2010 Seminars
Hui Huang (University of British Columbia, Vancouver)
Reconstructing images, surfaces and points
Tuesday, 14 December 1.30pm
Room 446, Erskine Building
The date of this seminar has been changed. It was originally scheduled for Friday December 10 at 2:10pm.
Abstract. The goal of this talk is to introduce effective techniques for solving inverse problems arising from image, surface and point cloud reconstruction. Both computational and theoretical issues will be discussed.
We first consider implicit image recovery that involve solving linear systems at each iteration. An adaptive Huber regularization functional is used to select the most reasonable model and a global convergence result for lagged diffusivity is proved. Two mechanisms--multilevel continuation and multigrid preconditioning--are proposed to improve efficiency for large-scale problems. Next, explicit image recovery involving the construction of an artificial time-dependent differential equation model followed by forward Euler discretization are analyzed. A rapid, adaptive scheme is then proposed, and additional hybrid algorithms are designed to improve the quality of such processes.
For surfaces, we discuss techniques for faithfully reconstructing triangular meshes with different features. Some models contain a lot of small yet visually meaningful details, and others consist of large flat regions, long sharp creases and distinct corners. For models with significant intrinsic texture, we methodically develop a fast multiscale anisotropic Laplacian smoothing algorithm. To reconstruct other piecewise smooth CAD-like models, we design an efficient hybrid algorithm based on specific vertex classification, which combines K-means clustering and geometric a priori information. Hence, we have a set of algorithms that efficiently handle smoothing and regularization of meshes large and small in a variety of situations.
In addition to images and surfaces, we also study unorganized point clouds, which contain outliers, noise, non-uniformities and without normal information. Based on point positions alone, we propose a weighted locally optimal projection operator to consolidate raw scan data, and introduce an iterative framework for robust normal estimation. The new measure that combines Euclidean and angular distances with propagation directions can successfully handle the challenging close-by surface sheet problem. We demonstrate at the end how a well-consolidated point cloud steers conventional surface generation schemes towards a proper interpretation of the input data.
Thomas Hangelbroek (Vanderbilt University, Nashville)
Approximation and Interpolation on Manifolds with Kernels
Monday, 13th December, 2:10pm
Room 446, Erskine Building
Abstract. Using kernels to fit data on spheres and Euclidean domains has been an active field of research in approximation theory for at least three decades. Many aspects of treating large scale, gridded or nearly gridded data on spheres or at regions are, by now, well-understood. Employing kernels to approximate highly unstructured data is not nearly as well-understood. Similarly, kernel approximation on esoteric domains like graphs, groups and manifolds has only recently become an active field of study.
In this talk, I’ll introduce the basic tools for interpolating and approximating with kernels on Riemannian manifolds, with examples from $R^d$ , $S^d$ and $SO(3)$. The goal is to discuss two major challenges in this area.
- constructing effective bases: well-conditioned, local bases that scale according to the spacing of the data
- treating highly unstructured data: data with large gaps and regions of accumulation, or having density that varies spatially
and to give an overview of recent progress in tackling these problems.
Maarten McKubre-Jordens (University of Canterbury)
Real Analysis in Paraconsistent Logic
Thursday, 25 November 3.10pm
Room 446, Erskine Building
Abstract. This talk presents recent work on analysis of the real line using an inconsistency-tolerant (paraconsistent) logic. A basic introduction to paraconsistent logics will be given. We show that basic field and compactness properties hold, by way of novel proofs that make no use of consistency-reliant inferences. It is shown that while inconsistency at the level of algebraic operations on the real number field cannot be contained (at least in the current formulation), it does not necessarily trivialize the system of real numbers, leaving open other prospects for non-nonsensical contradiction.
This seminar is a preview of the talk I will be giving at the NZMS Colloquium. This means that you have a chance to see what I'll be talking about so that you don't have to come to my talk there!
Michael DeGiorgio (University of Michigan)
Explaining Worldwide Patterns of Human Genetic Variation using a Coalescent-based Serial Founder Model of Migration Outward from Africa
Thursday, 18 November 3.10pm
Room 446, Erskine Building
Abstract. Studies of human populations have discovered multiple trends in genetic variation in relation to geography. Two of these trends are the decrease in genetic diversity and the increase in linkage disequilibrium with increasing distance from Africa. In this talk, I will describe a model of human demographic history, termed the serial founder model, and show that it explains these trends in genetic variation observed in human data. In contrast, we find that a complementary demographic model, which we term the archaic persistence model, produces opposite trends. We then develop a simpler model to illustrate that the feature that permits the serial founder model, but not the archaic persistence model, to explain these trends observed with increasing distance from Africa is its incorporation of a cumulative effect of genetic drift as humans populated the world. We also show that patterns in genetic diversity observed empirically in humans, and patterns observed in the serial founder model through simulations, can be analytically predicted using coalescence-time distributions under the model.
Helmut Schwichtenberg (Universität München)
Proofs and Programs
Thursday, 11th November 3.10pm
Room 446, Erskine Building.
Abstract. We address the delicate question of how to ‘decorate’ proofs in order to optimize their computational content. It is shown that a unique optimal such decoration exists and some examples are discussed.
Stefano Grassi (University of Perugia, Italy)
Topics in Unobserved Components Models
Tuesday, 9th November 3.10pm
Room 446, Erskine Building.
Abstract. The seminar is composed by three parts.
The first part of the presentation focuses on recent development in Dynamic Factor Models put forward by Grassi et al. (2010) and Jungbacker et al.(2009). In our paper we implement the maximum likelihood estimation method for high-dimensional dynamic multi-factor models in presence of large amount of missingness. The exact treatment of missing values with reduction technique proposed by Jungbacker and Koopman (2008) allows the estimation of the factors and a large number of parameters in very fast and efficient way. We use this new methodology to analyze the evolution of the degree of global and regional interdependence over the period 1950-2007. We decompose aggregate output for 150 countries into factors that are (i) common across all countries, (ii) common across regional areas and (iii) specific to each countries. The paper provides a systematic assessment of the estimation strategy and discusses the empirical evidence in the light of the previous literature. Finally we provides new results about global and regional convergence.
The second part relates to a recently proposed Bayesian model selection technique, known as stochastic model specification search. As shown in Grassi and Proietti (2010) the methodology can be quite successfully applied to discriminate between stochastic and deterministic trends. In particular, we formulate autoregressive models with stochastic trends components and decide on whether a specific feature of the series, i.e. the underlying level and/or the rate of drift, are fixed or evolutive.
In the third part I will provide some ideas how to merge stochastic model specification search and Dynamic Factor Model. Moreover other ideas to improve of the testing procedure will be provided.
Sebastian Boecker (Friedrich Schiller University, Germany)
Solving Hard Problems in Bioinformatics: Identifying Unknown Metabolites using Tandem Mass Spectrometry
Thursday, 4th November 3.10pm
Room 446, Erskine Building.
Abstract. Computationally hard problems often lie at the core of bioinformatics research. Parameterized algorithmics tries to circumvent the complexity of these problems by exploiting problem-specific parameters. Here, we seek exact algorithms that compute optimal solutions with provable worst-case running time polynomial in the size of the problem. In my talk, I will present several examples of NP hard problems for which we have successfully developed efficient parameterized algorithms, such as Cluster Editing for clustering objects, or Flip Supertrees for computing phylogenetic supertrees. Even though these problems are computationally hard, we can optimally solve surprisingly large instances in practice.
I will pay particular attention to some problems that arise in the context of identifying novel metabolites using tandem mass spectrometry. The structural elucidation of organic compounds in complex biofluids and tissues remains a significant analytical challenge. The automated identification of compounds is generally limited to searching databases in spectral libraries. In my talk, I will present a method for interpreting tandem mass spectra by computing fragmentation trees that establish not only the molecular formula of the compound and all fragment ions, but also the dependencies between fragment ions.
Roberto Baragona (Sapienza University of Rome)
A Class of Non-linear Non-stationary Functional Autoregressive Model Building by Genetic Algorithms
Tuesday, 2nd November 3.10pm
Room 446, Erskine Building.
Abstract. In this seminar, a class of functional autoregressive models will be presented that may be useful in a wide range of applications. These models may be structured so that they are able to take into account non-linearity and non-stationarity features that may be present in the data. Identification and estimation of such models is a demanding task for which using genetic algorithms is suggested. Examples of application to an environmental data set and a set of financial data are worked out to both illustrate the model building procedure and show the effectiveness of genetic algorithms to handle the difficult task of detecting multiple structural breaks and modelling them according to their non-linearity or non-stationarity characteristics. The presentation includes the following:
- the functional autoregressive models are illustrated and detailed specification of a rather general class enclosed in their set is shown
- the genetic algorithms are devoted a brief illustration
- the Canadian lynx data series is modelled by such a class of models which support the existence of non-linearity in the level
- the Dow Jones industrial average index for years 2005-2009 (daily closure values) is modelled and results suggest that in the whole time span under consideration non-linear and non-stationary features cannot be overlooked while high volatility shows in this series only after mid- September 2008.
Bill Taylor (University of Canterbury)
Optimal Strategies for Symmetric Win/Loss Games
Thursday, 28th October 3.10pm
Room 446, Erskine Building.
Abstract. We look at zero-sum games where the only payoffs are ± 1 (or 0), and otherwise symmetric. Equivalently, completed round-robin tournaments. The basic case is Scissors-Paper-Rock.
The surprising theorem is proved, that any optimum mixed strategy can only be supported by an ODD number of pure strategies.
(This talk is identical to the one given at the Wellington maths colloquium 3 or 4 years ago.)
Allan Willms (University of Guelph, Canada)
Parameter Range Reduction for ODE Models Using Monotonic Descretizations
Thursday, 21st October 3.10pm
Room 446, Erskine Building.
Abstract. Many models of physical and biological processes are given in terms of systems of ordinary differential equations involving possibly large numbers of parameters whose values are unknown. In many cases, the values of these parameters must be inferred from data representing the solutions of the model equations. The standard method is to iteratively integrate the system of equations at specified parameter values, compare the model solutions with the data and then adjust the parameter values in the "down hill" direction. As an alternative, or precursor to seeking this kind of "best point estimate", we present a scheme where the parameters are represented by ranges and the data are used to eliminate portions of these ranges that yield model solutions which are inconsistent. The talk will present a brief overview of the scheme and will focus on several aspects of the overall problem. In particular, the talk will discuss the types of discretizations of the ODE that make the scheme efficient and will describe a method to represent the recorded data in terms of ranges using a piecewise linear band.
John Cleary (University of Waikato)
Surviving Programming in a Parallel World
Thursday, 14th October 3.10pm
Room 446, Erskine Building.
Abstract. Since the end of the first Moore era in 2004, the number of transistors on chips have continued to grow exponentially but the clock speed and number of transistors in a CPU core has remained constant. This leads to a hardware world where the number of cores in CPUs is growing exponentially and where non-traditional architectures such as GPUs, FPGAs and ASICs are becoming available and commercially viable. This poses an enormous challenge for software engineers. Modern programming languages don't do parallel or architecture independence well. It is possible that the parallel hardware world will fail because of an inability to cost effectively write software.
In this talk I will examine a highly abstract programming language that aims both to be more expressive and easier to program than existing languages and to effectively deal with both parallelism and architecture independence. I will describe its semantics and how it can transform its surface representations into different target data structures. I will then show how we intend to retarget it to architectures such as many-core CPUs and GPUs.
John Lewis (WETA Digital)
Applications of Scattered Data Interpolation and Machine Learning in VFX
Tuesday, 12th October 3.10pm
Room 446, Erskine Building.
Abstract. This talk will first survey several applications of scattered data interpolation in visual effects, including texture synthesis, creature skinning, and inpainting. Radial basis functions, Poisson interpolation, and Gaussian processes are the underlying mathematical approaches to be mentioned. A second part of the talk will note the connection between scattered interpolation and machine learning (e.g. classification/regression) problems and show applications in inpainting and facial tracking. Examples of unsolved problems will be mentioned with the aim of stimulating discussion.
James Oxley (Louisiana State University)
Communicating Mathematics
Thursday, 7th October 3.10pm
Room 446, Erskine Building.
Abstract. In the Mathematics Department at Louisiana State University, all postgraduate students are required to take a two-semester course with the above title. Most mathematics postgraduate students at LSU are required, as part of their paid assistantships, to teach undergraduate students. Our course aims not only to prepare the postgraduate students for their jobs as teachers, but also to give them basic training in mathematical exposition readying them for the task of writing a thesis or a paper and for giving a conference talk or a departmental seminar. This talk will describe the details of this course, which has now been running for more than a decade.
Sharleen Forbes (Victoria University & Statistics New Zealand)
New ways of visualising official statistics
Tuesday, 5th October 3.10pm
Room 446, Erskine Building.
Abstract. Official statistics provide the evidence base for much of government policy but these have traditionally been released in simple and standard tables and graphs. The ability to harness the power of the internet together with new graphical techniques has led to a burst of creativity in a number of national statistics offices. New static and dynamic graphs and maps, combined interactive graphs and tables and graphs and maps that allow users to interrogate and interact with data in new ways will be demonstrated. Examples given include multidimensional scatterplots, cartograms, a CPI kaleidoscope, interactive maps, dynamic population pyramids and commuter flows and Hans Rosling's Gapminder. A word or two of warning on the possible limitations of data visualisation will also be given.
Douglas S. Bridges (University of Canterbury)
Weak-operator Continuous Functionals - Constructively
Thursday, 30th September 3.10pm
Room 446, Erskine Building.
Abstract. In this talk I shall first clarify what is meant by constructive mathematics (à la Bishop). Then I shall look at the constructive version of the characterisation theorem for weak-operator continuous linear functionals on the space of bounded operators on a Hilbert space.
This will involve my presenting (without proof!) some of the substantial lemmas - sometimes ones with little or no classical content - that are developed en route to obtaining the desired characterisation.
Geospatial Research Centre Presentation (University of Canterbury)
An Introduction to the Geospatial Research Centre and its Capabilities
Thursday, 23rd September 3.10pm
Room 446, Erskine Building.
Abstract. The presentation will show the Geospatial Research Centre (GRC) as a collection of people and tools available to provide complex data-gathering in the geospatial environment.
Professor Kevin Furlong (Penn State University & UC Erskine Fellow)
Putting the 2010 Canterbury Earthquake into Context: The Why and How
Thursday, 16th September 3.10pm
Room 446, Erskine Building.
Abstract. Earthquakes such as the September 4 Canterbury event reflect New Zealand's position astride the Pacific-Australia plate boundary.
Although we tend to focus on the major, highly visible fault lines, such as the Alpine Fault, it is important to remember that many secondary structures can host large and damaging earthquakes. This event provides evidence of the ongoing plate motions but also the manner in which plate boundary deformation is distributed.
This presentation will focus on both details of the actual event and where it fits into the bigger picture of plate tectonics in New Zealand.
Maarten McKubre-Jordens (University of Canterbury)
Infinity in Computable Probability: Logical Proof that William Shakespeare probably was not a Dactylographic Monkey
Thursday, 19th August 3.10pm
Room 446, Erskine Building.
Abstract. The "Infinite Monkey Theorem" has frequently been used to emphasize the danger of reasoning about probability when infinity is involved. The theorem states that at least one of infinitely many monkeys, producing a character string equal in length to the collected works of Shakespeare by striking typewriter keys in a uniformly random manner, will with probability one reproduce the collected works.
The bad news: classically, finding the champ chimp will take A Very Long Time. But there is worse news, constructively. Surprisingly, it is possible to assign to an infinite number of monkeys probabilities of reproducing Shakespeare's collected works in such a way that while it is impossible that no monkey reproduces the collected works, the probability of any finite number of monkeys reproducing the works of Shakespeare is arbitrarily small. This result potentially destroys any hope of completing such a project with as much computing power as we could ever hope to have.
However, the news is not all bad. For sufficiently large troops of monkeys, the fraction among all possible probability distributions of such pathological distributions is vanishingly small; so the chances of running into one of these distributions in practice is negligible.
Koen Struyve (Ghent University)
Projective Lines, Trees, Valuations and Beyond
Thursday, 12th August 3.10pm
Room 446, Erskine Building.
Abstract. We start by studying trees where the ends correspond with the points of a projective line PG(1,K) over a field K. When the group PGL(1;K) acts on this tree inducing its natural action on the ends of the tree, a beautiful argument by Jacques Tits shows how this tree arises from a valuation of the field. Vice versa, given a valuation of the field one can construct such a tree. These trees are useful to study properties of projective lines (or projective spaces in general). In particular we will show how these lead to "natural" epimorphisms of projective lines.
In the final part of the talk, we briefly indicate how this fits in with the wider viewpoint of spherical Moufang buildings (which include the classical and exceptional projective and polar spaces, generalized polygons,... ). For example, one obtains the following result:
Theorem (K.S.) Epimorphisms of a spherical Moufang building of rank at least 2, defined over a field and which is not an exceptional generalized quadrangle of type E8, correspond to valuations over the field of definition satisfying certain compatibility conditions.
Thomas Forster (Cambridge University)
Yablo's Paradox
Thursday, 5th August 3.10pm
Room 446, Erskine Building.
Abstract. The heyday of logical paradoxes was the crisis in foundations of 100 years ago. However, although the peak of new discoveries has passed, stragglers can still appear. This is one of them.
Yablo's paradox concerns the countably family of propositions y_i, for i in N, where each y_i is the assertion that all y_j with j > i are false. The point that Yablo was making with this is that contradiction is achieved here without self-reference. This allegation has been contentious, but in any case the paradox can be used to make other and equally interesting points, and I will elaborate on some of them in this talk.
Hans Feichtinger (University of Vienna)
Mathematical Foundations of Gabor Analysis
Thursday, 29th July 3.10pm
Room 446, Erskine Building.
Abstract. Gabor Analysis can be viewed as a sub-field of time-frequency analysis, which essentially is making use of a time-localized version of the Fourier transform, using a certain (localization) window. D. Gabor has suggested in 1946 to use the Gauss-function, due to its optimality property for the Heisenberg relation, and sample it over the integer lattice in TF-domain.
We will give a summary of results obtained in the last 30 years concerning Gabor families, in particular (redundant) Gabor frames and Gabor Riesz bases, and will try to relate these findings to applied problems, e.g. in audio (MP3) and mobile communication, but also mention the role of so-called Banach frames, Gelfand triples for other branches of modern analysis.
Konstantin Mischaikow (Rutgers University)
A Combinatorial Framework for Nonlinear Dynamics
Thursday, 22nd July 3.10pm
Room 446, Erskine Building.
Abstract. Much of the focus of dynamical systems is on the existence and structure of invariant sets. What we have learned over the past century is that this existence and structure is extremely rich - from the perspective of applications perhaps too rich. For example, any valid description of the global dynamics of a multiparameter nonlinear system arrived at through numerical simulation or the accumulation of experimental data requires that the structures being described can be represented via a finite amount of data and are robust with respect to perturbations. We shall outline a combinatorial approach to dynamics that has these properties. In particular, using a simple example multiple parameter problem arising from population dynamics we will build a database that provides a coarse rigorous description of the global dynamics at every parameter value. We will describe the algorithms used to approximate the dynamics and how algebraic topological tools are used to provide rigorous interpretations of the underlying continuous system based on the approximate dynamics.
Lisa Carbone (Rutgers University)
Discrete Symmetries of Infinite Dimensional Lie Groups
Thursday, 15th July 3.10pm
Room 446, Erskine Building.
Abstract. Kac-Moody groups are natural generalizations to infinite dimensions of finite dimensional simple Lie groups. These infinite dimensional groups appear in the study of algebraic symmetries of general relativity and a theory known as supergravity, which incorporates both general relativity and supersymmetry. The discrete symmetries, namely forms of these groups over the integers, play a particularly important role. We discuss the problem of constructing these groups and characterizing their symmetries.
Ingram Olkin (Stanford University)
Meta-analysis: History and statistical issues for combining the results of independent studies
Thursday, 8th July 3.10pm
Room 446, Erskine Building.
Abstract. Meta-analysis enables researchers to synthesize the results of independent studies so that the combined weight of evidence can be considered and applied. Increasingly meta-analysis is being used in medicine and other health sciences, in the behavioral and educational fields to augment traditional methods of narrative research by systematically aggregating and quantifying research literature.
Meta-analysis requires several steps prior to statistical analysis: formulation of the problem, literature search, coding and evaluation of the literature, after which one can address the statistical issues.
We here review some of the history of meta-analysis and discuss some of the problematic issues such as various forms of bias that may exist. The statistical techniques that have been used are nonparametric methods, combining proportions, the use of different metrics, and combining effect sizes from continuous data.
Ingram Olkin (Stanford University)
Probabilistic proofs of matrix inequalities
Wednesday, 7th July 3.10pm
Room 446, Erskine Building.
Abstract. Probabilistic inequalities often have the advantage of providing intuitive proofs. Furthermore, many inequalities achieve equality for two-point distributions, in which case sharpness is readily exhibited. This talk provides an exposition of a variety of probabilistic inequalities, and their counterpart matrix inequalities. Examples are the Ky Fan inequality, bounds for the product, difference and ratio of quadratic forms, bilinear forms, the Hadamard inequality, and more.
John Bamberg (University of Western Australia)
Hemisystems of Generalised Quadrangles
Thursday, 24th June 3.10pm
Room 446, Erskine Building.
Abstract. Generalised quadrangles are objects in finite geometry which are closely related to groups of Lie rank 2, and hemisystems of the generalised quadrangles we are interested in are certain sets lines giving rise to interesting strongly regular graphs, two-weight codes and partial quadrangles. Segre (1965) showed that there exists a hemisystem of the classical generalised quadrangle H(3,9), and it was conjectured by J.A. Thas in 1995 that no hemisystem of H(3,q2) exists for q>3. Ten years later, Cossidente and Penttila proved that for every q odd, there exists a hemisystem of H(3,q2). The only known generalised quadrangles with the same parameters of H(3,q2), q odd, are the flock generalised quadrangles. We will present in this talk an improvement of Cossidente and Penttila's results to flock generalised quadrangles. The speaker will give an overview of the basic notions in finite geometry needed to understand the main result.
(Joint work with Michael Giudici and Gordon Royle)
Elena Moltchanova (National Institute for Health and Welfare, Finland)
The effect of prenatal stress on long-life health. Survival Analysis applied to Helsinki Bombings
Tuesday, 8th June, 2:10pm
Room 446, Erskine Building.
Abstract. Many studies suggest that prenatal stress might have an effect on lifelong health, especially in terms of psychological outcomes and cardiovascular disease. Among sources of severe stress are natural disasters and wars. The availability of detailed follow-up data on all the people born in Helsinki during the years 1934-1944, i.e., just before and during the Second World War, allows us explore, whether war-time stress has had a noticeable effect on life-long health of our cohort.
Individual hazard functions were estimated based on Weibull distribution, and a smoothing conditional autoregressive model was applied to the distribution parameters along the time axis. The exact timing and extent of the recorded bombings was also taken into account. A Bayesian analysis was then performed.
I will mention the challenges often encountered in large-scale epidemiological cohort studies, and explain the different types of results, which can be derived from our model.
Laura Boykin (Lincoln University)
Biosecurity, Species Delimitation-Is There a Holy Grail?
Thursday, 3rd June, 3:10pm
Room 446, Erskine Building
30 minutes, followed by a break and 20 minutes of discussion.
Abstract. What is a species? This question has plagued scientist for many years and has a long and complicated history. For most well studied and apparently distinct evolutionarily groups this is rarely an issue. However, for recently diverged taxa, distinguishing and naming a species has impact beyond academic questions such as the tree of life or a scientist’s favourite study organism. Identifying the point where populations become a distinct species has direct impacts in real-world applications such as global biosecurity. When regulators are faced with an infected shipment at the borders putting a name on an exotic organism is of vital importance. Without a name there is no information on its biology and therefore its potential invasiveness. Treatment, surveillance and eradication plans cannot be actioned and the potential economic fallout from damage to New Zealand’s primary industries or trading partner complications cannot be anticipated. Often these intercepts are cryptic forms, such as the immature life stages of invertebrates or just disease symptoms on plants. The demand for quick species identification in these cases has resulted in adoption of molecular genetic methods. An increasingly popular method is DNA barcoding, taking advantage of the influx of new publically available genetic data.
Species delimitation using this approach can be straightforward in many cases. However, it has also highlighted that some very important high risk exotic pests cannot. That, together with cryptic morphologies and indistinct biological traits, effectively questions their taxonomic status. Examples will be given where species are easily defined and species identifications are straightforward. We will also highlight where DNA barcoding fails and other techniques are needed. So what are the rules about how to genetically define a species when it really matters? Statistical and bioinformatic methods, rather than biology in isolation, may hold the key to applications such as biosecurity.
Scott Graybill (University of Canterbury)
Modelling nephron dynamics and tubuloglomerular feedback
Friday, 28th May, 2:00pm
Room 446, Erskine Building
PhD presentation.
Abstract. The TGF mechanism is an autoregulatory mechanism unique to the kidney that maintains approximately constant blood flow to the organ despite wide fluctuations in pressure. It is present in each of approximately one million small tubules called nephrons in each human kidney. Oscillations in pressure, flow, and sodium chloride concentration have been attributed to the action of the TGF mechanism through a number of experimental studies.
A mathematical model of a single nephron from Holstein-Rathlou et al. uses a partial differential equation (PDE) model for the tubule and a second-order differential equation (DE) for the TGF feedback. The use of this second-order DE is uninformative as it is inherently oscillatory. The second order DE was replaced by a first order DE, which represents relaxation to a target value. This model is oscillatory due to the delays in the system.
The computationally expensive PDE model was simplified to an ordinary differential equation model by assuming a spatial profile. This model exhibits much of the same qualitative behaviour as the PDE model including sustained oscillations for similar ranges of parameter space. This model is less computationally expensive than the PDE model and allows analysis that was unfeasible with the PDE model.
Kevin Hannah (UC, Education Plus)
The Secondary Numeracy Project: Taking the Guesswork out of Mathematics
Thursday, 27th May, 3:10pm
Room 446, Erskine Building
30 minutes, followed by a break and 20 minutes of discussion.
Abstract. The Secondary Numeracy Project was first piloted in 2005. It is a program of professional development for secondary mathematics teachers that emphasises the use of mental computational strategies to solve numeric problems and aims to help secondary students develop a deeper understanding of mathematics. It is hoped that students' structural thinking about number might then be exploited to develop their understanding of algebra.
In this talk Kevin Hannah will give a brief overview of the Project and its possible implications in the university context: what skills and understanding will future students bring to university? what are the roles of visual representations and algorithms in students’ thinking?
Kevin is Team Leader of the Mathematics and Numeracy group at UC Education Plus and is also National Coordinator of the Secondary Numeracy Project.
Greg Reid & Niloofar Mani (University of Western Ontario)
Interactive Environment for Differential Equations on Manifolds
Tuesday, 25th May, 3:10pm
Room 446, Erskine Building
Abstract. There are software packages that support high-level physics-based modelling and simulation. One of the latest is MapleSim based on the mathematical manipulation language Maple which allows you to build component diagrams that represent physical systems in a graphical form. Models are automatically generated by dragging and dropping components from menus. In particular, this software automatically generates model differential equations with constraints (so-called differential-algebraic equations or DAE on manifolds). The simulations include striking 3D videos of mechanisms arising in electro-mechanical modeling. Unlike other approaches it enables the equations to be treated in analytical form.
I will give an introductory talk on this material, and sneak in some material on exciting open problems in the geometry of differential equations, such environments raise. Live demos of the software will be included.
Glen van Brummelen (Quest University, Canada)
The Mathematical Study of Historical Numerical Tables: Successes, Failures, Issues
Thursday, 20th May, 3:10pm
Room 446, Erskine Building
30 minutes, followed by a break and 20 minutes of discussion.
Abstract. Numerical tables, often relegated to the appendices of the history of mathematics, have nevertheless been crucial in the development of science and mathematics. In pre-modern cultures their appearances and roles have been evaluated periodically, but more careful studies of the tables themselves have been attempted only infrequently. Several successful analyses have allowed us to peer behind the curtain at the largely unrecorded computational culture that supported table-making. A couple of efforts have been made to produce systematic tools for analyzing tables, and these methods have led to successful analyses in diverse tables spanning millennia. Problems in this emerging field range from technical to cultural: certain statistical difficulties in studying mathematically-generated data can arise, and a few uncontrolled studies of tables that claimed dramatic but false conclusions have caused historians to view statistical methods with apprehension.
Beata Faller (University of Canterbury)
Combinatorial and Probabilistic Methods in Biodiversity Theory
Wednesday, 19th May, 2:10pm
Room 446, Erskine Building
PhD presentation.
Abstract. A central question in conservation biology is how to predict and maximize biodiversity as species face extinction. There are numerous ways to measure the biodiversity of a group of species, and one which recognizes the evolutionary linkages between species is phylogenetic diversity (PD). Briefly, given a subset of taxa, the phylogenetic diversity of that subset is the sum of the evolutionary distances of the edges of the minimal phylogenetic tree that connects this subset. The aim of my PhD research has been to develop and study models that are based on PD and can be used to forecast or optimize future biodiversity. This talk gives an overview of our models and findings. We discuss the computational complexity of optimization problems that aim to find species sets with maximum PD in different scenarios, and examine random extinction models under various assumptions to predict the PD of species that will still be present in the future.
Keith Martin (Royal Holloway, University of London)
The Cryptographic Toolkit II
Tuesday, 18th May, 1:10pm
Room 315, Erskine Building
Abstract. We will discuss further applications of cryptography. This talk will be reasonably independent of the previous one.
Keith Martin is Professor of Information Security at Royal Holloway, University of London. He is currently a Visiting Erskine Fellow in the Department of Computer Science and Software Engineering.
Keith Martin (Royal Holloway, University of London)
The Cryptographic Toolkit I
Friday, 14th May, 2:10pm
Room 446, Erskine Building
Abstract. Cryptography provides a "toolkit" of mathematical techniques for providing the fundamental security services that are required for electronic applications. We will review the essential components of this toolkit, focusing on their utility and application, rather than on any of the mathematical details behind them. This talk is intended to illustrate the use of cryptography in the "real world" (wherever that is!) rather than the theory behind it. It will be accessible to undergraduates who have taken or are taking MATH221 Algebra and Cryptography.
Keith Martin is Professor of Information Security at Royal Holloway, University of London. He is currently a Visiting Erskine Fellow in the Department of Computer Science and Software Engineering.
Pen Holland (Landcare Research)
Modelling the marsupial menace: when do possums kill trees?
Thursday, 13th May, 3:10pm
Room 446, Erskine Building
30 minutes, followed by a break and 20 minutes of discussion.
Abstract. The millions of brushtail possums inhabiting New Zealand's forests get the bulk of their diet from foliage, and have been implicated in the dieback and mortality of several native tree species. The mechanisms under which browse damage may impact on native forests operate over multiple scales, from the biting and chewing of single leaves by individual animals, to the decline in health and death of trees, and the loss of entire species from regions. Generalisable, quantitative predictions of tree mortality as a result of possum density have thus far been elusive, in part because foliage consumption rates vary hugely among trees as a result of palatability and local food availability. Data on browse damage and canopy health are plentiful, but come in the form of indices, rather than explicit physical quantities. In this talk, I will present a process-based model of foliage growth and possum intake which establishes the conditions under which consumption can be implicated directly, and indirectly, in individual tree mortality. I derive a theoretical relationship between empirical indices and foliage biomass and density, and the distribution of feeding on leaves within the canopy, in order to tie model results to empirical data. I will describe how key behavioural processes of possums must be incorporated into the model in order to replicate core qualitative and quantitative patterns of browse damage and mortality of kamahi trees in forests of the North Island.
Xin Zhao (University of Canterbury)
Extreme Value Modelling with Application in Finance
Wednesday, 12th May, 2:10pm
Room 446, Erskine Building
PhD presentation.
Abstract. Financial returns typically show clusters of observations in the tails, often termed ‒volatility clustering– which creates challenges when applying extreme value models, since classical extreme value theory assume independence of underlying process. An explicit model on GARCH-type dependence behaviour of extremes is developed by implementing GARCH conditional variance structure via the extreme value model parameters in the research. The combined model is better suited to explain the extreme quantiles and shows the advantages in making inferences and accounting for all uncertainties as a one stage model.
Another challenge in extreme value modelling is the threshold choice. To tackle the challenge, in particular with the generally asymmetric distributed financial data, a two tail GPD mixture model is proposed with Bayesian inference to capture both upper and lower tail behaviours simultaneously. The two tail GPD mixture model provides a very flexible model for capturing all forms of tail behaviour. A new Value-at-Risk (VaR) estimation method is then constructed by adopting the proposed mixture model and two-stage method: where volatility estimation using a latent volatility model (or realized volatility) followed by the two tail GPD mixture model applied to independent innovations to overcome the key issues of dependence, and to account for the uncertainty associated with threshold choice.
Jeanette McLeod (University of Bristol)
Asymptotic enumeration of integer matrices
Monday, 10th May, 2:10pm
Room 446, Erskine Building
50 minute specialised research seminar.
Abstract. Let 
= (s1, s2,...,sm) and 
= (t1, t2,..., tn) be vectors of non-negative
integers. Let M(m, ; n, ) be the number of m × n matrices over {0, 1, 2,...}
with the ith row summing to si and the jth column summing to tj . We are
interested in determining the asymptotic value of M(m, ; n, ) as m, n → ∞
under suitable conditions on and . In this talk we will survey the work
that has been done in the area and explore some of the techniques used.
In particular, we will present the calculation for estimating the number of
n × n symmetric matrices over {0, 1, 2,...} with zeros on the main diagonal
and each row and column summing to 
, for sufficiently large . Some time
will also be spent considering variations of this enumeration problem and the
application to the analysis of random matrices.
Liangyi Zhao (Nanyang Technological University & Max-Planck-Institut Für Mathematik)
On the Low-lying Zeros of Families of L-functions
Friday, 7th May, 2:10pm
Room 446, Erskine Building
50 minute specialised research seminar.
Abstract. In this lecture, I will present some one level density theorems for the low-lying zeros of families of L-functions. The families under consideration are those of L-functions associated with holomorphic Hecke eigenforms of level 1 and weight k twisted with quadratic Dirichlet characters, cubic and quartic Dirichlet L-functions and L-functions associated with elliptic curves. These results are from separate joint works with S. Baier and P. Gao.
Volkmar Liebscher (University of Greifswald)
Modelling of (small) gene regulatory networks by piecewise deterministic Markov Processes
Thursday, 6th May, 3:10pm
Room 446, Erskine Building
30 minutes, followed by a break and 20 minutes of discussion.
Abstract. We present an introduction into piecewise deterministic Markov processes occurring as model of promoter noise in gene regulatory mechanisms. Those processes are a suitable generalisation of Markov chain models for chemical reactions (Gillespies algorithm) to situations where only some of the reactants are present in small numbers.
We show by simulation that this class of processes is extremely rich in dynamical phenomena.
Besides simulation, also solution theory of PDE for the derivation of the stationary distribution, and ODE for the moments of the process are available to answer questions from biological experiments.
Carolyn Chun (Victoria University)
Fragility in matroids
Thursday, 29th April, 3:10pm
Room 446, Erskine Building
30 minutes, followed by a break and 20 minutes of discussion.
Abstract: For a matroid M with a minor N, we say that M is N-fragile if, for every element e in the ground set of M, either M/e or M\e does not contain N as a minor. Understanding the structure of N-fragile matroids is necessary for thinking about Rota's conjecture. In this talk, we present a characterization of the binary, Fano-fragile matroids.
Thomas Hangelbroek (Texas A&M University)
Surface Splines, the polyharmonic Dirichlet problem and boundary layer potentials
Monday, 19th April, 2:10pm
Room 446, Erskine Building
50 minute specialised research seminar.
Abstract. The presence of a boundary poses a fundamental challenge to most approximation methodologies. For approximation with kernels these effects are easily observed whilst being theoretically understood: error and instability are known to increase sharply in a neighborhood of the boundary. Novel approximation schemes using surface splines (a leading type of kernel approximation), and based on a family of potential theoretic integral representations, have been shown to deliver theoretically optimal rates of convergence while isolating the detrimental boundary effects in easy-to-manage integrals. In this talk I'll discuss the solution of the polyharmonic Dirichlet problem via boundary layer potentials, and I'll demonstrate how this key result furnishes the integral representations at the heart of these new approximation techniques.
Mina Teicher (Director, Emmy Noether Research Institute for Mathematics, Bar-Ilan University, Israel & Vice-President ICMI)
Braid Group and its Applications
Thursday, 15th April, 3:10pm
Room 446, Erskine Building
Abstract. In this talk, I will give an overview on the braid group. I will introduce a topological model for the braid group and discuss applications to Algebraic Geometry (especially algebraic surfaces and fundamental groups), Cryptography and, possibly, computer vision.
Ángel Ruiz (Vice-President ICMI, Costa Rica)
Pupils’ Beliefs About Mathematical Problems in Costa Rican Secondary Schools
Thursday, 1st April, 3:10pm
Room 446, Erskine Building
30 minutes, followed by a break and 20 minutes of discussion.
Abstract. The talk will describe the results of a survey done in Costa Rica with high school students in relation to their perception about what is a mathematical problem – a research associated to problem solving strategies.
It will include a preamble about the basics on similar perceptions about mathematical problems documented in other countries.
Abdulla Firag (University of Canterbury)
Statistical Analysis of Wireless Relaying Systems
Thursday, 25th March, 3:10pm
Room 446, Erskine Building
30 minutes, followed by a break and 20 minutes of discussion.
Abstract. Wireless multiple-input, multiple-output (MIMO) relaying systems have recently been given considerable attention due to their many advantages. Apart from increasing the range with low power at the transmitter, MIMO relaying systems can also achieve more capacity and better diversity.
In this talk, I will first give a brief overview of wireless relaying systems and proposed mathematical models, followed by statistical performance analysis of these systems.
Bill Barton (University of Auckland)
Have five colleagues watch me lecture? Why would I agree to that?
Thursday, 18th March, 3:10pm
Room 446, Erskine Building
30 minutes, followed by a break and 20 minutes of discussion.
Abstract. At Auckland we have begun a project aimed at lecture development using videos of ourselves lecturing. We watch the videos together as a group and talk about what is going on and why. We have tried videoing the same lecture during a later semester, and are about to extend the project to include more lecturers.
There is, understandably, so anxiousness about this process–even for an old school master like myself who was first videoed in 1972 during pre-service teacher training.
I'll talk about the project, the theory behind it, how we try to manage the sensitive issues, what it has raised on the emotional level, and the insights it has given us on our practice. Although the project is not over, we are impressed by the positive impact it has already had.
Peter Smith (University of Cambridge & UC Philosophy Erskine Fellow)
Kleene’s Proof of Gödel’s Theorem
Thursday, 11th March, 3:10pm
Room 446, Erskine Building
30 minutes, followed by a break and 20 minutes of discussion.
Abstract. Gödel’s First Incompleteness Proof can, of course, be established in a number of different ways. One of the less widely known proofs is due to Kleene, and doesn’t involve the construction of a ‘self-referential’ sentence or the arithmetization of syntax. I make no claim that it delivers new insights: but it is cute enough to be worth presenting.
Fabio Pardi (LIRMM, Montpellier, France)
Distance-based tree reconstruction: the importance of earnest assumptions
Thursday, 4th March, 3:10pm
Room 446, Erskine Building
30 minutes, followed by a break and 20 minutes of discussion.
Abstract. Several methods to infer phylogenetic trees (or cluster dendrograms) are based on a matrix of pairwise distances between species (or any kind of objects): the objective is to construct a tree with branch lengths so that the pairwise distances between the leaves in that tree are as close as possible to the distances in input. Due to their computational efficiency, these methods are very popular.
The fundamental step of distance-based tree reconstruction is to fit the branch lengths of a tree of fixed structure to the given matrix of distances. This step implicitly depends on a number of statistical assumptions on the distances in input. The most commonly used methods (including neighbor-joining) differ principally for the variances they assume for the input distances.
In this talk, I will discuss a number of tree reconstruction methods, showing my work on them and showing in particular how their properties (such as their robustness to noisy data) are affected by the variance model they assume. I am currently investigating variance assumptions leading to objective functions that can be optimized very rapidly. This has the potential to lead to a tree reconstruction algorithm as fast as the fastest available distance methods, but more accurate.
Dominik Schmid (Institute of Biomathematics and Biometry, Helmholtz Zentrum, München)
Scattered Data Approximation and Marcinkiewicz-Zygmund inequalities on SO(3)
Thursday, 25th February, 3:10pm
Room 446, Erskine Building
30 minutes, followed by a break and 20 minutes of discussion.
Abstract. Scattered data approximation problems naturally arise in various fields in science and engineering. After introducing such problems, we briefly present well-known approaches to handle such questions. By considering one of these approaches in more detail, we encounter so-called Marcinkiewicz-Zygmund inequalities. These inequalities are a very powerful tool to answer important questions that come along with the approximation of scattered data and are well-studied in the classical settings like the torus or the Euclidean sphere. However, in various applications scattered data approximation problems on more general structures naturally arise. Often certain matrix groups play an important role and, without doubt, the rotation group SO(3) is one of the most important groups in this regard. So, in the second part of the talk, we turn to scattered data approximation problems on the rotation group SO(3) and show how Marcinkiewicz-Zygmund inequalities can be established in this setting.
Zach Weber (University of Sydney & University of Otago)
Paraconsistent Mathematics
Wednesday, 24th February, 3.10pm
Room 446, Erskine Building
Abstract. Paraconsistent mathematics is the study of mathematical objects and structures, in which some contradictions are allowed. Tools from formal logic are used to make sure inconsistency is contained and that the overall theories are not absurd. I will give a brief overview of the background history and motivation for investigating paraconsistent mathematics, and the state of current research in geometry, arithmetic, and real analysis. Mainly, I will focus on describing recent developments in paraconsistent set theory, which is a formalized version of the original naive set theory of Dedekind, Cantor and Frege. I show on the one hand how the textbook theorems of ordinal and cardinal arithmetic can be proved. On the other hand, any contradictions are controlled and even studied for their interesting properties and consequences‒most notably, answers on the continuum hypothesis and the axiom of choice.
Robert D Russell (Simon Fraser University)
Adaptive Mesh Generation and Moving Mesh Methods
Tuesday, 9th February, 3:10pm
Room 446, Erskine Building
Abstract. Over the last several decades, many mesh generation methods and a plethora of adaptive methods for solving differential equations have been developed.
In this talk, we take a general approach for describing the mesh generation problem, which can be considered as being in some sense equivalent to determining a coordinate transformation between physical space and a computational space. Some new theoretical results are given that provide insight into precisely what is accomplished using mesh equidistribution (which is a standard adaptivity tool used in practice).
As well, we discuss two general types of moving mesh methods for solving time dependent PDEs, those based upon a variational formulation of the mesh generation problem and those which target mesh velocity. Among the methods in the latter class are those which solve the Monge-Ampere equation and the optimal mass transport problem, an area which has seen intense research activity of late.
Tomas Johnson (Computer-assisted Proofs in Analysis Group, Uppsala University, Sweden)
Dynamics of the Universal Area-Preserving Map Associated with Period Doubling
Friday, 22nd January, 3:10pm
Room 446, Erskine Building
Abstract. Universality phenomena are common in dynamical systems. Originally discovered in the one-dimensional setting, universality has now been observed for many different kinds of systems. In 1984 Eckmann, Koch, and Wittwer gave a computer-aided proof of existence of a fixed point of the renormalization operator associated with period doubling for area-preserving maps. This fixed point is a “universal” area-preserving map ‒ a map with orbits of all binary periods 2k, k ∈ N. We consider maps in some neighbourhood of the universal map and study their dynamics.
We first demonstrate that the universal map admits a “bi-infinite heteroclinic tangle”: a sequence of periodic points whose stable and unstable manifolds intersect transversally. A corollary of these results is the existence of unbounded and oscillating orbits. We also show that the third iterate for all maps close to the universal one admits a horseshoe. We use distortion tools to provide rigorous bounds on the Hausdorff dimension of the associated locally maximal invariant hyperbolic set.
Finally, we consider infinitely renormalizable maps ‒ maps on the renormalization stable manifold in some neighbourhood of the universal map. For all such infinitely renormalizable maps in a neighbourhood of the fixed point we prove the existence of a “stable” invariant Cantor set, i.e. a set such that the Lyapunov exponents are zero. We also show that there exists a submanifold of finite codimension in the renormalization local stable manifold, such the stable sets of the maps in this submanifold are “weakly rigid”: i.e. the dynamics of any two maps in this submanifold, restricted to the stable set, is conjugated by a bi-Lipschitz transformation that preserves the Hausdorff dimension.