We develop a model of phonological contrast in natural language. Specifically, the model describes the maintenance of contrast between different words in a language, and the elimination of such contrast when sounds in the words merge. An example of such a contrast is that provided by the two vowel sounds "i" and "e", which distinguish pairs of words such as "pin" and "pen" in most dialects of English. We model language users' knowledge of the pronunciation of a word as consisting of collections of labeled exemplars stored in memory. Each exemplar is a detailed memory of a particular utterance of the word in question. In our model an exemplar is represented by one or two phonetic variables along with a weight indicating how strong the memory of the utterance is. Starting from an exemplar-level model we derive integro-differential equations for the evolution of exemplar density fields in phonetic space. Using these latter equations we investigate under what conditions two sounds merge causing words to no longer contrast. Our main conclusion is that for the preservation of phonological contrast, it is necessary that anomalous utterances of a given word are discarded, and not merely stored in memory as an exemplar of another word.

150210110338

NOTE DAY, TIME AND VENUE OF THIS SPECIAL SEMINAR
We consider the similarity relations of isometry and homeomorphism for Polish metric spaces. We survey known results on the descriptive complexity of such relations. For instance, Gao and Kechris showed that isometry is "orbit complete", i.e. as complex as possible, while Gromov proved that for compact spaces isometry is smooth, so simple. The exact complexity of homeomorphism for compact metric has only recently been determined. Using a result of Camerlo and Gao, we show that, in the computable setting, it is Sigma-1-1 complete for equivalence relations.

Another interesting similarity relation is having Gromov-Hausdorff distance 0, which intuitively means isomorphic up to error epsilon, for each positive epsilon. This is related to the Scott rank in continuous logic.

150223092908

NeSI is New Zealand's computing research infrastructure, providing high performance computing and support services. The University of Otago is a NeSI investor, and so Otago researchers can access most NeSI resources at no charge. I will describe the structure of NeSI, the services and hardware resources it makes available, how to access those resources, and the ways in which statisticians and others are using them.

141212140915

The classical displacement-based finite element method when modeling the deformation of shell structures can suffer from several computational difficulties even within the framework of linear elasticity. Usually these problems appear when the thickness of the shell tends to zero; that is, when the shell becomes extremely thin. This phenomenon is referred to as the "numerical locking effect". One of the possibilities to overcome these numerical difficulties is to use a multi-field, dual-mixed variational formulation. In this talk a new dimensionally reduced axisymmetric shell model will be presented for modeling time-dependent and elastostatic problems. The formulation is based on a three-field dual-mixed variational principle, where the independent fields are the a priori non-symmetric stress tensor, the displacement vector and the infinitesimal rotation vector. The numerical performance of the developed h- and p-version axisymmetric shell elements is tested through investigating representative cylindrical shell problems.

150204130034

Regular polytopes serve as building stones for fractal constructions, see attached. In $\mathbb{R}^4$ exist six regular polytopes. Five of them generalise the platonic solids and one, which is called 24-cell, has no lower dimensional analogue. We will present several fractal constructions based on the 24-cell and on other regular polyhedra in three and four dimensions. Fractals in $\mathbb{R}^4$ cannot be illustrated, but we can visualise their three-dimensional intersections with hyperplanes. We call such an intersection a slice. To illustrate slices we used the cutting plane method which can be applied to a class of self-similar sets generated by homotheties with scaling factor an inverse of a Pisot unit $\beta$ and translations in $\mathbb{Q}(\beta)^n$. From the algebraic point of view our method generalises a result on $\beta$-representations stated by Schmidt in 1979.

The second part of the talk concerns the application of fractals in, for example, physics and life sciences. Interactions of fractal structures and electromagnetic waves promise new technologies, e.g. for producing energy or for cloaking. Concepts of fractal geometry can also be applied to the description of biological tissues and their cells.

141021135340

There are a number of standard constructions of C*-algebras associated to groups, including the full group C*-algebra, which can be thought of as the biggest C*-algebra encoding the algebraic structure of the group. The problem of associating C*-algebras to semigroups has been left unresolved a lot longer than its group counterpart, although a recent construction of Xin Li has been widely accepted for a large class of semigroups.

We will begin our talk with a brief recap of C*-algebras, before looking at some of the attempts to build full semigroup C*-algebras, including Li's construction. We will then look at two classes of semigroups that give rise to interesting C*-algebras. We will finish with a look at how the algebraic structure of a semigroup affects the structure of its associated C*-algebra.

141015111029

By a construction of B.O. Koopman (1931), every non-linear dynamical system can be completely described by a linear operator (in the discrete case) or a linear flow (in the continuous case) on a suitable function space. Spectral properties of this operator, in particular its eigenvalues, allow to identify structural components of the dynamical system. This theory has a wide range of applications. For instance, it underlies the celebrated Green-Tao theorem (2004) about arithmetic progressions in the primes. On the other hand, it also facilitates the calculation of characteristic quantities (e.g., of a turbulent flow of high vorticity) from empirical data. With my talk I want to give some insight into these connections.

141015111157

141009170928

Note day, time and venue of this Physics seminar
In 1963 several hundred Astrophysicists and Relativists met in Dallas to try to understand the enormously energetic objects and the centres of distant galaxies. My solution for a rotating Black Hole was presented there and very quickly became the standard model for such objects. The development of this paradigm will be discussed together with the many unsolved problems with it.

140923125735

When telecommunication infrastructures such as base stations or access points are not available, mobile ad hoc networks (MANET) can provide a useful platform for self-organized networking applications as multi-hop routes can be established on-the-fly. However, effective routing of real-time traffic on MANETs remains a challenging issue and many existing protocols are based on simple heuristics. We show that by using a Jackson network consisting of simple queueing models, an analytical solution can be found. Simulations results obtained from realistic video traces will be presented to support our proposed protocol. Finally, possible extensions to the protocol such as admission control and mobility support will also be discussed.

140626121931

Note - different day to usual for statistics seminar

The discipline of archaeology is dependent on the science of radiocarbon dating to build its chronologies. However, radiocarbon is not a straight-forward nor consistently reliable dating method. Variables that might impact radiocarbon measurement science include sample suitability, calibration curves and margins of error. In this seminar I review these variables as they affect radiocarbon chronologies of colonisation and change within the relatively short New Zealand archaeological sequence.

140725102528

The Painlevé equations are second order differential equations in the complex plane whose solutions have branchings only at some fixed points and extend to global multivalued meromorphic functions on the complement of these bad points. The singularities outside of these bad points are only poles, and they vary with the solution, they are "movable". The Painlevé equations I-VI had been classified in the early 20th century by Painlevé and Gambier. In the '70s and '80s they received new interest due to many applications in physics. McCoy-Tracy-Wu 1977 and Its-Novokshenov worked on real solutions of Painlevé III on the positive real line (for Painlevé III zero is the only bad point) and studied especially the asymptotics near 0 and infinity of single solutions. I will combine this with information on the moduli spaces of all solutions and extract results on the global geometry of the movable poles. As the moduli spaces are real 2-dimensional and the solutions live on the positive real line, many ideas and results in the talk can and will be illustrated by pictures.

140313103943

The seminar will explain a commonly used technique in empirical macroeconomic analysis, using econometric time-series methods. It is based on “letting the data talk” by studying the dynamic interactions of a set of variables, a vector of variables, regressed on their own histories (auto-regressions). Data issues such as unit roots and cointegration will be discussed briefly before focusing on an application. The example used in the seminar is a so-called structural vector-autoregression with recent monetary and fiscal data for Poland. Fiscal foresight, in the form of implementation lags, is accounted for with respect to both discretionary government spending and tax changes. The importance of combining monetary and fiscal transmission mechanisms is demonstrated. However, ignoring fiscal foresight has no statistically significant effects. Government spending multipliers take on values from 0.16 to 1.61, depending on how they are calculated. The tax multiplier is not very large.

140331164635

This talk will be about elastic beams and plates and how they vibrate under different forces and boundary/geometrical conditions. Many engineered structures are made up of beams and panels. These beams and panels are often modelled as elastic beams and plates, which can be approximated using the theory of linear elasticity. In this talk, I will introduce the theory of vibration of elastic beams and plates together with my work in acoustical behaviours of walls and floors in buildings. The mathematical models of beams and plates are also used to study composites of beams and plates, which can be found in lightweight structures. I will show how a beam-reinforced plate can be modelled using relatively simple coupling conditions between individual components. In recent years, the defect identification has been researched actively. I will introduce existing methods of defect identification and show how the Bayesian inference combined with the Markov-Chain-Monte-Carlo can be used to identify de-laminations in composite beams and plates.

140625134643

Joint mathematics and statistics seminar

A recombinant virus appears to be behind the loss of honeybee colonies to varroa mite infestation. Given genetic information about the viral recombinants in the honeybee from next generation sequencing, mathematical and statistical tools have been developed to determine both the recombinants present and their relative proportions. The method involves setting the problem geometrically and the use of appropriately constrained quadratic programming.

This seminar will present the background to the problem, together with the mathematical and statistical ideas that underpin the recombinant discovery. Output from software which runs the method, termed “MosaicSolver”, will be shown. This work is part of the “Insect Pollinators Initiative” currently underway in the UK.

140331164532

Chuen Yen Hong: Confidence intervals for the mean effect size in random-effects meta-analysis

Ben Atkins: The Trojan female technique: a revolutionary approach for effective pest control

Cain Edie-Michell: Ideal structure of Steinberg algebras

Chris Palmer: Actions of discrete groups on the Cantor set

Silong Liao: Glaucoma treatment: survival analysis

Baylee Smith: Spatial summaries of phase four Bronze Age burials at Ban Non Wat, Thailand

140523154928

Random fluctuations of an environment are common in ecological and economical settings. The resulting processes can be described by a stochastic dynamical system, where a family of maps parametrized by an independent, identically distributed random variable forms the basis for a Markov chain on a continuous state space. Random dynamical systems are a beautiful combination of deterministic and random processes, and they have received considerable interest since von Neumann and Ulam's seminal work in the 1940's. Key questions in the study of a stochastic dynamical system are: does the system have a well-defined average, i.e. is it ergodic? How does this long-term behavior compare to that of the state variable in a constant environment with the averaged parameter?

In this talk we answer these questions for a family of maps on the unit interval that model self-limiting growth. The techniques used can be extended to study other families of concave maps, and so we conjecture the existence of a "stochastic horseshoe".

140428094758

We apply the piecewise constant, discontinuous Galerkin method to discretize a fractional diffusion equation with respect to time. Using Laplace transform techniques, we show that the method is first order accurate at the $n$th time level $t_n$, but the error bound includes a factor $t_n^{-1}$ if we assume no smoothness of the initial data. We also show that for smoother initial data the growth in the error bound for decreasing time is milder, and in some cases absent altogether. Our error bounds generalize known results for the classical heat equation and are illustrated using a model 1D problem.

140403101218

Solving the discrete logarithm problem is key to breaking many cryptosystems: Given g, h in a cyclic group solve for x such that $h=g^x$. Two of the fastest methods for solving this are based on probabilistic curiosities, with Pollard's Kangaroo method based on the same principles as the Kruskal Count card trick, and Pollard's Rho method based on the Birthday Paradox. We explain these attacks and how they are related to the expected time until (self-)intersection of Markov chains. A very precise result is given which is the first method to accurately predict the differing performance of two attacks that are superficially similar: Pollard's Rho for discrete logarithm and Teske's additive version.

140305114124

NOTE EARLIER TIME

The order of a numerical method for initial value problems is a useful guide to its accuracy. To obtain accurate results with low computational cost, high order is usually preferable to low order.

It is interesting that the traditional theory of order, on which the numerical methods of Runge, Heun and Kutta were based, is incorrect for orders greater than 4 and a more rigorous theory will be presented. This modern order theory leads to an algebraic system based on the composition group for Runge-Kutta methods; its applications include the order of canonical methods and the order of so-called general linear methods.

140310160158

Von Neumann's ergodic theorem from 1931 originates in statistical mechanics and lies in the intersection of functional analysis and probability theory. Abstractly, one takes a linear contraction T on a Banach space X and considers the arithmetic means A_n of the first n powers of T:
A_n := 1/n (T + T^2 + ... + T^n)
Then the Mean Ergodic Theorem states that if X is reflexive then A_n(x) converges for each x in X.

In general, the convergence can be arbitrarily slow, i.e., no uniform convergence rate for all x in X can be expected. However, identifying x's with particular convergence rates leads to applications to strong law of large numbers with rates and central limit theorems for Markov chains. Deriennic and Lin raised the question whether (weak) convergence of certain power series in T on a vector x can ensure a rate for the convergence of A_n(x). In particular, they asked whether one has logarithmic rate for A_n(x) when x lies in the domain of the "one-sided ergodic Hilbert transform'' of T.

In recent work with Yuri Tomilov and Oleksandr Gomilko (Toruń) we developed a general approach to the matter and could answer the posed problem. In our work, power series whose Taylor coefficients form a so-called renewal sequence play a prominent role.

140305114008

Join Craig Bauling as he guides us through the capabilities of Mathematica 9.
Craig will demonstrate the key features that are directly applicable for us in teaching and research.

Topics of this technical talk include:
• Natural Language Input
• Market Leading Statistical Analysis Functionality
• 2D and 3D information visualization
• Creating interactive models that encourage student participation and learning
• Practical applications in Engineering, Chemistry, Physics, Finance, Biology, Economics and Mathematics
• On-demand Chemical, Biological, Economic, Finance and Social data
• Mathematica as a modern programming language

Prior knowledge of Mathematica is not required – new users are encouraged.
Current users will benefit from seeing the many improvements and new features of Mathematica 9.
This is a great opportunity to get faculty not experienced with Mathematica involved and excited. Students are welcome as well.

A joint seminar by the Department of Physics and the Department of Mathematics and Statistics

140324153217

In 1950's, William Leavitt constructed the first class of unital K-algebras in which they do not satisfy the Invariant Basis Property, i.e., R is isomorphic to the direct sum of n copies of R (for some n >1) as left R-modules. These algebras are now called Leavitt algebras and is denoted by L(1, n). Over the next 4 decades this work received modest attention in the mathematics community. In recent years, these algebras and their generalization (Leavitt path algebras) have received lots of interest from algebraist and C*-algebraist. A Leavitt path algebra is an algebra constructed from a directed graph (possibly infinite) based on the configuration of the vertices and edges. It turns out L(1, n) is the Leavitt path algebra associated to the class of directed graphs with one vertex and n edges.

One of the main open questions in the structure theory of Leavitt path algebras is determining if all simple Leavitt path algebras are classifiable (up to Morita equivalence) by algebraic K-theory. The aim of the talk is to give an overview on the recent progress of the classification program.

140305113917

C*-algebras were developed as models for quantum-mechanical systems, and have since been adapted as powerful representation-theoretic tools for studying algebraic and combinatorial objects. For example, C*-algebras associated to finite directed graphs have played an important role in invariant theory for symbolic dynamics.

When we model a quantum system with a C*-algebra, equilibrium states of the system are modelled by linear functionals called KMS states on the C*-algebra. Not every C*-algebra corresponds to a physical system, but it turns out that even then KMS states capture plenty of interesting information. I will outline some recent joint work with Astrid and Iain and Marcelo Laca on KMS states of C*-algebras associated to finite directed graphs.

140305113749

Yang Mills theory has had a profound effect on the development of differential geometry in the past four decades. A hallmark of the theory is the existence of gauge symmetries, namely a group of transformations that preserve the equations of motion. In some cases the process of quantisation can break the gauge symmetry and this renders the theory unphysical. It is therefore critical to remove such quantum anomalies and this puts strict constraints on the model. The purpose of this talk is to exhibit some of the rich mathematics underlying this phenomenon.

130724100502

Any location is characterised by a multivariate set of environmental conditions and these conditions are one of the filters determining which species occur at that location. Species are adapted to certain environmental conditions (their niche) and this niche may or may not be modified by the species’ interactions with other species. With changing environmental conditions, species either need to adapt to the new conditions or move to areas where suitable conditions remain. Insights into the spatial distribution and dynamics of past, current and future climate niche conditions inform our understanding of a species’ ecology and evolution. In this seminar I will illustrate recent approaches for quantifying the spatial distribution of climate niche conditions for a range of environments and spatial and temporal scales.

130712100640

Network rewiring as a method for producing a range of graph structures was first introduced in 1998 by Watts and Strogatz (Nature 393, 440-442). This approach allowed a transition from regular through small-world to a random network. The subsequent interest in scale-free networks motivated a number of methods for developing rewiring approaches that converged to scale-free networks. This paper presents a rewiring algorithm for undirected, non-degenerate, fixed size networks that transitions from regular, through small-world and scale-free to star-like networks. Applications of the approach to models for the spread of infectious disease and fixation time for a simple genetics model are used to demonstrate the efficacy of the approach.

130815113430

Note day, time and venue different from our usual
A lecture in the 10X10 series: ten speakers at ten locations in ten months. How mathematics is helping to find solutions to today's problems.

Mathematics has always been an important tool in a scientist's toolkit, but today the importance of mathematics to drive scientific enquiry and industrial innovation is even greater.

In this lecture, Professor James Sneyd will show that mathematics can give insight into how human cells work and how chemical networks inside human cells are regulated.

James Sneyd is a Professor of Applied Mathematics at the University of Auckland. In 2005 he was elected a Fellow of the Royal Society of New Zealand.
Visit our website for full details of the 10X10 Lecture Series, including audio and video recordings of previous speakers in the series. http://www.royalsociety.org.nz/events/10-x-10-lecture-series/
The lecture is free and open to the general public. However, to ensure a seat, please register to obtain a ticket via website.

131001141125

Statistics
Claire Flynn : Adaptive kernel density methods for estimating relative risk in geographical epidemiology
Yuki Fujita : Analysis of bird count data

Mathematics
Jack Cowie : Groupoids, partial actions and Baumslag-Solitar groups
Rotem Edwy: The integer Heisenberg group acts self-similarly
Calum Rickard : Unbounded extension of the Hille-Phillips functional calculus
Emily Irwin : HNN extensions, normal forms and Baumslag-Solitar groups

Note day and time of this event

120904102920

Dynamical systems of Lorenz type are similar to the famous Lorenz system of just three ordinary differential equations in a well-defined geometric sense. The behaviour of the Lorenz system is organised by a chaotic attractor, known as the butterfly attractor. Under certain conditions, the dynamics is such that a dimension reduction can be applied, which relates the behaviour to that of a one-dimensional non-invertible map. A lot of research has focussed on understanding the dynamics of this one-dimensional map. The study of what this means for the full three-dimensional system has only recently become possible through the use of advanced numerical methods based on the continuation of two-point boundary value problems. Did you know that the chaotic dynamics is organised by a space-filling pancake? We show how similar techniques can help to understand the dynamics of higher-dimensional Lorenz-type systems. Using a similar dimension-reduction technique, a two-dimensional non-invertible map describes the behaviour of five or more ordinary differential equations. Here, a new type of chaotic dynamics is possible, called wild chaos.
Joint work with: Bernd Krauskopf (University of Auckland), Eusebius Doedel (Concordia University, Montreal) and Stefanie Hittmeyer (University of Auckland)

130513120811

The standard Gibbs sampler (a.k.a. Glauber dynamics and local heat-bath thermalization) is fundamentally equivalent to Gauss-Seidel iteration, when applied to Gaussian-like target distributions. This explains the slow (geometric) convergence of the Gibbs sampler, but also indicates how to accelerate it using polynomials.

The potential to accelerate prompts our interest in the Gibbs sampler for an application of capacitance tomography to bulk-flow monitoring in industrial processes. We have been able to build a near-analytic Gibbs sampler in the broader class of inverse equilibrium problems by utilizing the graph-theoretic construction of circuit theory.

130712101909

This presentation gives an overview of the geodetic studies of the geodynamic processes in New Zealand, currently being undertaken by the School of Surveying. Primarily using Global Navigation Satellite Systems (GNSS) methods, the studies include regional earth deformation for seismic hazard research, the Southern Alps uplift experiment, sea level rise, coseismic and post-seismic deformation studies.

While the position (coordinate) time series analyses and linear regression models used are well defined, the stochastic models (white noise, flicker noise, random walk) are not so well understood or as easily implemented. Another issue concerns optimal methods for detecting deformation transients in the event of slow slip (or slow earthquake) events. This has implications for real time Network RTK systems used by the Surveying industry.

130712102355

Note day, time and venue
New computer methods have been used to shed light on a number of recent controversies in the study of art. For example, computer fractal analysis has been used in authentication studies of paintings attributed to Jackson Pollock recently discovered by Alex Matter. Computer wavelet analysis has been used for attribution of the contributors in Perugino's Holy Family. An international group of computer and image scientists is studying the brushstrokes in paintings by van Gogh for detecting forgeries. Sophisticated computer analysis of perspective, shading, colour and form has shed light on David Hockney's bold claim that as early as 1420, Renaissance artists employed optical devices such as concave mirrors to project images onto their canvases.

How do these computer methods work? What can computers reveal about images that even the best trained connoisseurs, art historians and artists cannot? How much more powerful and revealing will these methods become? In short, how is computer image analysis changing our understanding of art? This profusely illustrated lecture for non-scientists will include works by Jackson Pollock, Vincent van Gogh, Jan van Eyck, Hans Memling, Lorenzo Lotto, and others. You may never see paintings the same way again.

130919142051

This presentation is to inform the audience about the state of the art in forensic geographical provenancing. The presentation will impact on the audience and make it aware of the potential and issues of forensic geochemical profiling and will probably lead to more usage of the technique There is an increasing need for the ability to geographically provenance natural products, manufactured goods and humans in forensic casework. The global mobility of goods has led to large scale counterfeiting with serious financial and biosecurity consequences. In the case of commercial goods like food products, claims of geographical origin based on classical supply chain traceability information can easily be falsified. In cases where materials are non-compliant with a stated origin, or simply of unknown origin, tools are required that attribute these to most likely source regions using a scientific measure of probability. A similar approach is required for forensic evidence materials that could help reconstruct a crime or provide intelligence in counter-terrorism or military pursuits. In recent years it has become evident that a number of geochemical parameters are well suited to support legal expert opinions about the geographical origin of natural materials. The natural elemental and isotopic composition of water and soil provides a base to make inferences about agricultural products and most materials derived from such. In recent years it has become evident that the hydrogen and oxygen isotopic composition of rainwater is related to a limited number of well understood spatial parameters like latitude and altitude. Models of the isotopic composition of the precipitation have been validated globally and now the regional composition of even ground water and products in the foodchain can be predicted with a useful level of accuracy, enabling discrimination of latitudinal distances in the 200-mile range1. As the precipitation models roughly provide latitudinal bands of distinction other parameters are sought to give longitudinal discrimination and/or a higher scale of general spatial resolution. Any parameter that can be linked to existing information already captured in maps is desirable. Recent research has shown that especially the radiogenic isotopic composition of an element like strontium in soil extracts can provide information about the isotopic composition of the local foodweb2. The often relatively well-understood behavior of the isotope systems allows researchers to make spatial predictions of the isotopic profiles in target tissues and objects. The art of making such predictions has actually led to the term “isoscape” meaning the isotope landscape of a biological tissue or natural product of interest3. Hydrogen and oxygen isotope isoscapes are now applied in bird migration studies, provenancing of unidentified human remains and food authentication3,4. Our own research groups are working towards oxygen, strontium and lead isotope isoscapes for human provenancing and authentication and bio-security intelligence for a number of food products in Europe, Middle East, Asia and Oceania. Despite all these efforts a more formal probabilistic approach relevant for application in criminalistics is still very much under development. When trying to provenance questioned materials a probabilistically approach combining different isotope systems and other relevant case information will provide a more accurate prediction of the more and less likely regions of origin. This presentation will discuss the state of the art using casework and give guidelines for the interpretation and presentation of results in a forensic context.
REFERENCES [1] vdVeer et al, Journal of Exploration Geochemistry, 101, 175-184, 2009. [2] Voerkelius et al, Food Chemistry 118, 933-940, 2010. [3] West et al, Isoscapes, Springer, 2009. [4] Meier-Augenstein, Stable Isotopes Forensics. CRC Press, 2010.

130712102002

Students have very different learning outcomes and experiences in mathematics despite being in the same classroom, having the same teacher, and working on the same task. This is because students have unique relationships with the subject of mathematics. Naomi will explore these relationships and discuss their use in thinking about students' journeys throughout their mathematics education.

Naomi is a lecturer in mathematics education at the University of Otago, College of Education and has been presenting her research to both academic audiences and schools as the Bevan Werry speaker 2011-2013 for the New Zealand mathematics teachers' association.

NOTE : DAY, TIME AND VENUE OF THIS SEMINAR ARE DIFFERENT FROM USUAL

130829110745

In 1805, Professor August Yulevich Davidov claimed that:
"The outstanding contributions made by Poisson and by Laplace to the mathematical theory of capillary phenomena have completely exhausted the subject and brought it to such a level of perfection that there is hardly anything more to be gained by its further investigation."

In 2011, I set out to prove Davidov wrong. My thesis re-examines the phenomenon of capillarity through the powerful language of differential geometry, allowing us to describe fluid interfaces attached to solid surfaces with exotic geometries. The embedding formalism frees the interface from the artificial constraint of selecting a coordinate system in advance, and provides a direct link between the principle of least action and the numerical approximation of the solution, in this case by Finite Elements.

In this talk, I will describe the motivation for the project, its aims, and the initial derivation of the equations to be solved. I will then discuss some of the more detailed aspects of the mathematical formulation, as well as the numerical scheme.

130719112128

A famous and difficult theorem of Szemeredi asserts that every subset of the integers of positive density will contain arbitrarily long arithmetic progressions; this theorem has had four different proofs (graph-theoretic, ergodic, Fourier analytic, and hypergraph-theoretic), each of which has been enormously influential, important, and deep. It had been conjectured for some time that the same result held for the primes (which of course have zero density). I shall discuss recent work with Ben Green obtaining this conjecture, by viewing the primes as a subset of the almost primes (numbers with few prime factors) of positive relative density. The point is that the almost primes are much easier to control than the primes themselves, thanks to sieve theory techniques such as the recent work of Goldston and Yildirim. To "transfer" Szemeredi's theorem to this relative setting requires that one borrow techniques from all four known proofs of Szemeredi's theorem, and especially from the ergodic theory proof.

NOTE : DAY, TIME AND VENUE OF THIS LECTURE ARE DIFFERENT FROM USUAL

The Maclaurin Lectureship is a new reciprocal exchange between the New Zealand Mathematical Society and American Mathematical Society. A New Zealand and a United States-based mathematician will tour each other’s countries on alternate years, with the lecturers to be chosen by both societies.

Terence "Terry" Tao FRS was born on 17 July 1975 in Adelaide. Terry is an Australian mathematician working in harmonic analysis, partial differential equations, additive combinatorics, ergodic Ramsey theory, random matrix theory, and analytic number theory. He currently holds the James and Carol Collins chair in mathematics at the University of California, Los Angeles, and has joint US and Australian citizenship. He was one of the recipients of the 2006 Fields Medal, the mathematical equivalent of the Nobel Prize.

Terry was a child prodigy. According to Smithsonian Online Magazine, he could carry out basic arithmetic by the age of two. He remains the youngest winner of each of the bronze, silver and gold medals in the history of the International Mathematical Olympiad, at ages 11, 12 and 13 respectively. He published his first paper at age 15, and received his bachelor's and master's degrees at the age of 16 from Flinders University. From 1992 to 1996, Terry was a graduate student at Princeton University under the direction of Elias Stein, receiving his PhD at the age of 20. He joined the faculty of the University of California, Los Angeles in 1996. He was promoted to full professor at UCLA at 24, and remains the youngest person ever appointed to that rank by that institution.

Terry has won numerous honours and awards. He received the Salem Prize in 2000, the Bôcher Memorial Prize in 2002, and the Clay Research Award in 2003, for his contributions to analysis including work on the Kakeya conjecture and wave maps. In 2005 he received the American Mathematical Society's Levi L. Conant Prize with Allen Knutson, and in 2006 he was awarded the SASTRA Ramanujan Prize.

The lectureship is named after Richard Cockburn Maclaurin (1870 – 1920), who studied at Auckland University College – now The University of Auckland – and Cambridge University, and won the Smith Prize in Mathematics and Yorke Prize in Law. He was Foundation Professor of Mathematics at Victoria University College, as well as Dean of Law and Professor of Astronomy. In 1908 he became President of the Massachusetts Institute of Technology (MIT) and helped transform that institution into a world-class research-based technological university.

130628150503

If allowed to cross regional borders, invasive species pose one of the greatest threats to global biodiversity, environment, economic activity and human and animal health. When there are many thousands of well recognised invasive species that could cross any border, it is difficult to know where to start. Self Organising maps (SOMs) can be used to analyse species assemblages of a large number of global geographic regions as well as a large number of invasive species to give regional invasive species profiles that can provide very useful information for pest risk assessment.

Predicting future establishment and spread is integral to invasive species risk analysis. Increasingly, different classes of models are used to integrate the high dimensional array of climate and biotic information required to gain greater predictive precision. Such models are used when detailed functional relationships between a species and its environment are lacking. A range of modelling approaches will be illustrated that are designed to both predict and mitigate the impact of invasive species in new areas. Additionally, important issues that need to be resolved to improve these models to establish good practice and sensible modelling protocols for risk assessments, will be discussed.

130712101756

We consider Oldroyd type fluids in bounded domains with a sufficiently smooth boundary. An Oldroyd type fluid can be described as a parabolic-hyperbolic coupled system of partial differential equations. After a short introduction to fluid dynamics we discuss various approaches to attack this problem.

Then we present a recent result on existence and uniqueness of global strong solutions in the $L_p$-setting for small initial data. The idea of the proof is to investigate spectral properties of the associated Oldroyd operator. Depending on the type of the fluid, $0$ is either in the resolvent set or in the spectrum of the Oldroyd operator. Whereas it is rather easy to show the existence and uniqueness of global solutions in the first case the second case is more involved.

130807114403

This talk will be an introduction to paraconsistent mathematics. A paraconsistent logic is one in which local contradictions do not always imply global absurdity, so that it is possible to study inconsistent structures in a coherent way. As an example, the original `naive' theory of sets is very intuitive, but also famously paradoxical; a paraconsistent logic allows a full investigation of this theory. We will look at techniques for understanding 'naive' sets, and how such objects may be useful in proving new results.

130719111529

The lasso is a wildly popular technique for data analysis, with thousands of diverse applications across a wide variety of areas. At the core of the lasso is the ℓ₁ regularization problem $$\min_{\boldsymbol{\beta}} \|\mathbf{X} \boldsymbol{\beta} - \mathbf{y}\|_2 \mbox{ subject to } \| \boldsymbol{\beta} \|_1 \leq \lambda,$$where λ is a regularization parameter controlling the sparsity of the solution. For fixed λ the optimization is easy, in practice we need to consider the range of possible λ values. This is typically done using the ‘Homotopy Algorithm’ of Osborne et al. (2000) which traces a path of optimal solutions β as λ changes. The algorithm forms the basis of Efron et al’s famous ‘LARS’ method as well as compressed sensing techniques in signal analysis.

Despite its widespread use, the homotopy algorithm suffers from two significant problems: it can’t handle underdetermined systems and it breaks down in the presence of ties (hence the need for jittering). I will describe a version of the algorithm which addresses both issues, both for the lasso and the non-negatively constrained positive lasso.

130725090210

In this talk I will discuss the problems of multi-touch touchscreen tracking and focus on one particular system that uses linescan cameras and retro-reflective strips. Such a system is very good for single-touch tracking, but complexities arise when multi-touch tracking is needed. I show that a particle filter incorporating local search provides an accurate and efficient solution. The system achieves over 95% correct tracking with an error tolerance of 1mm for up to 8 touches. I will also briefly discuss some of the limitations of particle filters in practice.

130712100318

Scientists have now mapped the human genome. The next frontier is understanding human epigenomes: the ‘instructions’ which tell the DNA whether to make skin cells or blood cells or other body parts. Apart from a few exceptions, the DNA sequence of an organism is the same whatever cell is considered. So why are the blood, nerve, skin and muscle cells so different and what mechanism is employed to create this difference? The answer lies in epigenetics. If we compare the genome sequence to text, the epigenome is the punctuation and shows how the DNA should be read.

Advances in DNA sequencing in the last five years have allowed large amounts of DNA sequence data to be compiled. For every single reference human genome, there will be literally hundreds of reference epigenomes, and their analysis could occupy biologists, bioinformaticians and biostatisticians for some time to come.

This lecture is free and open to the general public. However, to ensure a seat, please obtain a ticket at www.royalsociety.org.nz Enquiries to: lectures@royalsociety.org.nz or 04 470 5781

130530132221

Genetics Otago, in association with the Department of Mathematics and Statistics

The reconstruction of ancestral states on evolutionary trees is now a standard method of making inferences about character evolution. Increasingly, ancestral-state reconstructions have been used to support arguments for unexpected violations of "Dollo's Law", including reacquisition of eyes and pigment in cave-adapted organisms, re-evolution of sexuality from parthenogenetic ancestors, and the reacquisition of wings in stick insects. Methods for inference of ancestral states have become increasingly sophisticated, with maximum-likelihood and Bayesian methods largely replacing earlier reliance on maximum parsimony. However, despite the apparent rigor of these stochastic-model approaches, a number of aspects of their application are unsatisfying. These include model misspecification and inadequate data for accurate model-parameter estimation. In this talk I will review a number of studies that have employed ancestral-state reconstruction methods, and present the results of simulation studies that attempt to answer the question of whether the conclusions from these studies can be trusted. Future directions that attempt to overcome the limitations of existing methods will be outlined (probably too vaguely).

Professor David Swofford is the author of PAUP (Phylogenetic Analysis Using Parsimony), a computational phylogenetics program for inferring evolutionary trees (phylogenies). http://paup.csit.fsu.edu/about.html

130605132444

Potential theory has, thus far, been a missing chapter in the development of constructive mathematics. Here we take some steps to remedy that.

First, we present a constructive proof of the existence of the weak solution to the Dirichlet problem when the domain is internally approximable by certain compact subsets. The extra hypotheses we pose are constructively strictly stronger than the usual classical requirement, yet are typically satisfied in that setting.

Second, we present Brouwerian examples showing that the existence of solutions in the general case of the Dirichlet problem is an essentially non-constructive proposition. A corollary of this is that there can be no universal algorithmic procedure for computing solutions to the well-known Navier-Stokes equations for fluid flow.

130215134915

For a directed graph $E$, we associate a projection to each vertex and a partial isometry to each edge, and form the universal $C^*$-algebra $C^*(E)$, called graph $C^*$-algebra, which is generated by a collection of those projections and partial isometries that satisfy certain relations. One of the benefits of studying graph $C^*$-algebras is that we can visualize very abstract properties of $C^*$-algebras via concrete characteristics of underlying graphs.

As higher dimensional analogues of directed graph, a $k$-graph can be viewed as a $k$-colored graph with a certain property, called factorization property. As done for a directed graph, we can similarly form the universal $C^*$-algebra associated to a relatively nice $k$-graph. The theory of $k$-graph $C^*$-algebras carries more complexity but the $k$-graph $C^*$-algebras provide concrete examples of many classifiable $C^*$-algebras and an important testing ground for complicated mathematical structures. In this talk, we first discuss the elementary properties of $k$-graphs and their $C^*$-algebras, and describe the ideal structure of them. In particular, we describe the primitive ideals of a row-finite sourceless $k$-graph $C^*$-algebra in terms of a maximal tail and a character of its periodicity group.

130215135022

In recent years there has been a quiet revolution from signal processing to probabilistic inference as a framework for studying perception. We and other animals behave as if our neurons compute the posterior density of state variables relevant to the tasks we perform. Neurons are highly specialized, energetically expensive cells that cannot have evolved unless there was a utility gradient for sense data or behaviour to be packaged into brief, energetic pulses. What does the ecology of the time when neurons first appeared, at the Ediacaran-Cambrian boundary 543 million years ago, tell us about the utility of spiking neurons? Analysis of simple models indicates that the evolution of neurons probably had nothing to do with any advantage in being Bayesian, but spiking neurons provided both an opportunity and a mechanism for Bayesian inference that could be exploited immediately. The ecology of ancient oceans may hold the key not only to explaining why we are natural born Bayesians, but how we do it.

130225153241

In my research I have often made use of skew product graphs. I will describe their construction and a few of their basic properties. Then I will talk about certain connectivity properties of graphs and how they behave under the skew product construction. Finally I will indicate how the graphical results which I have presented apply to my research.

130328134251

We have studied the k-mer spectra of more than 100 species from Archea, Bacteria, and Eukaryota, particularly looking at the modalities of the distributions. As expected, most species have a unimodal k-mer spectrum. However, a few species, including all mammals, have multimodal spectra. These species coincide with the tetrapods. Genomic sequences are clearly very complex, and cannot be fully explained by any simple probabilistic model. Yet we sought such an explanation for the observed modalities, and discovered that low-order Markov models capture this property (and some others) fairly well.

Joint work with David Horn, Nick Goldman, Yaron Levy, and Tim Massingham.

130227121431

In this talk various type of the convexity concepts are introduced, and some Bernstein-Doetsch type results are presented.

The celebrated result of Bernstein and Doetsch (1915) shows that the Jensen-convexity of a function under a weak regularity property assumption (e.g. locally boundedness from above at a point) implies the continuity and convexity of the function.

First we extend this result to the so-called approximately convex functions, a notion which was introduced by Hyers and Ulam (1952). We show that the approximate Jensen convexity under some natural assumptions implies the approximate convexity of the function. We use Takagi-type functions in our proofs, yielding optimal estimates.

Finally, we investigate some generalizations of convex functions (e.g. (k,h)-convex, h-convex, s-convex in first (Orlicz) and second (Breckner) sense, P-functions, Godunova-Levin functions) and prove some properties of these functions. We show that every local minimizer of a (k,h)-convex function (similar to the usual convexity) under some assumptions is a global one.

130205144027