Let K be a closed subgroup of a locally compact group G (but thinking of G as infinite discrete is OK). Given a unitary representation U of K on a Hilbert space, there is a concrete procedure, dating back to Mackey and Blattner, of inducing U to a unitary representation of G. Mackey's imprimitivity theorem identifies the representations of G that are equivalent to a representation induced from K. Rieffel's influential reformulation of Mackey's imprimitivity theorem is in terms of the Morita equivalence of two C*-algebras A and B: this is an A-B bimodule X carrying the necessary structure to induce Hilbert-space representations of B to Hilbert-space representations of A, and back again. The C*-algebras of Rieffel's reformulation are those of the group K and a crossed-product algebra. This is an expository talk in which I will explain what crossed products of C*-algebras are, what Morita equivalence is all about, and how the two combine to give imprimitivity theorems.

100430152817

Here we consider two models for the large amplitude planar motion of a cantilevered beam with a feedback force and/or torque (proportional and acting opposite to the velocity/angular velocity respectively) applied at the free end. We derive nonlinear equations of motion for each model based upon certain natural geometric assumptions. We then use the relationship between the models to prove the existence of classical solutions and show that all classical solutions are uniformly exponentially stable.

100427150241

A graph G is said to have property E(m, n) if it contains a perfect matching and for every pair of disjoint matchings M and N in G with |M|= m and |N|= n, there is a perfect matching F in G such that M  F and N  F = { }. It is well known that no planar graph is E(3,0) or even E(2,1). If we consider even planar triangulations and demand that the edges are suitably far apart we can improve on these results. In this talk we will look at some of these improvements and consider the properties E(m, n) for planar triangulations when various distance restrictions are imposed on the edges to be included and the edges to be avoided in the extension.

100415102320

The space of representations of a locally compact group comes equipped with a natural topology -- the Fell topology -- though it can be very degenerate. In this talk I will give an introduction to the "noncommutative topology" of group representations, using operator algebras and K-theory. This approach leads to questions in equivariant index theory. I shall thus describe methods for analysing differential operators on the flag variety of a complex semisimple group, particularly the group SL(3,C).

100408164510

Quite recently, Katsura defined the notion of a topological graph which unifies countable directed graphs and dynamical systems. This enabled him to introduce a new class of C*-algebras generalizing the classes of graph C*-algebras and homeomorphism C*-algebras. Here we are interested in the non-selfadjoint tensor algebras associated to topological graphs. These algebras generalize non-selfadjoint operator algebras associated to countable directed graphs or dynamical systems already present in the literature (e.g. Arveson’s analytic semi-crossed product C(X) x|σ , where (X, σ) is a continuous dynamical system). +􀁝 Arveson/Davidson-Katsoulis proved the following rigidity result: two dynamical systems (X, σ) and (Y, τ) are conjugate if and only if the algebras C(X) x|σ and C(Y) x|τ +􀁝+􀁝 are isomorphic. In the topological graphs case, we also hope to be able to classify the associated tensor algebras. We had to introduce a notion of local conjugacy between topological graphs. We show that if two tensor algebras of topological graphs are algebraically isomorphic, then the graphs are locally conjugate. Conversely, if the base space is at most one dimensional and the edge space is compact, then locally conjugate topological graphs yield completely isometrically isomorphic tensor algebras (joint work with K.R. Davidson).

100408164326

The Metropolis (Hastings-Green) algorithm for MCMC was invented in the 1950ʼs and has changed little since. In contrast, algorithms for linear algebra that were also invented in the 1950ʼs have seen dramatic improvement, as is evident from the robust and efficient numerical packages such as LAPACK, EISPACK, and others. Our goal is to steal the developments in linear algebra and apply them to sampling of probability distributions. Some important sampling algorithms, such as Gibbs sampling of Gaussian distributions, correspond exactly to stationary iterative methods used for solving linear systems. We state and prove a theorem to that effect. Further, the convergence rates are exactly the same. This means that novel and efficient Gibbs samplers can be produced, with convergence established, by looking in an introductory numerical analysis text. For large linear problems, stationary iterative methods are considered to be very slow, with the state of the art being Krylov-space methods with polynomial accelerators. We show how polynomial acceleration can be applied to give very fast Gibbs sampling algorithms, that sets a new state of the art for samplers.

100329144416

I resume former discussions of the question whether a General Relativistic configuration with two aligned axisymmetric black holes in vacuum can be in equilibrium. To answer the question I formulate a boundary value problem for two separate (Killing-) horizons in axisymmetry and stationarity and apply th e inverse scattering method to solve it. Finally, I use a universal inequality for sub-extremal black holes and the positive mass theorem to show that two (sub-extremal or degenerate) black holes cannot be in equilibrium.

100318165915

Drug dilution (MIC) and disk diffusion (DIA) are two common tests used by clinicians to determine pathogen susceptibility to antibiotics. For each of these tests, two drug-specific breakpoints classify the unknown pathogen as either susceptible, intermediate, or resistant to the drug. While MIC breakpoints are largely based on the pharmacokenetics and pharmacodynamics of the drug, comparable DIA breakpoints are not as straightforward to calculate. Current Clinical and Laboratory Standards Institute (CLSI) guidelines require a scattergram of test results for numerous pathogens and the DIA breakpoints are based on limiting the classification discrepancies. This approach, however, does not account for certain test properties and experimental errors. I will discuss this as an error-in-variables problem and then describe a hierarchical model, which factors in the uncertainty of both tests, the drug-specific relationship between the two tests, as well as the underlying distribution of pathogens. For the drug-specific relationship between these two tests, I propose both a parametric and nonparametric approach. A loss function is then used to determine the DIA breakpoints. This is joint work with the CLSI Subcommittee on Antimicrobial Susceptibility Testing.

100312100200

Fuchsian equations form a class of partial differential equations which allow to construct solutions with in principle prescribed singular behaviors. This has important applications for instance in general relativity and allows to get deeper insights into outstanding problems, like the BKL- or cosmic censorship conjectures. In this talk I will point to certain drawbacks of the current Fuchsian theory and present a new framework. In particular this leads to a new existence proof of a singular initial value problem of certain second-order hyperbolic Fuchsian equations which makes direct use of the hyperbolic structure. In particular it yields a numerical scheme with strong control of convergence. This is joint work with P. LeFloch from the University of Paris.

100308124838

A short introduction to non-relativistic elasticity theory is given with emphasis on the partial differential equations describing the theory.

Based on this, I will discuss various existence theorems.

If time permits, I will make some remarks about the corresponding problems in general relativity.

100301144954

The theoretical analysis and numerical simulation of the propagation properties of light in optical media are important to many applications. The mathematical modelling of electromagnetic devices is based on the wave equations which are derived from the Maxwell’s equations by elimination of either the electric or the magnetic field. In electromagnetic modelling, the analysis and computation of eigenvectors are prevalent activities; even the wave equations (monochromatic light) appear as eigenvalue problems. The way the eigenmodes are chosen depends on the geometry of the media; the cases of structures which are invariant or periodic in one, two or three directions will be considered. The kernel of the curl operator in the wave equation has infinite dimension; this can create some numerical stability issues for standard continuous finite element methods and other eigensolvers. Discontinuous finite element methods (edge elements) are now widely used to solve the wave equations; their success is partly due to their ability to incorporate discontinuous gradient functions.
Electromagnetic waves can travel over arbitrarily large distances however finite element methods are limited in their ability to solve problems over large or unbounded domains. Semi-analytical techniques, based on the natural modes of the photonic media, combine the features of analytic and purely numerical techniques. They can be efficiently applied to obtain the solutions of problems over domains with arbitrary extent although they may lack some of the generality of numerical techniques. They can provide better understanding of the physics of complex photonic systems.
Applications of numerical and semi-analytical techniques to photonic devices will be discussed:
- Optical fibres: guided modes, dispersion curves
- Photonic crystals: Bloch modes, band structures, defect modes, couplers
- Diffraction gratings

100301144742

One of the greatest challenges facing theoretical physics is the
reconciliation of Quantum Theory with General Relativity. I propose a
discrete mathematical structure to underlie the continuum, a causal set, and show that a classical precursor to its dynamics yields causal sets which share many features in common with de Sitter spacetime. This leads to a sketch of how quantum gravity may resolve some of the current puzzles of cosmology.

100301144243

We discuss some versions of a numerical method for the discretization in time of an initial value problem for a parabolic equation in a Banach space framework.

The method applies a quadrature rule to a contour integral representation of the solution in the complex plane, typically based on Laplace transformation.

For each quadrature point an elliptic problem is solved, and this may be done in parallel. The error bounds obtained may be used in the analysis of fully discrete methods obtained by application of our time discretization method to a spatially semidiscrete finite element version of an initial-boundary value problem for a parabolic partial differential or a fractional order diffusion equation.

100112132126

Sam Fernando
Infectious Diseases and Fractional Reaction-Diffusion

Chris Laing
Game Theory

Harish Sankaranarayanan
Potential Theory in the Complex Plane

Padarn Wilson
Constructing a Brownian Motion

THERE WILL BE A BREAK FOR REFRESHMENTS AFTER TWO PRESENTATIONS.

091013123110

This will be a general interest presentation on the nature of gravity waves and detection strategies.
What are gravity waves and how do we look for them?
How do we currently measure displacements ~O(10-19 m)?
How much better than that can we expect to do?
How far out in space and back in time will we be able to explore?
What do we hope to learn about the universe?

Footnote
Natalia Zotov gained her Ph.D. at Otago in 1974, under the supervision of Prof. W. Davidson.
The subject of her thesis was the comparison of cosmological models with observational data.

090929163312

The following question will be considered:
given any positive integer n, how many distinct groups are there of size n ? Those in the know use the abbreviation gnu(n) to denote this number – the group number of n. A more indigenous number, denoted moa(n), will also be considered.

The quantity gnu(n) is influenced not by the size of n but rather by the number of primes (counting repetitions) of which n is a product. Given the long history of groups (initiated by Cayley in 1854), it is perhaps surprising that only in the last decade has a sizeable table of group numbers become available. The talk will be largely non-technical.

090917100743

For decades biologists believed that animals walk because the brain calculates the motion trajectories for the limbs. More recently a new hypothesis about animal locomotion suggests the contrary. Animal are built to walk “naturally” in the first place, such that they can walk without relying on controls. Instead, the walking gait is generated simply by the interaction of gravity and inertia, in a stable, naturally emergent limit-cycle, known as passive dynamic walking. The feasibility of passive dynamic walking had been demonstrated for biped system consisting of only a pair of legs. In this talk we will look into "full-body" passive dynamic walking models including not just legs, but also a body and upper-limbs. In particular, we will discuss the contribution of the upper-body to the stability and efficiency of passive biped walking, and show that several simple and animal-like mechanical linkages can generate a walking gait by using only gravity. They can also recover from small perturbations without the need of a controller input. These results suggest that the role of locomotion control is to provide stability, rather than driving the limb along a pre-calculated trajectory.

090914165409

The famous question of Kac is whether one can hear the shape of a drum. Or more precisely, whether all eigen frequencies of a drum determine the drum. In general the answer to the latter question is negative. The eigen frequencies are equal if and only if there exists a unitary operator which maps the Laplacian on the first drum onto the Laplacian on the second drum. In this talk we discuss what happens if the unitary operator is replaced by an order isomorphism, i.e., if it maps positive functions to positive functions. Or equivalently, if the diffusion processes on the two drums are equal.

This is a joint work with M. Biegert and W. Arendt.

090910095933

The need to decarbonise the world’s energy supply has renewed motivation in the study of ocean wave energy converters (WECs). Two key challengesthe technology must meet are to withstand the harsh marine environment, and to deliver reliable power at a competitive cost.

I am investigating a scheme where small-scale WECs (“point absorbers”) are installed in a linked chain, rather than individually to the sea bed. This scheme offers potential advantages relating to both aforementioned challenges, and also presents an interesting dynamical system.

I am exploring an idealised model, which yields a set of coupled first-order nonlinear integro-differential equations. This is solved numerically in the time-domain using MATLAB. In this talk I will discuss some basic principles of wave-body interactions and apply these to my system, present a few early results, and attempt to make what I’m doing palatable to an audience of mathematicians.

090826085912

Semiclassical general relativity and quantum field theory in curved spacetimes is a first step towards combining Einstein's purely classical theory of gravity with the quantum nature of matter. Implementing semiclassical general relativity requires one to calculate the expectation value of quantum operators in curved spacetimes, which can be quite difficult. On the other hand, string theory proposes that our universe contains extra dimensions. In this talk I will revisit the problem of calculating the vacuum polarization in a black hole spacetime, but for arbitrary dimensions.

090818141123

Thomas Harriot (1560-1621), the Elizabethan mathematician and scientist, was educated at Oxford, but worked outside the universities, mainly in the households of Sir Walter Raleigh and ‘The Wizard Earl’, Henry Percy, the ninth Earl of Northumberland. According to D.T. Whiteside, the editor of Newton’s mathematical papers, Harriot “possessed a depth and variety of technical expertise which gives him good title to have been England’s – Britain’s – greatest mathematical scientist before Newton”.

Among Harriot’s surviving papers are notes on the so-called maximal intercept problem: with A fixed on the lower circumference of a circle, to choose X on the circle so that the segment (YX) cut off by the horizontal diameter is largest.

Harriot knew that this occurs when YX is bisected by the vertical diameter, but it is not known how. I will discuss this question, and give an outline of Harriot's life.

090810144607

Various countries maintain germplasm collections, whose intent is to preserve living genetic material for the indefinite future. For example, the Ames Iowa collection preserves over 6,000 variants of maize, 1,700 variants of mustard, 172 variants of cucumbers, 130 variants of Echinacea, as well as many other species. These are preserved as dry seeds stored at controlled temperature and humidity. However, seeds die even in optimal storage conditions. So, curators irregularly test seed germination. Seed lots of maize are tested approximately every 8 years. When germination falls below a species-dependent threshold, e.g. 50% or 85%, the seed lot is regenerated by growing out plants and collecting their seeds. The current schedule of seed testing has two problems:
long-lived seed lots may be tested repeatedly until the threshold is reached
short-lived seed lots may not be tested until long after the threshold is reached.

We were asked if we could use available germination data to develop a better decision rule for when to test seeds and to detect threshold crossing. Doing this involved:
modeling expected seed germination over time (selecting an appropriate mean function)
choosing whether to pool data across variants (random effects or fixed effects)
choosing a reasonable error structure (linear mixed model or GLMM)
estimating quantiles of predicted distributions (bootstrap resampling for a LMM)
assessing alternative decision rules (receiver operating characteristic curves)

Each of these issues will be illustrated and discussed.

090727111907

Mathematica is the main mathematical package used at Otago. One of our graduates, Jason, is in the core development team and is visiting as part of an Australasian tour. Mathematica has mathematical expertise and can do mathematical as well as numerical computation. Mathematica poineered the natural document interface that is becoming the norm in scientific software. The documents can be manipulated by Mathematica itself and this has been extended to graphics and interactive interfaces. This gives it a unique capacity to communicate scientific concepts. There are now over 5000 demonstrations posted on the web site. Large systematic databases of scientific information such as DNA, chemical, financial and geographic are built in. Jason's talk will cover the recent developments of Mathematica and demonstrate its main features. Participants will come away with a comprehensive understanding of Mathematica’s key capabilities and core design principles.

090722114722

In this talk I want to present the work I am doing in my PhD thesis. My work is inspired by the problems which are still present in the numerical simulations of general relativistic systems. First I will give a motivation for my work, followed by a description of the two numerical methods to improve the accuracy of the numerical results. These methods are spatially discrete action functionals and a symplectic integrator for the time evolution. Later I introduce so-called Wave Maps and the Bianchi IX system, which I use for testing the new approaches. Wave Maps are geometric wave equations and the Bianchi IX
system is a cosmological model for a spatially homogeneous universe. At the end of the talk I will give an outlook for some future tasks.

090720155110

Exact determination of the vibration of a composite structure becomes impossible beyond the low-frequency range due to uncertainties in the structure. Therefore, the prediction model in the mid- to high-frequency range must include the effects of the irregularities. I will show how the power spectral density of an irregularity in a component can be included in the model. A lightweight floor/ceiling system is used as an example. Effects of twisting joists or varying stiffness of the floor show up as a simple summation in the original forward operator. Understanding random irregularities in composite structures have become important, because any manufacturing process introduces randomness, and traditional modelling techniques such as the finite element method or statistical energy analysis have not been very successful.

090720154222

Every zeta function counts something. The most fundamental question in arithmetic is: How many primes 2,3,5,7, . . ., or how many points with coordinates in the algebraic closure of a finite field of a given set of polynomial equations with coefficient in this field do we have up to a given bound? I will explain how zeta functions answer these questions.

090511104424

There is a diverse range of interesting mathematical questions that arise in conservation biology, including many of a combinatorial and probabilistic nature. Phylogenetic diversity is a prominent notion for measuring the biodiversity of a collection of species. This talk will present a few of the challenges that we have been facing when studying future phylogenetic diversity using extinction models and combinatorial optimization methods.

090515140559

Human lifetimes have an increasing failure rate, or force of mortality; as people age they are more likely to die. In contrast, exponentially distributed variables have constant failure rates and a characteristic lack-of-memory property: for exponentially distributed inter-event times, new events occur purely at random, unaffected by elapsed time. Here we consider the durations of rule of Chinese emperors, or the ‘Sons of Heaven’. We expected their reigns would cease due to accumulated social, political or economic tensions and thus have increasing failure rates. Using a new approach we show this is not the case. In fact we demonstrate the durations of rule were exponential. Moreover we find that a constant-parameter exponential distribution characterizes those reign lengths for over two millennia, from 221 BC to 1911 AD. So there was a constant probability distribution of Chinese emperors’ reigns ceasing over that entire period, and such cessations occurred unexpectedly, with no predictable historical influence. We estimate the “half-life” of Chinese emperors’ reigns to be 10 years. Hence even stable societies, such as Imperial China, do not imply stability of rule. Certainly the reasons for each specific change of Chinese emperor have been documented. Yet we demonstrate that overall, reigns ceased in a time-homogeneous manner without predictable accumulation of stress or damage. We explain such instability of rule by viewing the stresses upon the position of emperor as a stochastic process. We assume any emperor was replaced only when stress on the position exceeded some high threshold. For a stationary stochastic process such exceedances are known to form a Poisson process, which implies an exponential distribution for the time between successive emperors. Thus the probabilistic behaviour of the succession of Chinese emperors is the same as that of, e.g., the emission of alpha particles, first demonstrated by Rutherford a century ago.

090507162752

Swirling sweepers give computer graphics artists a natural tool for defining and controlling deformations of complex shapes. Being derived from diffeomorphisms they are potentially able to avoid the topological degeneracies that confuse users of other deformation tools. The practical implementation uses a hybrid discretisation and we will describe pictorially the simple methods that establish bounds on mesh and stepsize in order to preserve the topology.

090508160701

Accurate theoretical predictions of the gravitational-wave (GW)
signals produced by the merger of two black holes may be
central to the first direct detection of GWs; they will certainly
be necessary in many cases to accurately estimate the parameters
of GW sources. The GWs predicted by Einstein's equations for
black-hole mergers can only be calculated by computer simulation.

Performing such simulations raises issues for mathematical
relativity (for example the construction of initial data),
numerical analysis (reformulating Einstein's equations in a
numerically stable form), applied mathematics (writing accurate
and efficient 3D simulation codes) and data analysis (once we have theoretical waveforms to use in real detector searches). I will review recent progress in each area of this rapidly advancing
field.

090504104525

The numerical solution of the Einstein equations for general relativity has recently shown a great deal of promise in handling a number of difficult physical and mathematical problems. The principle application of these techniques has been in the study of dynamical spacetimes involving binary black holes, where accurate models of the spacetime evolution are required to inform the detector community. Beyond the astrophysical interest, however, the mathematical and numerical techniques required to accurately evolve spacetimes are of interest in themselves. The Einstein system, essentially a non-linear wave equation, must be formulated as a well-posed initial-boundary value problem with appropriate coordinate conditions for black hole spacetimes. The non-linearities make it particularly difficult to formulate appropriate boundary conditions, and complicate the measurement of physical observables. A promising avenue is to take advantage of conformal compactifications of the spacetime, where there is a well developed formalism for treating variables at null-infinity. This talk will discuss these issues and present results from current numerical implementations which aim to provide invariant measures of gravitational waves in a dynamical spacetime."

090504104330

Singular structures in solutions of Einstein's field equations have always attracted great interest, from the earliest discoveries of black hole singularities in the Schwarzschild spacetime, the big bang singularities in Friedmann-Robertson-Walker solutions, to much more severe phenomena in the last decades such as spiky features in Gowdy spacetimes. One has obtained a certain phenomenological picture of the properties of generic singularities, summarized by the so-called cosmic censorship and BKL conjectures. But it is still fair to say that we do not at all have a complete understanding of all possible complexities. It should be noted that not only the geometry itself, but also choices of coordinates or other gauge freedoms, deliberately or not, can lead to singularities in the solutions, which are, however, often interpreted as unphysical. Nevertheless, the exploitation of the gauge freedom in such a way can sometimes be a trick to make a problem more tractable, even though singularities are introduced. Examples are the usage of symmetry adapted coordinates, which often get singular at symmetry axes, or compactifications of spacetimes yielding singularities at "infinity". Under suitable assumptions, the latter singularities can be removed by means of conformal rescalings leading to Friedrich's conformal field equations. In many other cases, with both "physical" or "unphysical" singularities, the field equations reduce to the so-called Fuchsian form. Under certain assumptions, such equations allow a well-posed initial value problem with "data" on the "singularity". In my talk, after having introduced the background, I will explain the analytical results underlying the well-known theory of first order Fuchsian systems, and some applications in general relativity together with their heuristics and numerical insights. Then I will elaborate on new understandings of second order Fuchsian systems which are more adapted to Einstein's field equations. This is work in progress by LeFloch and myself. With the applications in general relativity in mind, I explain the new analytical theory and first numerical experiments. If there is still time, I will moreover discuss my results obtained earlier by means of alternative techniques to such singular initial value problems, in particular conformal regularizations underlying the conformal field equations mentioned above.

090504105326

According to general relativity, the fabric of spacetime near a
singularity generally behaves violently, stretching one way and then the other from one moment to the next. The dynamics is called "Mixmaster" after a brand of food mixer. It was thought that the dynamics is nonetheless mildly inhomogeneous. i.e. spatial derivative terms are much smaller than time derivative terms. Recent evidence of formation of spiky structures suggests otherwise. Exact solutions for the spikes have been found by applying a solution-generating transformation to known solutions.

Numerical simulations indicate that general solutions converge to the spike solutions near a singularity, and that the spikes recur.

090504104214

Ocean waves travel under sea-ice as ice-coupled waves, the motion of which is governed by coupled differential equations for the water and ice domains. These waves are scattered by changes in ice thickness and over long distances this induces approximately exponential decay, but at these lengths viscosity is also important. I'll describe how scattering by long transects of viscoelastic ice may be modelled and present some results for wave decay under Arctic sea-ice.

090427105059

The distribution of the position of a random walker making independent, identically distributed random movements is given in the scaling limit by a stable distribution. If the jump distribution is not power-law (heavy-tailed) then the stable distribution is the Gaussian. However, for jump distributions that are heavy tailed over several orders of magnitude the random walker is very well modelled by a non-Gaussian stable distribution (until the median gets close to the end of the power-law, then it slowly changes to Gaussian). We investigate the behaviour (governing equations, solutions, particle tracking solutions, etc.) of a random walker whose heavy-tailed jump distribution has been exponentially tilted; i.e. the jump distribution looks like a power-law that eventually turns into an exponential.

090420100712

Black hole solutions of the Einstein equations of general relativity have been studied for over 90 years. Traditionally, the simplest types of black hole solutions have been studied, but over the past 20 years there has been an explosion of interest in more complicated black holes which arise when the Einstein equations are coupled to different types of matter field. These more complicated black holes are known as “hairy” black holes. In this talk we describe some black hole solutions of the Einstein equation with a particular type of matter (a Yang-Mills gauge field), in which the black hole solutions can have unlimited amounts of “hair”, which we call “furry” black holes.

090326105352

In this talk we will give an overview of various aspects of cryptography and cryptanalysis, from the historical (Caesar cipher, enigma machine) up to modern day techniques (block ciphers, substitution-permutation networks).

We will then discuss some applications of group theory to both public and private key cryptography. Specifically, we will look at some elegant examples of how groups and their automorphisms can be used to encrypt and decrypt data.

No prior knowledge of group theory or cryptography will be assumed.

090318152425

Although high up on engineers' agenda are surfaces capable of holding an air layer, these are barely understood from a more scientific point of view. Air water interfaces can be described as constant mean curvature (cmc) surfaces. But boundary value problems with respect to cmc surfaces – as needed for investigations on air water air interfaces – are a rather exotic field of research in mathematics, too. This talk aims at giving a basic introduction to this interesting topic.

090311092042

Two projects involving undergraduates and collaborators are described briefly. The projects come from two different approaches to statistics education. On campus, an interdisciplinary research team modeled challenging data provided by a linguist to assess the effect of predictors on quantifying accents. This project occurred under the auspices of our Center for Interdisciplinary Research (CIR). The CIR has been supported for 5 years with a $1.3 million grant from the National Science Foundation (NSF). Over 30 collaborative projects have been completed over this time. The CIR has attracted a large number of students to the study of statistics and led to 40 students attending graduate study in statistics or a closely related field from a school of 3000 students over the past 4 years. Many of our statistics students are also involved in a second relatively unique program which provides students the opportunity to collaborate with World Health Organization (WHO) researchers in Geneve, Switzerland. A primary charge of WHO is to evaluate the global burden of disease (GBD) for an extensive list of diseases. Students on this project examined the effect of model choices on estimating GBD with some surprising results.

090305145952

In this talk we will give an overview of various aspects of cryptography and cryptanalysis, from the historical (Caesar cipher, enigma machine) up to modern day techniques (block ciphers, substitution-permutation networks).

We will then discuss some applications of group theory to both public and private key cryptography. Specifically, we will look at some elegant examples of how groups and their automorphisms can be used to encrypt and decrypt data.

No prior knowledge of group theory or cryptography will be assumed.

090309101133

This talk will assume no prior knowledge of quantum computing and concentrate on the mathematics of quantum computing. We will cover the basic ideas about qubits, the no cloning theorem and then two applications: teleportation and Deutsch’s algorithm (and its significance in attacking the RSA code).

090227155344

Significant wave height, Hs, is a measure of the variability of the ocean surface and is defined to be four times the standard deviation of the height of the ocean surface. Estimates of Hs can be considered as a two dimensional random field that develops over time. We propose a method of constructing models for estimates of Hs based on fitting random field models. The proposed model is parametric and the spatial parameters are estimated applying a new methodology based on the total variation on the Hs estimates from the TOPEX-Poseidon satellite.

For the temporal correlation of the field we assume a parametric covariance function, whose parameters are related to those of the spatial correlation through the velocity with which the field is drifting. The spatial and temporal models are then combined to give a stationary spatio-temporal model that is valid over small areas and for short periods of time.

090219112830

While Graph Theory draws from many areas and has become increasingly sophisticated, there are still serious and important problems that can be readily explained to the uninitiated. The solutions to these problems are often far from trivial and remain elusive for considerable periods. Still, with innovative application of elementary techniques progress can be made. In this talk we shall have a closer look at one such problem.

Let G be a connected graph with p vertices and q edges and define the parameter r = q  ¬p + 1. Denote by (r) the maximum number of cycles in such a graph. In 1981 it was noted by Entringer and Slater that the dimension of the cycle space of such a graph is known to be 2r and consequently, (r) ≤ 2r  1. In the same paper, the Möbius ladders were used to show (r) ≥ 2r  1 + r2  3r + 3. At the time it was conjectured that the true value of (r) should be closer to the latter bound. We discuss these bounds and some recent progress in the general case along with an effective resolution of the conjecture for planar graphs.

(Joint work with Carsten Thomassen)

090121122302

The Macdonald polynomials are an important family of orthogonal symmetric functions which generalize several classical families of symmetric functions including the Schur functions, the Jack symmetric functions and the Hall-Littlewood polynomials.

Although the Macdonald polynomials satisfy many nice properties, an explicit analytic formula for them remained an open problem ever since their introduction by Ian G. Macdonald in the 1980’s. In joint work with Michel Lassalle we succeeded in inverting the Macdonald Pieri formula and were thus able to obtain an explicit expansion of Macdonald polynomials in terms of modified complete symmetric functions or, by duality, in terms of elementary symmetric functions. Specialization yields similar expansions for monomial, Jack and Hall-Littlewood symmetric functions.

Our result can also be used to give an explicit (albeit complicated) formula for the q,t-Littlewood-Richardson coefficients.

090122145801

There are many mathematical formulations of dynamics. I will explain how the dynamical systems studied by operator algebraists arise from a more classical notion of dynamical system as a differential equation. The result will be a continuous action of a group G on a space X, and hence an action of G on the algebra C0(X) of continuous functions on X vanishing at infinity. I will survey the interplay of the dynamics of the pair (G, X) and the representation theory of an operator algebra built from the action of G on C0(X).

090128140537

Pattern classes will be introduced, with particular emphasis on those that can be produced by simple abstract permuting machines, or data types. Even quite simple machines can produce very complex classes, whose enumeration and structure is not at all clear. In particular the case of two stacks operating in parallel will be discussed, illustrating connections with formal languages, planar maps, functional equations, and a little bit of complex analysis.

090121122440

Let G be a graph with at least 2n+2 vertices, where n is a non-negative integer. The graph G is said to be n-extendable if every matching of size n in G extends to (i.e., is a subset of) a perfect matching. The study of this concept began in earnest in the 1980’s although it was born out of the study of canonical matching decompositions carried out in the 1970’s and before. As is often the case, in retrospect it is apparent that there are roots of this topic to be found even earlier. In this talk, we will begin with a brief history of the subject and then concentrate on reviewing results on n-extendability and closely related areas obtained in the last ten - fifteen years. These areas include “restricted” matching extension and the property E(m, n), matching extension in graphs embedded in topological surfaces, the study of bricks and braces and recent progress on Pfaffians and the enumeration of perfect matchings.

081020084228

GnRH neurons are key elements of the reproductive neuroendrokine system and play important central regulating role in the dynamics of the hormonal cycle. The aim of the study was to develop a simple conductance-based Hodgkin-Huxley type dynamic model of the ionic channels in GnRH neurons that can later be included in a hierarchical overall dynamic model of GnRH pulse generator. The model parameters were estimated using whole cell patch –clamp recordings, both voltage and current clamps. The experiments were performed in the Institute of Experimental Medicine of the Hungarian Academmy of Sciences.
The developed model includes a Na+, three types of K+, and two types of Ca+ channel together with a static leakage current that form parallel conductances. The proposed model is highly nonlinear in its parameters, and contains a very large number of parameters (over 40). Therefore, a special iterative method is proposed for parameter estimation that utilizes the characteristic parts of the patch –clamp recordings sensitive to certain parameters to decompose the estimation problem.
The calibrated model is able to reproduce well the measured data but in the case of the current parameter set it is not sensitive enough to the parameters of the Ca+ channels that links the membrane Ca+ conductances to intracellular Ca+ dynamics, which is then controls the hormone secretion. Moreover, the Hodgkin-Huxley type models of the ionic channels are not suitably parametrized for parameter estimation. In the case when only voltage clamp and current clamp data without the application of specific channel blockers are available, the parameters are weakly identifiable. Therefore, further work will be directed towards model simplification and finding alternative, better parametrized model forms for the remaining ionic channels.

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The standard formulation of quantum mechanics associates to each system a complex wavefunction in Hilbert space. These wavefunctions allow us to predict the probability of the system being in a certain state. We associate experimentally measurable quantities with operators acting on these wave functions.

This leads to several questions. What are the possible wavefunctions? What are the operators and how are they connected to experimental measurements? What are the formulations of quantum mechanics and what do they tell us?

I will attempt to answer these questions by looking at the mathematical formalism of quantum mechanics. I will start with the modern theory of integration and then some general properties of operators. Next I will cover some of the formulations of quantum mechanics. Finally I will give a quick introduction to scattering.

081001104835

Significant wave scattering occurs in the zones of relatively thin sea-ice that surround the Polar Regions, and the coupled motion of the water and ice presents a mathematically challenging problem.

Consequently, many unphysical assumptions about the ice-cover are made in order to facilitate solution.

I will outline a method that allows us to produce results using a more realistic model in which the ice may vary in thickness and is partially submerged.

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