The Lax Milgram Lemma from 1954 opened an easy - now standard - approach to establish existence and uniqueness for elliptic equations. But it also allows one to prove without effort parabolic results. More precisely, the corresponding elliptic operator generates a holomorphic semigroup on a Hilbert space.

In order to obtain results on L^p spaces, a simple strategy consists in proving that the corresponding semigroup is contractive for the L^p norm. This works very well in virtue of a criterion for the invariance of convex sets due to Beurling-Deny for positivity and the sub-Markovian property, and to Ouhabaz for arbitrary convex sets. This gives a near complete picture of the autonomous situation.

Lions proved an important theorem establishing existence and uniqueness for time-dependent equations governed by a form. In the talk we will discuss this theorem in detail. An important problem of maximal regularity is still open but recently some progress was made (paper [1] below). With regard to the invariance of convex sets, final results were obtained [2] which improve the known results, even in the autonomous case.

Form methods are very efficient and cover a variety of interesting and relevant boundary value and initial value problems. In this talk I will try to give an easy introduction to these methods illustrating them with examples of parabolic equations.

[1] W. Arendt, D. Dier, H. Laasri, E.M. Ouhabaz: Maximal regularity for evolution equations governed by non-autonomous forms. Preprint 2012

[2] W. Arendt, D. Dier, E.M. Ouhabaz: Invariance of convex sets for non-autonomous evolution equations governed by forms. Preprint 2013

[3] W. Arendt, D. Dier, M. Kramar Fijavz: Diffusion in networks with time-dependent knot conditions. Preprint 2013.

130128151323

The origin of life is one of the most fundamental, but also one of the most difficult problems in science. Despite differences between various proposed scenarios, one common element seems to be the emergence of an autocatalytic set or cycle at some stage. However, there is still much disagreement as to how likely it is that such self-sustaining sets could arise "spontaneously". This disagreement is largely caused by the lack of mathematical models that can be formally analyzed.

In this talk, after a brief introduction to the origin of life problem itself, I will introduce a formal framework of chemical reaction systems and autocatalytic sets, and then present both theoretical and computational results which indicate that the emergence of autocatalytic sets is highly likely, even for very moderate (and plausible) levels of catalysis. Furthermore, I will present a new way to identify and classify autocatalytic subsets, which elucidates possible mechanisms for evolution and emergence to happen in such sets. Finally, I will end with some speculation on how this formal framework might be extended to areas beyond the origin of life, and possibly be applied to living cells, entire ecologies, and perhaps even the economy.

121207110007

We present a fully numerical approach to compact Riemann surfaces starting from plane algebraic curves. For a given algebraic equation with two variables, the code in Matlab computes the branch points and singularities, the holomorphic differentials, and a base of the homology. The monodromy group for the surface is determined via analytic continuation of the roots of the algebraic equation on a set of contours forming the generators of the fundamental group. The periods of the holomorphic differentials are computed with Gauss-Legendre integration along these contours. The Abel map is obtained in a similar way. The performance of the code is illustrated for many examples. As an application we study quasiperiodic solutions to certain integrable partial differential equations.

121105091536

Do you know what a C*-algebra is? Algebras of continuous functions and direct sums of matrix algebras are typical examples. In fact there are no other commutative or finite-dimensional C*-algebras. But there are lots of non-abelian and infinite-dimensional C*-algebras. More than we can understand!!! One special class of C*-algebras, now called Kirchberg algebras, are of particular interest because they are 'classifiable' (by K- or KK-theory), a result obtained by Kirchberg and Phillips in the mid 1990’s. Many of the naturally occurring examples of Kirchberg algebras arise from dynamical systems. In joint work with M. Rørdam we are particularly interested in dynamical systems where a discrete group is acting on a commutative C*-algebra. As an application of our results we show that every discrete countable group admits a action on the Cantor set such that the corresponding crossed product C*-algebra is a Kirchberg algebra if and only if the group is non-amenable and exact. I will talk about classification of C*-algebras and if time allows explain how one gets a Kirchberg algebra out of each discrete non-amenable exact group.

120725152350

The study of sequence diversity under phylogenetic models is a classic. Theoretical studies of diversity under the Kingman coalescent appeared shortly after the introduction of the coalescent in 1982.

In this talk I revisit the topic under the multispecies coalescent, an extension of the single population model to multiple populations. I derive exact formulas for the sequence dissimilarity under a basic multispecies setup, discuss the effects of relaxing some of the model assumptions, and show some simple usages for real data.

120921153135

This talk aims to give a quick but elementary introduction to the homological dimensions of rings, looking at rings with small dimension. If time permits, I'll say a little about the recently-solved Bazzoni-Glaz conjecture on the weak dimension of Gaussian rings, the subject of MSc student Eman Alhassan's thesis.

120725152305

1.30 Del Nawarajan : Quantum algorithms
1.45 Yue Wang : Simulating fractional reaction-diffusion models
2.00 John Holmes : An investigation into models of a cysteine metabolism reaction
2.30 Ilija Tolich: C*-algebras generated by power partial isometries
3.00 Sam Paulin : Fuchsian theory and the relativistic Euler equations
3.30 Kelly O'Connell : Bratteli diagrams and their Leavitt path algebras
4.00 Sam Stewart : Soliton theory
4.30 Chris Stevens : Colliding gravitational plane waves

Note day and start time of this event. Individual times are a guideline only.

120510124714

This talk will use visualizations of examples such as Rubik's cube to introduce the basic algorithms of group theory. The results will be applied to the canonicalization of expressions in tensor algebra.

120820162952

A SNP (single nucleotide polymorphism) is a location on the genome where members of a population have difference nucleotides (A,C,G,T) at a single sequence position. Databases of SNPs are used primarily as road maps of the genome, helping researchers identify associations between genes, traits and diseases. In this talk I will discuss the use of SNPs to infer relationships between different populations or species, including the estimation of ancestral population sizes. I'll introduce the biological and statistical models used, and introduce SNAPP, a likelihood algorithm which uses numerical and combinatorial tricks to compute the relevant likelihoods. Previously only Monte-Carlo estimates had been available.

This work has been published in:
Bryant, D., Bouckaert, R., Felsenstein, J., Rosenberg, N., RoyChoudhury, A. Inferring species trees directly from biallelic genetic markers: bypassing gene trees in a full coalescent analysis. Molecular Biology and Evolution 19(8):1917-1932

Software is available at snapp.otago.ac.nz

120921152902

The standard modelling approach of the interactions between ocean waves and sea-ice combines the thin-elastic plate theory (for the ice floes) and a potential flow in the fluid, forming the standard theory of linear hydroelasticity. This theory is also relevant in the design of very large floating structures, such as floating runways. Both research areas have acknowledged the critical lack of experimental data to support the expanding corpus of theoretical research. As a remedy, a unique set of wave tank experiments has been conducted to validate a three-dimensional model of linear water wave scattering by one and two floating thin-elastic disks. The experimental procedure has been designed to reproduce the limitations of the model, such as linearity and harmonic motion, as a means to conduct a proper validation analysis.

A description of the standard hydroelastic model and a summary of the solution method will be given. A number of technical solutions in the experimental setup and the data processing technique required to enforce the restrictions of the model, will be presented. Using the natural modes of vibration of the circular plate as comparison objects, we will show that reasonably good agreement is found over a fairly large frequency spectrum, although clear trends of discrepancy exist. Results involving a single disk and the interaction between two disks will be presented. Extensions to the standard model will be considered to account for additional physical processes that may influence the motion and explain the discrepancy.

120725152138

A function f that is entire, that is, analytic in the entire complex plane, is represented everywhere by its Taylor series. Valiron and Wiman introduced the idea of analyzing the behaviour of f using two auxiliary functions associated with the Taylor series: the maximum term and the degree of the maximum term. The theory has been developed since by others, notably Kövari, Clunie and Hayman. One widely applied result is an asymptotic relation for the derivatives that can be used to estimate the growth of solutions to ODEs. These ideas have recently been extended to analytic functions in the finite disc and to functions of several variables, but I will focus on the entire case.

120725151954

The mathematical modelling of sea ice – ocean wave interaction provides rich terrain for a number of analytical and numerical problems. Many popular formulations involve coupling potential flow to a fourth order elasticity PDE, a system which is analysed using a diverse variety of frameworks. An overview of some of these approaches will be presented, with particular discussion of the merits and demerits of the Finite Element Method, as an adaptable technique for this application.

Lessons learned in the visualization and presentation of this work will be discussed, alongside the underlying innovative, open source technologies employed during the course of the project, and a tour of creative methods available to render results for public consumption. Software from the FEniCS Project (fenicsproject.org) will be featured in this talk.

120820162706

Phylogenetic methods are used to infer ancestral relationships based on genetic and morphological data. What started as more sophisticated clustering has now become a more and more complex machinery of estimating ancestral processes and divergence times. One major branch of inference is maximum likelihood methods. Here, one selects the parameters from a given model class for which the data are more likely to occur than for any other set of parameters of the same model class. Most analysis of real data is executed using such methods.

However, one step of statistical inference that has little exposure to application is the goodness of fit test between parameters and the data. There seem to be various reasons for this behaviour, users are either content with using a bootstrap approach to obtain support for the inferred topology, are afraid that a goodness of fit test would find little or no support for their phylogeny thus demeaning their carefully assembled data, or they simply lack the statistical background to acknowledge this step.

Recently, methods to detect sections of the data which do not support the inferred model have been proposed, and strategies to explain these differences have been devised. In this talk I will present and discuss some of these methods, their shortcomings and possible ways of improving them.

120817094134

Classical robotics is about using motors to over-ride inertial, elastic and dissipative forces acting on mechanical structures in order make them do what we want. The future is about combining inference and control with the design of mechanical linkages whose dynamics are exploited, not over-ridden, to move quickly, accurately and efficiently. Stochastic dynamical systems theory and computational modelling can join the dots from the reproductive strategies of sponges to the dynamics of squash rackets, helping us to understand how brains and bodies coevolved for agile movement, and showing how to build better robots and train better athletes.

120725151738

A connected graph G is said to have the cycle extension property (or more briefly, "G is CEP") if it contains a 2-factor and for every cycle C in G with |V(C)| ≤ |V(G)|-3, there is a 2-factor F_c in G which includes C.

Besides explaining the terms above more fully, I will present some background to the problem and provide a characterization of all graphs which are CEP.

120725151643

Species distribution models are important in ecology and conservation. They typically predict the probability of occurrence of a species in geographical space from data on the presence or absence of the species at sites with known environmental characteristics. These models have no temporal interpretation, and therefore cannot tell us anything about population dynamics.

I will describe related models for population dynamics, based on categorical survey data. Categorical surveys record abundance categories (e.g. abundant, rare, not seen), and can be a good way to cover large numbers of sites quickly. From the resulting models, we can get predictions about species distributions with a temporal interpretation. I will apply this approach to data on two intertidal snail species in the UK.

120810101350

A self-similar group is a group which comes with a faithful action on a rooted tree. In this talk, we will make this definition precise, discuss some of the immediate consequences, and illustrate with two key examples. We will then try to explain why an operator-algebraist might be interested in such things.

The applications are from joint work with Marcelo Laca, Jacqui Ramagge and Mike Whittaker.

120725151545

Out of necessity, charts are used to study the gravitational singularities of Lorentzian manifolds. Since such singularities are located on the topological boundary of the range of charts, the principle of general covariance has no applicability. Nevertheless, some concept of coordinate independence is needed so that we can speak of the physical properties of a singularity. This talk will use the Curzon solution to provide an example of this problem and will discuss recent attempts to produce a solution.

120725151439

The singular initial value problem and Fuchsian equations have proved to be successful tools for exploring (analytically) the singular behaviour of solutions to Einstein's equations. In particular they are used to verify asymptotically velocity term dominated (AVTD) behaviour in some system of coordinates. I will present a theorem which says there exist unique solutions to the singular initial value problem for a certain class of quasilinear Fuchsian partial differential equations. This theorem may be applied to show AVTD behaviour in families of solutions to the Einstein equations with a torus symmetry, and we hope as well a U(1) symmetry.

120710165949

4th year final project presentation
Note day and time

In this seminar I will present the main results of my project on conformal mapping, in particular normal families, Riemann's Theorem and Schwarz-Christoffel formula.

120517121422

After a brief introduction to spectral geometry, we will discuss the degree to which the eigenvalue spectrum of the Laplace operator of a Riemannian orbifold encodes information about the orbifold. In particular, we ask, "How does the presence of mild singular points in orbifolds cause the spectral geometry of orbifolds to differ from that of manifolds?" Stated more informally, "Can you hear the singular set of an orbifold?"

120321095520

Imposing appropriate boundary conditions is a very delicate procedure in both analytical and numerical studies of partial differential equations. Incorrect imposition can lead to loss of convergence and time-instabilities. Thus, special care must be taken when imposing boundary conditions.

In this talk we will develop high-order finite difference numerical schemes for first and second order partial differential equations. The continuous derivatives will be approximated by central finite difference operators that satisfy a summation by parts (SBP) formula. For the implementation of the boundary conditions the SAT (simultaneous approximation term) penalty method will be used.

Finally, we will present several simple numerical examples to demonstrate that the SAT method preserves the designed accuracy of our numerical schemes and guarantees their Lax-stability.

120321095431

It has been known since 1980 that if a groupoid is both minimal and topologically principal, then its C*-algebra is simple (Renault). Whether the converse holds was an open question until now. In a recent paper with Brown, Farthing, and Sims, we answer this question in the affirmative. In this talk, I will start by defining the relevant terminology and describe how we associate algebras to groupoids. Then, I will outline how we were able to prove the converse of Renault's theorem. Our technique is new and gives a possible strategy for answering other hard C*-algebraic questions.

120321095342

120430122018

In numerical schemes for fractional in space partial differential equations, a first order Grünwald formula is used to approximate the fractional derivative. We develop higher order Grünwald-type approximations and discuss consistency, stability and smoothing of some of these higher order schemes with specific error estimates in L₁. We present the main tools used in the investigations, namely, a new Carlson-type inequality for Fourier multipliers and a technical result which gives sufficient conditions for multipliers associated with difference schemes approximating the fractional derivative to lead to stable schemes. The theory is then extended to fractional powers of operators on an abstract Banach space. We also give results of some numerical experiments including a third order scheme.

120321094751

NOTE - DAY AND TIME ARE DIFFERENT TO USUAL SCHEDULE

In this seminar I’ll explain the PAPRIKA method that I co-invented.* PAPRIKA is a partial acronym for ‘‘Potentially All Pairwise RanKings of all possible Alternatives’. The PAPRIKA method, implemented by 1000Minds software (www.1000minds.com), is for determining the weights for decision criteria used in additive Multi-Criteria Decision-Making (MCDM) models and Conjoint Analysis (or ‘Discrete Choice Experiments’). The weights represent the relative importance of the criteria to decision-makers. MCDM models (commonly known as points systems or point-count models) are used for ranking or prioritising alternatives in a very wide range of applications – e.g. prioritising patients for elective surgery, investment decision-making, helping students to choose their majors (see www.nomajordrama.co.nz), choosing a new home, etc. When you’re working on a decision, getting the weights ‘right’ is important because even if you’re applying the right criteria, unless your weights accurately reflect your preferences you’re likely to make the wrong decision!

* P Hansen & F Ombler, “A new method for scoring multi-attribute value models using pairwise rankings of alternatives“, Journal of Multi-Criteria Decision Analysis 2009, 15, 87-107. And for an overview search ‘PAPRIKA’ on Wikipedia.

120427135853

Let K be a field. Abrams and Aranda Pino introduced a graded K-algebra L(E), called the Leavitt path algebra, constructed from a directed graph E. We assume E satisfies certain finiteness conditions (row-finite, no sources) and characterize when the resulting Leavitt path algebra is semisimple: that is when L(E) is the direct sum of matrix algebras. As stated this result is due to Abrams et. al. 2010. We then show which semisimple K-algebras can be constructed from row-finite directed graphs with no sources. These results are a special case of results obtained in joint work with Astrid an Huef.

120321094716

A JOINT MATHEMATICS AND STATISTICS SEMINAR

The multinomial distribution describes the probability of observing a given number of outcomes of each of r different types over n i.i.d. trials. This distribution is ubiquitous in applied statistics, but surprisingly, no algorithms having useful performance guarantees were known for calculating the mode (most probable) vector(s) of a given distribution until recently. I will motivate the problem with a biological application, and then describe a simple O(rlog r) time, O(r) space algorithm invented by myself and Prof. Mike Hendy for calculating one or (a compact representation of) all modes. Our algorithm is numerically robust due to the absence of unbounded chains of floating-point arithmetic, and in practice it drastically improves on the performance of earlier algorithms.

120418123344

I will describe the numerical simulations of spherically symmetric spacetimes with dust and radiation. Grid points are staggered to avoid coordinate singularity at the origin. A suitable choice of variables is used to avoid blow-up at the origin. A new innovation is the lightlike outer boundary. As a result, the outer boundary conditions need not be specified. The lightlike boundary also provides accurate computation of redshift and luminosity distance.

120321091115

The role of modern probability will be discussed and illustrated with many examples from "real life", including gambling, parenthood, and the sinking of the Titanic.

NZMS Forder Lecture for 2012

120411093702

A number of 'exact' (and beautiful) solutions are known for two-dimensional systems of probability and physics. These systems have critical points, and certain special techniques have emerged for their study.

I will discuss the twin features of conformal invariance and universality, particularly in the context of the percolation model. A fascinating structure is becoming clear, with connections to analysis, geometry, and conformal field theory, but serious difficulties remain.

(NB change to our usual day, time and venue)

120402135619

Machine Learning (ML) (Michie et al., 1994; Mitchell, 1997) is a scientific discipline which is concerned with how algorithms are designed and developed to allow computers to build learning models based on different data sources such as empirical data, databases, or sensor data.

These models not only are able to learn and recognise complex patterns or relationships between observed variables within the data but also are able to make intelligent decisions based on these data sources. Furthermore they can also generalise from data which they have learned from to produce useful output in new cases.

ML theory, approaches, and techniques have been influenced by statistics and share a lot in common with this discipline although they seldom use the same terms. In this talk I will introduce what machine learning is, its relevance to statistics (Hastie et al.; 2011) and cover real-world applications of machine learning systems.

Michie, D., Spiegelhalter, D. J. and Taylor, C. C. (1994). ,Machine Learning, Neural and Statistical Classification. Ellis Horwood.
Mitchell, M. T. (1997). Machine Learning. MacGraw-Hill.
Hastie T., Tibshirani R. and Friedman J. (2011). The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Fifth Edition, Springer.

120227165458

The gravitational radiation of objects like systems of stars or black holes is usually analysed in the context of solutions to Einstein's field equations which are asymptotically flat. R Penrose suggested that the conformal structures of such solutions extend smoothly to null infinity, the ideal place where a natural notion of gravitational radiation can be defined for the non-linear theory. In this talk we consider time reflection symmetric data, discuss recent results which seem to indicate that static data play a special role in their analysis, and present a new result which characterizes the gap between conditions which are necessary and conditions which are sufficient for the field to develop the desired smoothness of the asymptotic structure.

120307160552

Algorithmic ideas from computational optimization provide great inspiration for building efficient simulation algorithms – that draw samples from a desired distribution – for use in MCMC. I will show how polynomial acceleration of MCMC is simple in concept, and explicitly construct an accelerated Gibbs sampler for Gaussian-like distributions. The past few years have seen development of a number of other sampling algorithms based on Krylov space and quasi-Newton optimization algorithms that achieve spectacular performance in some settings.

120227161656

We introduce the characteristic form and partition function form (pff) transferable utility cooperative games and some of their characterizing properties (incl. externalities). We define and demonstrate the concept of the recursive core of a partition function form game to analyze the stability of partitions.

We consider two possible applications: firstly, a pff game over routing networks and secondly, a pff game over electrical power transmission networks. In both cases we demonstrate positive and negative externalities and the possibly arising subadditivity. Furthermore we demonstrate that the recursive core may be empty in both cases.

120224083926

Natural systems are characterized for having both spatial and temporal heterogeneity, which in the case of microbial communities are intricately associated with biogeochemical cycles. It is well accepted that microbial communities are both responsive to changes and highly diverse, yet integration of analytical methods for developing or validating microbial ecology models are lacking. To address this shortcoming we sampled an agricultural soil (hay field) over a 2-month period of time to examine the effects that natural (seasonal driven) changes would have on the microbial component of nitrogen and carbon cycling dynamics in surface soils. Total soil DNA and RNA were used to assess the abundance of targets (gene copies representing enzymatic potential and transcript numbers representing expressed potential, respectively) for multiple carbon (C) and nitrogen (N) cycling genes. Soil samples were collected every 6 hours for 48 hours. Multiple additional samplings were performed in the ensuing weeks. The results show a dynamic community that responds to changes in temperature and moisture. However, due to the complex nature of microbial communities, and the many variables that can influence microorganisms, more thorough statistical analyses are required.

120227121623

For a group G and a set of generators S of G, the Cayley graph Γ(G, S) is defined to have vertex set G and g, h ∈ G are adjacent if and only if gs = h or hs = g for some s ∈ S. The diameter of Γ(G, S) is the maximum distance among pairs of vertices; equivalently, the diameter is the minimum number d such that every group element can be written as a word of length at most d in terms of the elements of S and their inverses. The diameter problem may be interesting for a particular group and set of generators (how many turns do we need to solve Rubik’s cube?), but the mathematically most challenging questions are about estimating
diam(G) := max_S{diam(Γ(G, S))}
with the maximum taken over all sets of generators of G, and for G in an appropriate family of groups. The challenge driving most recent activities is Babai’s conjecture, which states that for all finite nonabelian simple groups, diam(G) < (log|G|)^c, for some absolute constant c. The conjecture was proven by Pyber, Szabó and Breillard, Green, Tao in 2011 for Lie-type groups of bounded rank, but the case of alternating groups cannot be handled by their machinery. For alternating groups A_n, Babai’s conjecture requires a polynomial, n^c, diameter bound. We can prove a slightly weaker quasi-polynomial result:
diam(A_n) < exp(O((log n)⁴ log log n)).

This is joint work with Harald Helfgott (ENS, Paris).

120131094635

If rotating dust stars would exist in general relativity, this would represent an example of a complete balance between the attractive Newtonian force and the typically repulsive gravitomagnetic "force". Although such a balance is considered as improbable by all experts, till now all attempts for a non-existence proof for rotating dust stars failed.

I present a new attack for this problem by considering the level lines of the quasi-Newtonian potential U in the Bardeen metric form for stationary and axisymmetric systems. If rotating dust stars would exist, one U-level-line would run along the axis in the dust, would bifurcate at the poles of the star, would close in the exterior vacuum region, and would enclose a U-minimum there. Hereby the question of existence of rotating dust stars is reduced to the concrete but still open mathematical problem whether the potential U can have a minimum in the vacuum region.

120125121757

In this talk, we explore a family of polynomials whose roots are related to the Mandelbrot set. These roots correspond to the k-periodic points of the iteration defining the Mandelbrot set. We discuss some of a variety of approaches to compute the roots of these polynomials; classical iterative schemes, eigenvalues of companion matrices and a novel family of recursively defined matrices. Time permitting, we look at an experimentally-discovered asymptotic series for the largest-magnitude roots.

Joint work with Piers W Lawrence and David J Jeffrey.

120207115912

We extend the Multi-Level Monte Carlo (MLMC) algorithm in order to quantify uncertainty in the solutions of multi-dimensional hyperbolic systems of conservation laws with uncertain initial data and sources. The algorithm together with the novel load balancing procedure is presented and the scalability on the massively parallel hardware is verified. A new code ALSVID-UQ is described and applied to simulate uncertain solutions of the Euler equations, ideal magnetohydrodynamics (MHD) equations and shallow water equations. Numerical experiments showing the robustness, efficiency and scalability of the proposed algorithm are presented.

120118120348

Centroidal Voronoi tessellations (CVTs) are special Voronoi diagrams for which the generators of the diagrams are also the centers of mass (with respect to a given density function) of the Voronoi cells. CVTs have many uses and applications, several of which we discuss. These may include data compression, image segmentation, clustering, cell biology, territorial behavior of animals, resource allocation, grid generation in volumes and on surfaces, meshless computing, hypercube sampling, and reduced-order modeling. We also discuss a little bit of the theory associated with CVTs and deterministic and probabilistic methods for determining CVTs, including some probabilistic methods that are amenable to parallel processing.

120126110847

In the talk we will consider Hamiltonian perturbations of hyperbolic systems of partial differential equations (PDEs) with one spatial variable. The deformation theory of integrable PDEs will also be discussed. We also consider the behaviour of solutions to such systems at the point of "phase transition" from regular to oscillatory behaviour.

111205111904

The evolution of genomes has traditionally been thought to follow a single tree-like pattern of ‘vertical’ inheritance from ancestors to descendents. However, there are a number of biological reasons that individual genes or parts of genes within a genome might not have evolved along the same tree. Identifying discordant gene trees allows biologists to quantify the importance of ‘horizontal’ evolution and to identify the different relationships that constitute the history of genomes. I will present two methods for assessing disagreement between trees. First, I will describe a hierarchical likelihood ratio test that is powerful, but computationally intensive. Second, I will present a clustering method that is more suitable for larger data sets. As the number of complete genome sequences grows, methods for evaluating discordance are becoming increasingly important to evolutionary biologists.

111102092630

The total magnetic charge of particle-like solutions of the EYM equations are determined by the asymptotic behaviour of the Yang-Mills fields. By examining this behaviour we can investigate the question of whether magnetically charged particle-like solutions exist. I will introduce the various ideas and explain what we know about this so far and what remains to be determined.

111028101702

With the success of the Hamilton-Perelman program for proving the Geometrization Conjecture, Ricci flow has become a mathematical celebrity. Almost all discussions of Ricci flow focus on the behavior of the flow on families of three and four dimensional geometries. Why are two dimensional geometries neglected? After a brief review of Ricci flow, we discuss some interesting questions involving surfaces, and answer some of these questions.

111011090644

Eman Alhassan
Dedekind Domains

Fatemah Al Kalaf
Conformal Mapping

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

Richard McNamara
Parseval Frames

Sam Primrose
Leavitt Path Algebras

111010113545

One of the motivators for noncommutative geometry is the "functional definition" of Riemannian distance. This result shows how the distance of a Riemannian manifold can be recovered from the interplay of the Dirac operator, the algebra of functions on a spin manifold and the Hilbert space of square integrable spinor fields. Since the Riemannian metric can be reconstructed from the distance this result tells us that a Dirac operator, an algebra and a Hilbert space are, collectively, an analogue of geometry.

There exists a similar functional definition for the Lorentzian distance on globally hyperbolic manifolds. Since global hyperbolicity is a restrictive condition a more general functional definition would give much needed insight into how to formulate noncommutative geometry on Lorentzian manifolds. In this talk I will present a new result, due to Adam Rennie and myself, which extends the existing functional definition to the largest class of Lorentzian manifolds that it applies to.

I will begin the talk by reviewing the Riemannian results and contrasting these against the existing Lorentzian results. In particular, by discussing the differences between these results I will illustrate the issues encountered when attempting to extend the globally hyperbolic function definition. Lastly, if time permits, I will review the techniques used in the proof of Rennie's and my extension.

110727143832

Around the Antarctic continent is a region of loosely-spaced ice known as the Marginal Ice Zone. As waves enter this region, the individual ice floes bend and flex, dispersing and reflecting some of the incident wave energy. Through this mechanism, they provide some protection for land-fast ice from erosion. As such, the understanding of these dispersive interactions forms an important component in the consideration of Antarctic contributions to global climate dynamics.

Arising from this phenomenon is the mathematical problem of a floating elastic in a fluid basin, which forms the basis of the current research. The computational domain is discretised for finite element solution of the governing equations. A time-stepping approach is employed, with a predictor-corrector method applied to find the combined motion of the beam and fluid at each iteration. This leads to an extensible formulation able to incorporate a wider variety of physical properties than is possible using the frequency domain techniques predominant in sea-ice research. Thus we provide for more robust, physically realistic models of polar ice floes.

110718104530

Molecular phylogenetics is the art of constructing phylogenies (evolutionary trees) from biological sequence data. Using some simple mathematical tools I have been endeavouring to turn this art into a science.

I will introduce some of my findings including Hadamard conjugation and the closest tree selection algorithm.

110713083845

Sequential analysis is the theory of cumulative sums of random variables. The central result in sequential analysis, Wald's Fundamental Identity, can be used to calculate absorption probabilities in random walks with barriers.

The Moran process from mathematical biology is a birth-death process used to model the spread on mutant genes in a population. This process can be used to calculate the probability that a beneficial mutation will spread to an entire population. The Moran process is the cumulative sum of random changes in the population state, and is therefore amenable to sequential analysis.

We have used sequential analysis to develop an analytical approximation to a simple simulated ecosystem. An analytical model gives us insights not available from the simulation results, and sequential analysis allows us to build our approximate model using a single conceptual tool.

110915095127