Mathematics Colloquium

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 signi cant 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 803 (12 Wally's Walk roof top)
Title:  TBA
Abstract:  TBA

Week 9 - 12 May 2017

Date:  Friday 12 May 2017
Speaker:  Dr Joshua Ross (University of Adelaide)
Venue:  E7A 803 (12 Wally's Walk roof top)
Title:  TBA
Abstract:  TBA

Week 10 - 19 May 2017

Date:  Friday 19 May 2017
Speaker:  Dr Tanya Evans (Auckland, New Zealand)
Venue:  E7A 803 (12 Wally's Walk roof top)
Title:  TBA
Abstract:  TBA

Week 11 - 26 May 2017

Date:  Friday 26 May 2017
Speaker:  Dr Christopher Lustri (Macquarie University)
Venue:  E7A 803 (12 Wally's Walk roof top)
Title:  TBA
Abstract:  TBA

Week 12 - 2 June 2017

Date:  Friday 2 June 2017
Speaker:  A/Prof. Zihua Guo (Monash University)
Venue:  E7A 803 (12 Wally's Walk roof top)
Title:  TBA
Abstract:  TBA

Week 13 - 9 June 2017

Date:  Friday 2 June 2017
Speaker:  Dr Natalie Thamwattana (University of Wollongong)
Venue:  E7A 803 (12 Wally's Walk roof top)
Title:  TBA
Abstract:  TBA

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 o shore 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 e ectively exploit, older females whose fertility was declining left more descendants by subsidizing grandchildren and allowing mothers to have new o spring 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 tradeo s, and the possibility of grandmother help. Using this model, we show how the grandmother e ect 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 e ects can slow down and prevent escape from the ancestral condition, and it allows us to investigate the e ect 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 modelsAbstract: 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)

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 di erential equations. In particular, we will discuss how to characterize certain function spaces related to second order di erential operators in terms of cancellation conditions.

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