## Mathematics Colloquium

## Mathematics Colloquium

The Department Colloquium are held on Friday afternoons, from 3:00 to 4:00 pm, during teaching semester and are followed by refreshments.

#### 2017 Series...

###### 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
signicant 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 eld
theory. The talk should be accessible to senior mathematics majors.

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

**Date:** Friday 10 March 2017 **Speaker:** Dr Emily Riehl (Johns Hopkins University)**Venue: **E6B 149**Title:** TBA**Abstract:**
TBA

###### 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.

### Past Colloquia

#### 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.