fertility. We develop a probabilistic agent-based model that incorporates two sexes and mat- ing, fertility-longevity tradeos, and the possibility of grandmother help. Using this model, we show how the grandmother eect could have driven the evolution of human longevity. Simulations reveal two stable life-histories, one human-like and the other like our nearest cousins, the great apes. The probabilistic formulation shows how stochastic eects can slow down and prevent escape from the ancestral condition, and it allows us to investigate the eect of mutation rates on the trajectory of evolution.

**Date: **19 August 2016**Speaker:** Dr John Power (University of Bath)**Venue:** AHH Lecture Theatre 1.200**Title**:*Category theoretic semantics for theorem proving in logic programming: embracing the laxness***Abstract: **I shall first outline the central ideas of logic programming, in particular the concept of SLD-resolution. I shall then discuss category theoretic semantics: first of propositional logic programs, then of more general ones. The central mathematical concept is that of a coalgebra, and the central construct is that of the cofree comonad on an endofunctor; in order to extend from propositional logic programs to more general ones, one needs to consider lax transformations between coalgebras if one is to model theorem proving. There is a natural category-theoretic alternative in terms of "saturated semantics", and if time permits, I shall discuss that too.

**Date:** 12 August 2016**Speaker:** Professor Moshe Haviv (Jerusalem University)**Venue:** AHH Lecture Theatre 1.200**Title:** *A rate balance principle and its application to queueing models***Abstract:** We introduce a rate balance principle for general (not necessarily Markovian) stochastic processes. Special attention is given to processes with birth and death like transitions, for which it is shown that for any state i, the rate of two consecutive transitions from i-1 to i+1, coincides with the corresponding rate from i+1 to i-1. This observation appears to be useful in deriving well-known, as well as new, results for the Mn/Gn/1 and G/Mn/1 queueing systems, such as a recursion on the conditional distributions of the residual service times (in the former model) and of the residual inter-arrival times (in the latter one), given the queue length. The talk is based on Oz, Adan and Haviv (2016) http://arxiv.org/pdf/1510.02779v1.pdf

**Date:** 5 August 2016**Speaker:** Dr Brett Wick (Washington University)**Venue:** AHH Lecture Theatre 1.200**Title: **Commutators, Factorization and Function Spaces**Abstract: **In this talk we will discuss the connection between function theory and operator theory by showing that certain operator theory concepts have natural analogues in function theory. This will be motivated by examples in spaces of analytic functions, results from harmonic analysis and partial dierential equations. In particular, we will discuss how to characterize certain function spaces related to second order dierential operators in terms of cancellation conditions.