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New Zealand Mathematical
Society Colloquium

Run by the New Zealand Mathematical Society
4 – 6 December 2012
Massey University, Palmerston North
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Plenary Abstracts

Astrid an Huef

University of Otago

Algebraic systems of isometries

An isometry on a Hilbert space is a distance-preserving linear transformation. The algebras generated by algebraic systems of isometries are surprisingly rigid. I will consider examples of such systems arising from directed graphs: there the vertices of the graph are represented by subspaces and the edges are represented by isometries between appropriate subspaces. The algebras generated by this system have a very attractive structure theory which is reflected in the combinatorial properties of the graph. I will give an introduction to the key ideas, and explain why the so-called graph algebras have attracted so much interest over the last 15 years.

Bill Barton

University of Auckland

The teacher-researcher interface: What is research-based teaching really?

Universities claim that their teaching is “research-based”. What does this mean in our field? How exactly does a mathematician’s research activity feed into their lecturing? I argue that it does, in a very strong way—but not in the way that might be expected—and that we would do well to celebrate this. To make my point I will borrow some theoretical constructs from mathematics education: constructivism; horizon mathematics; pedagogical content knowledge. Please bring your laptops and/or smartphones, and be ready to turn them on.

John Sader

Melbourne University & Caltech

Dynamics of Nanoscale Mechanical Devices in Fluid with Applications to Atomic and Molecular Sensing

Sensitivity of a mechanical device to environmental changes can be enhanced through miniaturization. This has led to some key advances in nanoscience, which include the imaging of surfaces with atomic and molecular resolution, measurement of inertial mass at the atomic scale and monitoring of biological processes in liquid. Controlling the effects of surfaces and fluid dissipation presents significant challenges to achieving the ultimate sensitivity of these devices. Particularly, the fluid-structure interaction of resonating microcantilevers in fluid has been widely studied and is a cornerstone in nanomechanical sensor development. In this talk, I will give an overview of work being undertaken in our group dedicated to exploring the underlying physical processes in these and related systems. This will include exploration of recent developments that focus on cantilever sensors with embedded microfluidic channels, gigahertz nanoparticle resonators in fluid and examination of the effects of surface stress on the resonant properties of cantilever sensors.

Kate Smith-Miles

Monash University

Footprints in Instance Space: steps towards a free lunch

The No-Free-Lunch Theorem tells us that, without prior knowledge of the properties of an instance of a problem, we cannot expect any single algorithm to outperform all others across all instances. If an algorithm performs exceptionally well on a certain class of instances, there will always be some other class of instances where it is outperformed by another algorithm. Understanding how the properties of an instance affect algorithm performance is the key to being able to articulate the strengths and weaknesses of an algorithm, and to anticipate when it is likely to be better than others.

In this talk I will present a new methodology for achieving this goal, and demonstrate its applicability to optimization, although it generalizes to other problem domains. The methodology involves: visualizing the set of all possible instances based on features that correlate with difficulty; statistical generalization of algorithm performance in this instance space, shown as a footprint where an algorithm's performance is deemed to be good; and then measuring the relative area of the footprint of different algorithms. The methodology is applied to provide insights into optimization algorithm performance on the Travelling Salesman Problem and graph colouring.

Ilze Ziedins

University of Auckland

Stochastic networks: queues, delays and bottlenecks

Queues and delays: we all experience them, never enjoy them, but there is usually little we can do about them. This talk will discuss some of the diverse applications where queueing models are used, concentrating particularly on healthcare modelling and optimization, and networks with selfish routing.

I will give an overview of some recent work and problems in healthcare modelling, including work with the cardiac unit at Auckland City Hospital, where we have built a simulation model of the intensive care unit that can be used to improve resource management and reduce patient delays. In contrast to this, I will also discuss networks such as transportation networks, where individuals can choose their own pathway through the system. Not all networks permit self-optimizing behavior, or selfish routing, but those that do may experience very poor performance.

What does queueing theory have to say about such networks, and what can we learn from it?