In recent years, considerable advances have been achieved in understanding the boundary value problem in general relativity. In particular, starting with the harmonic formulation of the evolution equations, it was found that there were large families of boundary data for which the system was strongly well posed, that is, stable. These conditions include many which were consistent with constraint propagation in the sense that, if initially the constraints were satisfied, they would remain so for the whole evolution and amongst these, there were some for which all the rest of the fields satisfied non-incoming radiation conditions. Thus they were all candidates for being good conditions to represent radiating isolated systems. Here we present such construction in some detail. But the gauge conditions that have been imposed are intermixed in all these conditions and it is not clear which ones are better for such a representation. One way to explore this set of conditions is to ask the following geometrical uniqueness questions: Can we specify instructions to impose boundary conditions in such a way that two versions of instructions for the same initial data would result in the same space-time? That is, would the resulting evolutions be linked to each other by a diffeomorphism? In this work we analyze the problem in the linearized version and see that there is a set of preferred conditions which satisfy certain geometrical uniqueness, yet which also contain ambiguity.

110718103048

The field of computational immunology is a relatively well established area of research these days. Over all, mathematical modelling of biological phenomena such as immune system with the aim of making accurate predictions has gathered pace and at present is a very attractive and interesting area of research.

The inflammatory response of tissue to injury or invasion is a complex phenomenon involving, vascular, humoural and cellular factors. In this work we have developed a spatio-temporal mathematical model consisting of a system of diffusion-advection-reaction equations, to capture some aspects of tissue inflammatory response. Two sorts of cell movement mechanisms are considered: 1. chemotactic as the major cytotaxic effect that leads the movement of leukocytes towards the site of infection/inflammation, 2. leukocytes’ random motility described via diffusion process. Cell motility mechanisms are formulated using the idea of reinforced random walks (RRW). RRWs are in particular suitable for modelling cell motility in response to one or more control substances.

The proposed model accounts for (1) antigen recognition, (2) the effector function (activation/inhibition), (3) innate immune response, (4) elimination of antigen and resolution of the infection and (5) returning the immune cells back to a steady state. 3-D MATLAB simulations have enabled us to visualise the dynamics of the immune cells and chemicals.

110718102819

Recent advances in technology have enabled large numbers of genetic markers to be assayed simultaneously for many species. For sheep, a product known as the Illumina OvineSNP50 BeadChip can assay over 50,000 single nucleotide polymorphism (SNP) markers at once, and this has been used to genotype more than 8000 New Zealand sheep. The main focus of this effort is to enable 'genomic selection' which allows prediction of genetic merit of an animal on the basis of genotyping alone. Collections of case-control animals have also been genotyped to confirm or map a variety of single gene traits including: horns, yellow fat and microphthalmia. Genotype results can also be used to characterize populations and to inform conservation decisions for rare breeds. A number of interesting statistical challenges arise when analyzing such data and these will be discussed.

110718102540

I will present existence results for the initial value problem associated with the Einstein field equations of general relativity, when weak regularity only is assumed on the initial data set and, therefore, on the spacetime itself. The curvature must then be understood in the weak sense, and the formulation of the initial value problem for the Einstein equations must be revisited.

Relevant papers are available at the link: philippelefloch.org

110718102151

The Leavitt path algebras, introduced by Abrams and Aranda Pino in 2005, are purely algebraic analogues of the C*-algebras of directed graphs. There are striking connections between the properties of the Leavitt path algebra and the underlying graph. The higher-rank graph C*-algebras, introduced by Kumjian and Pask in 2000, have provided many new and interesting examples of tractable C*-algebras. Here I will talk about an analogue of the Leavitt path algebras for higher-rank graphs which we call the Kumjian-Pask algebras. I'll start by explaining what a higher-rank graph is and how to visualise it. Then I'll discuss how the Kumjian-Pask algebra is defined, its universal property, and the uniqueness theorems which say when a representation of this algebra is injective. This is joint work with Gonzalo Aranda Pino, John Clark and Iain Raeburn.

110718101927

I will give an informal introduction to solving and simplifying systems of algebraic and differential equations. The talk is intended for a wide audience. It will be illustrated by examples, pictures, history and discussion of modern efforts in computer algebra.

The talk begins with some brief discussion of Galois' work. Galois established that higher degree equations were solvable in radicals provided they were symmetric enough. The Gauss algorithm reduces linear equations to triangular form. I will give an introductory discussion of Groebner bases which similarly triangularizes higher degree systems. In particular it reduces the solvability of such systems with finitely many solutions to that of single univariate polynomials. This affords an easy transition to our introductory discussion of an analogous reduction of linear partial differential equations to so-called differential Groebner bases. Again solvability is reduced to that of a univariate case.

I will give briefly discuss Lie's attempt to generalize Galois approach to differential equations. Again solvability is expressed in terms of symmetry.

110718101725

With the "genomics revolution" continuing to generate data in ever-increasing amounts, the disciplines of statistics and bioinformatics remain vital components in our drive to interpret and understand genomic data. Students in these disciplines are often exposed to a tantalizing glimpse of the problems encountered by "real world" users of genomic data sets, however the scale (and complexity) of these problems is generally greatly reduced to accommodate the limited resources available as part of a standard teaching programme. The skills learned in such courses also tend only to scratch the surface of what is required for in-depth genomic analyses, despite the fact that these techniques are often applicable to a much broader class of problems. In this talk I will describe some of the current genomics problems typically faced by statisticians and bioinformaticians, along with the statistical tools that are often applied. I will also describe two national infrastructure initiatives that the University of Otago is heavily involved in, with the goal of starting a dialogue on how we can begin to incorporate these important "research-led" initiatives into our teaching programmes to provide our students with the tools they need for dealing with large amounts of genomic data.

110718101545

A spherical building Δ is a special type of simplicial complex. It is made by sticking together a collection of Euclidean n-spheres; each sphere carries the action of a Coxeter group, which divides the sphere up into simplices.

Let Γ be a group of automorphisms of Δ and let Σ be a closed convex contractible subset of Δ which is stabilised by Γ. The Centre Conjecture of Tits states that if Σ is a subcomplex of Δ then Γ fixes some point in Σ (such a point is called a “centre” of Σ). I will discuss an approach to this conjecture using work of Kempf, Rousseau and Hesselink. The idea is to construct a centre explicitly as the maximum of a certain real-valued function on Σ.

This is joint work with Michael Bate and Gerhard Röhrle.

110729142746

NOTE - DAY CHANGED FROM USUAL SCHEDULE

Computer algebra packages for analysing a system of PDE to find its Lie symmetries have been available for some 25 years. Typically, such packages require PDE in 'orthonomic' form, i.e., certain derivatives are isolated on the left-hand side. For highly nonlinear PDE this is too restrictive, but extending the class of PDE to which the analysis applies raises a number of nontrivial issues in differential algebra. In this talk I will discuss these issues via a sequence of examples.

110707102753

The Neolithic transition through Europe was a key event in the development of human technology. It was the last subdivision in the series of stone ages, and heralded the invention of agriculture.

Echoes of the expansion into Europe might still be found in data from the diverse fields of archaeology, genetics and linguistics. However, conclusions concerning the time and the direction of the expansion vary, particularly when considering data from disparate sources.

Our intention is to use methods from the area of indirect inference to reconstruct the Neolithic transition. By combining information from the three sources of data, along with a population diffusion simulation model, we aim to quantify the uncertainty about demographic parameters of interest.

This is still a work in progress, but we illustrate our proposed approach with a selection of trivial toy examples.

110705100542

One of several forms of academe-industry interactions, Study Groups with Industry are high intensity workshops where mathematicians work in groups on industrial mathematical and/or modelling problems brought to the workshop by companies. Originating in Oxford in the 1960s, this activity now takes place on a regular basis in countries in Europe, North America, Africa and Asia. I will describe how these meetings unfold, and give a number of case stories, mainly from the European Study Groups with Industry (ESGI).

110607141022

Inaugural Professorial Lecture

110512115011

Eman Alhassan
Dedekind Domains

Boris Daszuta
Spectral Methods, Wave Equations and the 2-Sphere

Richard McNamara
Parseval Frames

Sam Primrose
Leavitt Path Algebras

110518162414

Every week, islands around New Zealand are subject to a barrage of invasions by four-legged creatures with sharp teeth and big appetites. These invaders are mammal pests, including rats, stoats, and mice, and they have plenty of tricks up their furry sleeves for reintroducing themselves to conservation sanctuaries. They are excellent and eager swimmers, hitch rides on boats, abound in resourcefulness, and can cost tens of thousands of dollars - each - to track down and remove when discovered on a sanctuary island.

Understanding where mammal invaders are coming from is pivotal to the long-term protection of sanctuaries. I will describe the statistics behind genetic assignment methods to estimate the origin of individuals, emphasising both the usefulness and the limitations of the techniques. To make best use of these tools, we need to coordinate efforts on a national scale. I will give a demonstration of some map-linked database software I am developing with programmer Sunil Patel, to coordinate data management from the initial trapping stages to the final data analysis.

The talk is intended to be accessible to biologists as well as of interest to statisticians and mathematicians.

110302095922

An entire function f has an asymptotic value a, which may be finite or infinite, if there is a curve joining 0 to ∞ on which f(z) → a as z → ∞. It is a theorem of Iversen that ∞ is an asymptotic value of every non-constant entire function. Denjoy conjectured that if f has k distinct, finite asymptotic values then f must have a certain minimum rate of growth, in the sense that

lim sup_r→∞ (log log M(r,f)) / log r ≥ ½k

Here M(r,f) = max_|z|=r |f(z)| is the maximum modulus of f. This was proved independently by Ahlfors (for which he received a Fields medal) and Beurling (in his thesis). I will discuss this result and work related to it, including a kind of “reverse” Denjoy theorem that arises from a still unresolved conjecture on gap power series.

110302100230

Just as the sinusoidal eigenfunctions of the second derivative operator represent waveforms on a vibrating string, the eigenfunctions of the Laplacian describe waves on a vibrating membrane or drum. Or thinking quantum mechanically, the eigenfunctions represent wavefunctions of particles in an infinite potential well. The corresponding eigenvalues represent frequencies of vibration, or quantum energy levels.

How much do we know about these eigenvalues? Not as much as one might suppose! I will describe old and new results, and open problems, about dependence of the eigenvalues on geometric properties of the membrane, such as area and moment of inertia. Especially, we will try to solve max/min problems on the (deceptively simple) class of triangular drums. One of these problems leads to a Parseval identity for a system of three vectors in two dimensions, known as the "Mercedes-Benz" tight frame. This identity extends to higher dimensions by way of the irreducibility of the isometry groups of Platonic solids.

110302093110

Inaugural Professorial Lecture Series 2011

110321124444

It will be demonstrated that some properties of the Kerr-Newman black hole solution, which describes a single axisymmetric and stationary charged black hole in electrovacuum, carry over to black holes with surrounding matter. In particular, it will be proved that for so-called sub-extremal black holes, the inequality (8 π J)² + (4 π Q²)² < A² holds, where J is the angular momentum, Q the charge and A the horizon area of the black hole. Moreover, an inner Cauchy horizon is found to exist in general, and the regularity of the interior region between this horizon and the event horizon will be studied. These investigations lead to a universal inequality between angular momentum and the areas of event and Cauchy horizon.

110302094312

The mathematical field of Inverse Problems is well-developed, with several scholarly journals dedicated to its study. The primary focus of scholarship within the field has been at its interface with analysis, numerical analysis, and scientific computing. However, in applications, inverse problems involve noisy data, suggesting that statistical techniques are also worth studying. Within the past decade, or so, a number of researchers have conducted pioneering work at the interface between inverse problems and statistics (e.g., University of Otago's Colin Fox), and it is now well-known that Bayesian statistical methods are particularly well-suited for use on inverse problems. In this talk, I will provide a general introduction to inverse problems, including issues that arise in their solution, and then discuss their connection with Bayesian statistics. Finally, I will present a MCMC method for computing samples from the posterior distribution of the unknown parameters in a linear inverse problem, which will allow us to compute estimates, as well as to quantify uncertainty in those estimates. Numerical examples will include image deblurring and computed tomography.

110302093852

Elliott's program for classification of C*-algebras is one of the main thrusts in the research of operator algebras. Given a special C*-algebra D, it is often important from the standpoint of Elliott's classification program to know when a given C*-algebra, A, is D- absorbing, that is, when A ⋍ A ⊗ D. Our research investigates when certain C*-algebraic constructions preserve the property of D-absorption. To be more specific, given a C*-algebra A and an automorphism (or endomorphism) of A, there is a natural action of ℤ (respectively ℕ) on A. Using this action, we can then construct a new C*-algebra called the crossed product and denoted C*(ℤ,A) (respectively C*(ℕ,A)). The goal of this talk is to find conditions on these actions guaranteeing that D-absorption passes to the crossed product. In particular, we prove that given an endomorphism of a separable unital D-absorbing C*-algebra A satisfying an analogue of the Rohklin property, the crossed product C*(ℕ,A) is also D-absorbing. This result generalizes a result of Hirshberg and Winter, which proves that crossed products by Rohklin automorphisms preserve D-absorption. Our level of generality is justified by the study of certain endomorphisms of the CAR-algebra. This is joint work with Ilan Hirshberg.

110302093709

A phylogenetic tree comprising clades with high bootstrap values or other strong measures of statistical support is usually interpreted as providing a good estimate of the true phylogeny. Convergent evolution acting on groups of characters in concert, however, can lead to highly supported but erroneous phylogenies. Identifying such groups of phylogenetically misleading characters is obviously desirable. I will present a procedure, developed in conjunction with Barbara Holland, Martyn Kennedy and Trevor Worthy, that uses an independent data source to identify sets of characters that have undergone concerted convergent evolution, illustrated using the problematic case of the cormorants and shags, for which trees constructed using osteological and molecular characters both have strong statistical support and yet are fundamentally incongruent. I will show that the osteological characters can be separated into those that fit the phylogenetic history implied by the molecular data set and those that do not. Moreover, these latter nonfitting osteological characters are internally consistent and form groups of mutually compatible characters or “cliques,” which are significantly larger than cliques of shuffled characters. I suggest, therefore, that these cliques of characters are the result of similar selective pressures and are a signature of concerted convergence.

110302093232

A matrix A is completely positive (CP) if it can be decomposed as A=BB^T, where B is a (not necessarily square) nonnegative matrix. The smallest number of columns of a matrix B in such a decomposition, is called the cp-rank of A. Obviously, a necessary condition for a matrix to be CP is that it is positive semidefinite and element-wise nonnegative but in general, this is not sufficient. In the talk I will survey some of the applications of CP matrices and what is known about them and about the cp-rank.

110228093240

A groupoid is a small category in which every morphism is invertible. The study of these very general objects has proven useful in a variety of mathematical settings. In functional analysis, a groupoid can be used to construct a C*-algebra so that properties of the groupoid correlate nicely with properties of the C*-algebra.

This class of groupoid C*-algebras is broad and includes many subclasses, including C*-algebras constructed from directed graphs. Another purely algebraic object constructed from a directed graph is a Leavitt path algebra. These algebras arose in 2005 and surprisingly, many connections have been made between the highly analytic graph C*-algebras and the purely algebraic Leavitt path algebras.

In this talk, we will examine groupoids and the construction of the C*-algebras associated to them. We will also describe on ongoing project to construct a groupoid algebra that generalises Leavitt path algebras.

110202165741

A tiling of the plane refers to a covering of R² by translates of a finite set of polygons that only intersect on their borders. A tiling T is said to be aperiodic if there is no non-zero vector x in R² such that T + x = T. One way to construct aperiodic tilings is by substitution; the most famous substitution tiling being the Penrose tiling.

Given an aperiodic substitution tiling T, satisfying mild conditions, a dynamical system can be constructed that is locally the product of a disk and a Cantor set. Anderson and Putnam found a relationship between this dynamical system and the theory of hyperbolic dynamical systems known as Smale spaces. Smale spaces were first introduced by David Ruelle as a purely topological description of the basic sets of Stephen Smale's Axiom A diffeomorphisms on a compact manifold.

In this talk I will use a dynamical system constructed from an aperiodic substitution tiling as an introduction to Smale spaces. No prior knowledge of dynamical systems will be required.

110201110952

A dynamical system consists of a set X describing the states of a system, and an action of time
(t, x) ⟼ t • x describing how the states of the system change. Intuitively, we think of x describing the state of the system now, and t • x the state of the system in t seconds. An irreversible dynamical system is one in which it is not possible to "back in time" to retrieve x from t • x.

We use C*-algebras to study irreversible dynamical systems by replacing the set X of states with a C*-algebra A, and replacing the irreversible action of time by an action α of a semigroup P of endomorphisms of A. We study systems (A, α, P) by representing them on Hilbert space, which means we assign operators to the elements of both A and P in a way that encodes the action of P on A. The semigroup crossed product C*-algebra of (A, α, P) is universal for such representations in the sense that every representation on Hilbert space factors through the representation into the crossed product.

In this talk I will run through the history of the crossed product of a C*-algebra by a single endomorphism. I will pay particular attention to Ruy Exel's 2003 construction of such a crossed product, and I will examine the connection between Exel's crossed product and the C*-algebras associated to directed graphs.

I will assume basic knowledge of bounded operators on Hilbert space, and will begin the talk with the definition of a C*-algebra.

110202170014

In the early 1990s a group of conservation biologists proposed that the diversity of a geographic region should not be restricted to counting the species present in the region but rather also incorporate the genetic information of said species. This led to the introduction of phylogenetic diversity. Though the measure was well received its use was limited due to lack of sufficient genetic data and proper software.

In recent years, both limitations have been addressed. With the advent of next generation sequencing, generating massive amounts of data for a geographic region has become feasible. Further, bioinformaticians have provided several packages for computing the phylogenetic diversity for a set of species from a phylogeny.

Here, I will present such a tool which employs linear programming to compute the phylogenetic diversity for a geographic region based on one or more genes for a set of species considered. I will demonstrate the power of the method on a data set for 700 floral species from the Cape of South Africa.

110112110142

The tight span, or injective envelope, is an elegant and useful construction that takes a metric space and returns the smallest hyperconvex space into which it can be embedded. The concept has stimulated a large body of theory and has applications in combinatorial optimisation and data visualisation (which was where I first got interested in it). In this talk I'll introduce diversities, which are similar to metrics except that they are defined on all subsets of a set instead of just pairs. A diversity (X,δ) satisfies, for all subsets A,B,C of X:
(D1) δ(A)≥ 0, and δ(A) = 0 if and only if |A| ≤ 1.
(D2) If B is non-empty then δ(A ∪ C) ≤ δ(A ∪ B) + δ(B ∪ C).
I will show how the rich theory associated to metric tight spans and hyperconvexity extends to a seemingly richer theory of diversity tight spans and diversity hyperconvexity.

This is joint work with Paul Tupper at Simon Fraser university.

101104102119

In 1994 Archbold introduced the upper and lower multiplicity numbers
of the irreducible representations of a C*-algebra. This will be
demonstrated with examples. In 2006 Archbold and an Huef showed that,
for a C*-algebra associated to a group acting freely on a space, the
upper and lower multiplicity numbers correspond to a topological
property of the action. In this talk we will give a brief overview of
C*-algebras and groupoids before discussing the generalisation of
Archbold and an Huef's results to the C*-algebras of principal
groupoids. If time permits the new level of generality will be
justified by demonstrating a large class of examples based on
groupoids constructed from directed graphs. This is joint work with
Astrid an Huef.

101104102205

101103160036

Continuing improvements in the through-put and cost efficiency of next generation sequencing technologies are fueling a rapid increase in the number and scope of metagenomics projects, which aim at studying consortia of microbes by sequencing, in environments as diverse as soil, water, gut, ancient bones and the human biome. The first three computational challenges in metagenomics are: (1) Taxonomic analysis, who is out there? (2) Functional analysis, what are they doing? (3) How do different metagenomes compare and are their differences correlated to environmental differences? This talk will give an introduction to metagenomics and will discuss some of the methods that have be developed to address these three challenges. Metagenomics is a very growing field and much work needs to be done to support the analysis of metagenomic datasets.

Biography: Daniel Huson is widely known for his work in computational biology. He is the author of many extensively used software packages including tools for phylogenetic analysis, data visualisation, meta-genomic analysis, and evolutionary music composition. After completing a PhD in mathematics in 1990, Daniel held various research positions in Germany and elsewhere including a two-year postdoctoral fellowship at Princeton working with Tandy Warnow and John Conway. Daniel was a senior scientist at Celera during the first sequencing of the human genome and has been professor of Bioinformatics at the University of Tübingen since 2002.


A JOINT SEMINAR WITH THE DEPARTMENT OF BIOCHEMISTRY

101022103346

101104095132

Computational neuroscience is about computational modelling of biological neurons and neural circuits. These entities can be represented as systems of differential equations because, in the case of neurons, we are dealing with changes of various variables, be it the membrane potential or synaptic efficacy. Numerical simulations enable us to replicate and thus understand better the dynamics of these complex systems.

101019121941

Metamaterials are artificial materials engineered to possess certain properties, including some that may not be found in nature such as a negative index of refraction. The development of metamaterials has led to several novel applications, including optical cloaking. The recent construction and demonstration of an actual cloaking device has conjured dreams of Harry Potter's cloak of invisibility and has garnered lots of media attention.

Transformation optics is a method of designing optical devices based on visualizing the desired behavior of electromagnetic fields; and basically constitutes an inverse problem: Given a desired electromagnetic field behavior, what material parameters are required to realize that behavior? Together, transformation optics and metamaterials are a powerful combination, as virtually any desired field behavior can now be realized by employing metamaterials.

Using the particular example of cloaking, I will describe metamaterials and discuss a newly developed approach to transformation optics. This new approach eliminates the hand waving that had been present in transformation optics while simultaneously making transformation optics even more useful by broadening its domain of applicability.

100917095143

Sports technology is a growth area within and beyond the university. Computational modelling has a key role.

For further details contact mike.paulin@stonebow.otago.ac.nz

101007143252

Wave maps are analogues of harmonic maps for Lorentzian manifolds. They provide simple examples for systems of non-linear wave equations and have therefore been studied thoroughly. In this talk I will present an introduction to wave maps, discuss some of the rigorous results and show how wave maps appear in general relativity. Finally, I will present numerical studies of a particular wave map, which is of relevance in cosmological situations.

100917095341

Darren Alexander
Correlated Binary Data: A methodological approach
Pharmaceutical studies often produce datasets in binary form. One such example is a study where overdoses of the drug, Venaflaxine, were witnessed in a 10 year long cohort study. The patients were monitored to see if a seizure occurred after a particular overdose, and provides the motivation for looking at binary data, whilst placing an emphasis on correlation and extreme proportions of binary success.

Ross Haines
Anticipating Penalty Kicks - Bayesian Modelling of Eye-Tracking Data
When the time-dependence of athletes’ gaze pattern data is utilised in analysis, useful information is extracted, and subtle behavioural characteristics that are not apparent with discrete summary statistics are revealed. To illustrate this, Markov models are fitted to gaze-behaviour data from a football penalty kick scenario.

100924115704

The diffusion equation is related to the Schrödinger equation by analytic continuation. The formula E² = p²c² + m²c⁴ leads to a relativistic Schrödinger equation, and analytic continuation yields a relativistic diffusion equation that involves fractional calculus. In this talk we develop stochastic models for relativistic diffusion based on continuous time random walks and equivalent differential equations with no fractional derivatives.

100917095247

I am going to use this talk to give an overview of the research that I have produced, with help from the rest of the maths/ice chaps, in my postdoc since I came to Otago 3 (very short) years ago.

Our work has allowed us to gain inferences about wave activity in the ice covered veneer of the Arctic Basin and the vast fields of ice floes in the Antarctic Ocean.

If I don't ramble on too much, I'll also tell you about what lies ahead next year.

100827154930

By virtue of its definition, there can be no singularities in a manifold, yet singular behaviour can be observed in the limit towards the 'edge' of a manifold. The structure of such singular behaviour is an important physically motivated question. There is, however, no a priori mathematical structure to work with. Thus new constructions which describe the 'edges' of a manifold are needed. I will talk about one modern construction called the Abstract Boundary.

The focus of this talk will be a conceptual introduction to the Abstract Boundary along with a discussion of three areas of recent research - the relationship to standard topological completions, an extension of the Penrose-Carter conformal boundary, and an application to the singularity theorems of General Relativity.

100901125335

I will introduce the notion of an approximate group. Roughly speaking, this is a set A together with a multiplication table in which xy lies in A for some fraction, rather than for all, choices of x and y in A. Approximate groups have been much studied in recent years and I will give a survey of this topic. Particular issues to be addressed include
1. Basic examples. 2. Structure theory in particular cases (abelian, nilpotent, matrix groups...) 3. A general conjecture 4. Applications

100818145426

This is about stable numeric-geometric methods for general systems of
differential equations with constraints (so-called differential-algebraic equations or DAE, or more generally partial differential algebraic equations or PDAE). This is joint work with my recent MSc student Niloo Mani and PhD student Wenyuan Wu.

There are software packages that support high-level physics-based modeling 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 3 D videos of mechanisms arising in electro-mechanical modeling. Unlike other approaches MapleSim enables the equations to be treated in analytical form.

Determination of constraints for DAE and PDAE is needed for the determination of consistent initial conditions and the numerical solution of such systems. This talk will concentrate on introduction of concepts from the (Jet) geometry of differential equations, illustrated by visualizations and simple examples.

This talk will be an introduction to stable numerical methods for such general systems. The corresponding problem for the non-differential case, that of approximate polynomial systems, has only recently been given a solution, through the works of Sommese, Wampler and others. The new area called numerical algebraic geometry, will also be described. Key data structures are certain witness points on jet manifolds of solutions, computed by stable homotopy continuation methods.

100826153027

Gelfand Triples (sometimes called rigged Hilbert spaces) are a very useful general construction that allows to talk of good functions (think of Schwartz test functions) and most general objects (such as tempered distributions) surrounding the central Hilbert spaces. Pairs of Sobolev spaces show a similar setting and are useful for the description of elleptic PDEs.

The theme of the talk is the presentation of a particular Banach Gelfand Triple, i.e. a Gelfand triple consisting of Banach spaces (actually isomorphic to the canoncial sequence spaces (l1,l2,linfty)) which arose in the context of time-frequency analysis resp. Gabor analysis. It is described by the integrability of the short-time Fourier transform of its elements (with respect to a Gaussian window). It is however not only useful in that original context, but has a wide range of applications, e.g. to give an optimal description of the Fourier transform, or allowing a kernel theorem (the analogue of a matrix representation of linear mappings between finite dimensional vector spaces), similar to the Schwartz kernel theorem.

This triple is just the prototype of a family of more general function spaces, called modulation spaces. They are quite similar to Besov or Triebel-Lizorkin and potential spaces, which are characterized among others by the continuous wavelet transform. Coorbit theory allows to view these spaces from a unifying point of view.

Material related to the topic of the talk is found at the NuHAG web-page www.nuhag.eu

100819084418

For millenia cryptography was an entirely symmetric affair, where sender and recipient would have to securely agree on some secret "key" by which messages could be encrypted and decrypted. This problem of key distribution was finally ameliorated in the mid '70s with the publication of Diffie and Hellman's much celebrated paper "New directions in cryptography", wherein asymmetric cryptography was born.

The Diffie-Hellman key exchange protocol was originally defined in terms of the multiplicative group of integers mod p and its effectiveness relies on the difficulty of computing discrete logarithms in this group. In the 1980s Elgamal developed this into a fully-fledged cryptosystem and since then a great deal of interest has arisen in the use of groups for public key cryptography.

I will discuss some of the major developments of group-based cryptography, in particular the MOR cryptosystem which generalises Elgamal encryption to an arbitrary group and uses an analog of the discrete logarithm involving automorphisms of this group.

100806142112

A wavelet is a function on the real line whose dilations and translations form an orthonormal basis for the Hilbert space of square-integrable functions. A famous theorem of Mallat (1989) describes a procedure for constructing wavelets starting from something engineers call a "quadrature mirror filter", which is a function on the unit circle.

We will say what all these words mean and describe an approach to Mallat's theorem which explains how it is that one gets from the circle to the line. The crucial step is a purely algebraic construction called a "direct limit".

This is joint work with Nadia Larsen (Oslo).

100802092241

Spatio-temporal modelling is an important area in statistics that is one of rapid growth at the moment, with various applications in environmental science, geophysical science, biology and others. Especially after all the recent technological advances such as satellite scanning that resulted in increasingly complex environmental data sets, estimating and modelling the covariance structure of a space-time process have been of great interest.

This is a two-folded talk. On the one hand we discuss general non-stationary spatio-temporal surfaces that involve dynamics governed by velocity fields. We start with spatial fields with a prescribed covariance function and by applying an extension of the standard moving average construction we obtain models which are stationary in time. The resulting surface changes with time but it is dynamically inactive since its velocities, when sampled across the field, have distributions that are centred at zero. We introduce a dynamical evolution to such a field by composing it with a dynamical flow governed by a given velocity field. This leads to non-stationary models. We demonstrate that for such a surface its dynamics is a combination of dynamics introduced by the flow and the dynamics resulting from the covariance structure of the underlying stochastic field.

These models are extensions of the discretized autoregressive models which have been used to model the significant wave height estimates from satellite data. Although, the apparent motion of the significant wave height fields is the composition of various velocities, we assume that each sea-state moves with the composite velocity, which we model using a flow of diffeomorphisms that are the solution to the transport equation. Finally, we present a method for estimating this velocity within a state-space model framework.

100722143059

I will outline how an applied problem can lead to interesting pure mathematical problems. A modelling question in subsurface contaminant transport - how to include the knowledge of the regional flow field into an anomalous advection-dispersion model - inspires the development of an unbounded functional calculus for group generators. Most importantly, after the abstract theory is explored some of the results find their way back to the application and can be used to formulate the model, show well-posedness and develop a numerical approximation. This is a joint work with Boris Baeumer (University of Otago) and Markus Haase (TU Delft).

100715155143

A three-dimensional model of linear water-wave scattering by a collection of circular elastic plates has been devised, which provides the basis of the numerical component of a project looking at the propagation of waves through a field of ice floes. The scattering response of a single floe combined with an interaction theory solves this problem in the frequency domain. The work will supplement wave tank experiments scheduled to take place in few weeks. For the purpose of designing the experimental process, a second model has also been developed in which the transient response of a group of ice floes in a 2D numerical wave tank is investigated. A short presentation of the facilities will be given, as well as preliminary results of the numerical models.

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STAT 480 Preliminary Presentations

Darren Alexander
Analysis of correlated binary data

Yahya Aljohani
"Statistics Anxiety": measuring its level and impact
on choice of statistical software

Ross Haines
Bayesian modelling of eye-tracking data

Crystal Symes
Multivariate analysis used to estimate stature of Prehistoric Thai people from Ban Non Wat

MATH 480 Final Presentation

Padarn Wilson
Constructing a Brownian motion

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In this talk I want to present Wave Maps, which are a kind of geometric evolution equations. These equations are generalizations of the wave equation to mappings from and into (semi)-Riemannian manifolds. First I will describe the geometric and analytic background and give some examples where they appear in Theoretical Physics. The next part will cover a 2+1 dimensional example, where I will show why it is interesting to do research on these systems. At the end I will talk about the numerical code which I am developing for solving the Wave Map equations.

100526104517

In this talk I will discuss the dispersion relation (DR) for an isotropic elastic plate floating on an inviscid incompressible fluid. The motion in the fluid is taken to be linear and irrotational, and can thus be determined from a potential that satisfies Laplace's equation. The solid motion is also linear, with the displacement satisfying Navier's equation, which can be written in terms of two coupled potentials that satisfy Helmholtz equations. These are also coupled to the fluid potential. The roots of the DR thus obtained are compared to those of the DR for a floating Euler-Bernoulli thin elastic plate, and of an in vacuo elastic plate (found by considering plane wave solutions to Navier's equation in the absence of gravity and any external stresses). The eigenvalues of the full problem are similar to those for both of the simpler problems, but with additional bifurcations. The motivation for solving this problem was to test the range of validity of the thin plate approximation (TPA) which is the standard model for sea ice. We find that the wavelengths for the propagating flexural waves are extremely close over the typical thickness range (1-10m) for sea ice, but are noticeably different at larger thicknesses typical for ice shelves (~200m). There are also additional propagating modes, both flexural and compressional, that are not predicted by the TPA. However, all but one of these occur at very high frequencies, and the mode that does exist at reasonable frequencies has an extremely high wavelength (>10km), so we are unsure how much energy would be transferred into this mode in an actual scattering problem.

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