Institution: The Australian National University
Dr. Benjamin Andrews
|
2000 |
2001 |
2002 |
| Funding |
$58,000 |
$60,000 |
$63,000 |
Title: Fully nonlinear geometric evolution equations
and global differential geometry.
Summary:
This project aims to apply fully nonlinear parabolic equations to obtain
new results in the global differential geometry of hypersurfaces and Riemannian
metrics. This work will exploit the observation that fully nonlinear equations
can allow more precisely controlled behaviour than is possible in other,
apparently similar equations, at the expense of only slightly greater analytical
difficulty. The research is expected to yield best-possible results describing
the global implications of local curvature conditions in a variety of situations,
as well as advancing understanding of the regularity and asymptotic behaviour of
important classes of fully nonlinear equations.
Institution: Queensland University of Technology
A/Prof. Vo Anh, Queensland University of Technology
Prof. Chris Heyde, The Australian National University
|
2000 |
2001 |
2002 |
| Funding |
$51,000 |
$51,000 |
$51,000 |
Title: Stochastic analysis of long-range dependent multifractals.
Summary:
With greater computing power and better measurements, recent studies
have found that data in diverse fields display long-range dependence and
multifractality. Existing methods based on the traditional Markov concept cannot
cope with these important phenomena. This project will develop new models and
tools to handle long-range dependent multifractals. These include stochastic
partial differential equations with fractional diffusion and multifractal input.
Asymptotics for rescaled solutions of these equations will be provided for
parameter estimation, numerical solution and simulation of the models. The
methodology will be applied to analyse and predict some major data series in
finance, air pollution and hydrology.
Institution: The University of Adelaide
Dr. Pier Bouwknegt
|
2000 |
2001 |
2002 |
| Funding |
$76,000 |
$75,000 |
$70,000 |
Title: Mathematical and physical aspects of quasi-particle
excitations in quantum many body systems.
Summary:
Quantum many body systems in low dimensions exhibit curious phenomena
such as the existence of collective excitations carrying quantum numbers which
are fractions of the quantum numbers carried by the microscopic degrees of
freedom in the system. An essential aspect of these so-called quasi-particles, is
that their statistics will be equally unusual. This project aims to study both
the mathematical structure behind these quasi-particles as well as physical
applications. Expected outcomes are a better understanding of the possible forms
of (exclusion) statistics, of two dimensional conformal field theories and of the
representation theory of quantum groups.
Institution: The University of Sydney
Dr. John Cannon
|
2000 |
2001 |
2002 |
| Funding |
$61,000 |
$61,000 |
$66,000 |
Title: A cohomological approach to
computing the fundamental invariants of a
finite group.
Summary:
The notion of symmetry plays a fundamental role both in mathematical and
physical theories. The natural symmetries of an object (e.g. molecule,
differential equation) will often lead to a considerable simplification.
Mathematically, the idea of symmetry is captured in the notion of a group. The
goal of this work is to find efficient means of determining the basic properties
of a finite group. Such properties include its subgroups and its possible
realisations in terms of matrices. This capability then enables us to exploit
knowledge of the symmetries in the analysis of a particular situation.
Institution: The University of New South Wales
Prof. Michael Cowling and Dr. James Wright
|
2000 |
2001 |
2002 |
| Funding |
$61,000 |
$61,000 |
$71,000 |
Title: Analysis and Geometry of Radon-type Operators.
Summary:
Certain mathematical problems involve a space filled with
lower-dimensional spaces, like sheets of paper rolled in a pad or wrapped in a
roll; it is easier to deal with the flat pad than with the curved roll. This
project aims to extend mathematical tools which are useful for dealing with the
flat case to the curved case. These tools are analytic in nature, but the
geometry of the lower-dimensional spaces plays an essential role in understanding
how to extend them.
Institution: The University of Sydney
Prof. Edward Dancer
|
2000 |
2001 |
2002 |
| Funding |
$51,000 |
$51,000 |
$52,000 |
Title: Bifurcations and secondary
bifurcations for ordinary and partial
differential equations.
Summary:
The applicant and coworkers have developed new techniques to study
nonlinear equations and used them to solve an old problem on water waves. We want
to develop these rather promising techniques much further and apply them to a
variety of problems, coming from a number of different applications (including
combustion theory, catalysis and water wave theory).
Institution: University of Tasmania
Prof. Robert Delbourgo, University of Tasmania
Dr. Habil Dirk Kreimer and Dr. Andrei Davydychev,
University of Mainz
|
2000 |
2001 |
2002 |
| Funding |
$51,000 |
$51,000 |
$56,000 |
Title: Geometrical and Topological Properties of
Multiloop Feynman Diagrams.
Summary:
Mathematically and physically, Feynman diagrams encapsulate quantum
mechanical processes arising in standard models of elementary particles, eg
electroweak theory and chromodynamics. Computational methods (numerical and
algebraic) of evaluating these diagrams are well-developed but lead to few
significant insights. We aim to obtain a geometrical interpretation of such
computations beyond one-loop level (at one-loop level the answer signifies a
solid angle) with connections to dual diagrams, their topology and knot theory.
By associating algebraic singularities with geometrical degeneracies we will be
able to elucidate some mysteries of the brute-force algebraic approach and
provide an alternative geometrical evaluation procedure.
Institution: The University of Adelaide
Prof. Michael Eastwood, The University of Adelaide
Prof. Charles Graham, The University of Washington
Dr Toby Bailey, The University of Edinburgh
Dr. Vladimir Ezhov, The University of Adelaide
|
2000 |
2001 |
2002 |
| Funding |
$20,000 |
$20,000 |
$20,000 |
Title: Symmetry and Analysis in Differential Geometry.
Summary:
Differential geometry is the study of shape using the calculus. This is
a fundamental research project in this area. In particular, it will investigate
geometries which are especially symmetric. Various classification problems will
be considered. Symmetry methods will be used to study integral geometric
transforms. Conversely, the integral geometry will illuminate the representation
theory for the symmetries. The analysis of differential equations (those defined
by involutive structures) and non-linear (the Einstein-Weyl equations) is a
further aim. The various aspects of the project are linked by the symmetries that
pervade both mathematics and physics.
Institution: University of South Australia
Prof. Jerzy Filar, A/Prof Vladimir Gaitsgory and
A/Prof Philip Howlett
|
2000 |
2001 |
2002 |
| Funding |
$51,000 |
$51,000 |
$51,000 |
Title: Asymptotic Analysis of
Singularly Perturbed Mathematical Programs.
Summary:
We propose a unified description of the asymptotic behaviour of
solutions to perturbed mathematical programs as the perturbation parameter
approaches zero. For problems defined by analytic functions we show solutions can
be written as Puiseux series in the perturbation parameter. We pay particular
attention to singular perturbations where the asymptotics are not correctly
represented by simply setting the perturbation parameter to zero. An important
theme is construction of a problem that describes the asymptotics and is
independent of the perturbation parameter. Knowledge of this true limit problem
is especially important for applications where the perturbation parameter is not
precisely known.
Institution: The University of Melbourne
Dr. Omar Foda
|
2000 |
2001 |
2002 |
| Funding |
$65,000 |
$65,000 |
$65,000 |
Title: Recursive Combinatorial Transforms in
Solvable Lattice Models.
Summary:
Computing one-point functions, in solvable lattice models, can be
reduced to enumerating, or q-counting, weighted combinatorial objects that
satisfy prescribed conditions. Examples of these objects are partitions,
tableaux, sequences and paths. Following Andrews, Bressoud, and Burge, the
applicant and collaborators developed and used Bailey-type combinatorial
transforms that recursively reduce complex q-counting problems to elementary
ones. We plan to systematically generalise this approach, and to apply it to
enumeration problems that are motivated by the representation theory of affine
and Virasoro algebras, and the related solvable lattice models.
Institution: The University of Queensland
Dr. Mark Gould and Prof. Anthony Bracken,
The University of Queensland
Prof. Vladimir Korepin , SUNY at Stony Brook
|
2000 |
2001 |
2002 |
| Funding |
$56,000 |
$53,000 |
$53,000 |
Title: Quantised Algebraic Structures and
New Supersymmetric Integrable Models.
Summary:
Integrable models are the subject of high-profile research activity
worldwide. They involve powerful new mathematics, in particular quantised
algebraic structures, and provide important insights into the behaviour of
complex physical systems. The project concerns the development and analysis of
new supersymmetric integrable models of correlated fermions, an area of intense
international research activity because of its relevance to high-Tc
superconductivity and related phenomena. We have already made significant
contributions to this rapidly expanding field and seek a full time researcher to
capitalise on this promising start.
Institution: The University of Melbourne
Prof. Anthony Guttmann
|
2000 |
2001 |
2002 |
| Funding |
$95,000 |
$103,000 |
$103,000 |
Title: Lattice problems in statistical mechanics and combinatorics.
Summary:
Lattice problems have, for some time, been one of the most actively
studied areas of statistical mechanics and combinatorics. Many seemingly simple
problems, such as counting the number of two-dimensional self-avoiding walks,
have proved intractable. Yet other problems have been comparatively
straightforward. We have recently devised an innovative method that allows one to
predict, whether a given lattice based model is going to be "hard" or "easy." My
purpose here is to develop and extend the method so that it becomes a standard
tool for workers in the field, and one that can be used to obtain insight and
solutions to previously unsolved problems.
Institution: The University of Melbourne
Dr. Craig Hodgson and Prof. Walter Neumann
|
2000 |
2001 |
2002 |
| Funding |
$51,000 |
$51,000 |
$51,000 |
Title: Hyperbolic 3-manifolds.
Summary:
Three dimensional manifolds are of central importance in topology,
algebra, and cosmology (providing models for the universe). Thurston's
Geometrisation Conjecture gives a beautiful conjectural picture of 3-manifolds in
terms of eight uniform geometries. Seven of these lead to well understood
situations. This project studies 3-manifolds modelled on the eighth geometry:
hyperbolic geometry. The detailed understanding of hyperbolic 3-manifolds is a
most fertile research area with many long standing open questions. This project
is aimed at making advances on fundamental questions in the following areas:
Deformation spaces, Representation spaces, Geometric invariants, Geodesics in
3-manifolds, Computation and Experiment.
Institution: Macquarie University
Prof. Sergei Konyagin
|
2000 |
2001 |
2002 |
| Funding |
$20,000 |
$20,000 |
$20,000 |
Title: Trigonometric series and exponential sums,
and their application to the problems of analytic number theory.
Summary:
A natural way to represent a function is as a trigonometric series. But
whether that series represents the function adequately depends on its
convergence, and divergence, properties. The project will settle deep questions
on the adequacy of these representations. To study the distribution of prime
numbers researchers have developed what might be called mathematical lenses to
allow them to bring into focus certain aspects of prime numbers. This project
will build such lenses. Bounds for exponential and character sums find
application in coding theory, encryption, and construction of pseudo-random
number generators. Improved estimates tailored to applications will be provided.
Institution: The University of Melbourne
A/Prof. Vikram Krishnamurthy
|
2000 |
2001 |
2002 |
| Funding |
$80,000 |
$78,000 |
$80,000 |
Title: Estimation of nonlinear stochastic dynamical systems.
Summary:
Nonlinear stochastic dynamical models are fundamental to most branches
of control, signal processing and telecommunications. This project will focus on
the design, analysis and implementation of three newly emerging paradigms for
estimating the state of such systems when the state is observed in noise. These
paradigms include stochastic sampling methods which have recently revolutionised
the field of applied statistics; iterative state sequence estimators and adaptive
filtering. We will focus on applying the resulting general methods to two
important applications (i) tracking the coordinates of manoeuvring targets given
noisy observations (ii) Interference suppression in telecommunication systems.
Institution: The University of Queensland
Prof. Geoffrey McLachlan and
A/Prof. Kaye Basford, The University of Queensland
A/Prof. Padhraic Smyth, University of California, Irvine
|
2000 |
2001 |
2002 |
| Funding |
$51,000 |
$56,000 |
$56,000 |
Title: Classification of Multiply Observed Features in
Terms of Fitted Densities.
Summary:
We aim to develop methodology for the classification of entities on
which multiple measurements are made. The problem is more complex than typical
classification since each entity must be classified on the basis of a density
rather than a single feature vector. Two parametric approaches (the estimative
and the predictive) and a novel nonparametric method are proposed and will be
investigated. They will be applied to three important applications that motivated
the research: 1) detection of iron-deficiency anaemia2) classification of PAP
smear slides in cervical cancer screening 3) identification of iron types in
mineral processing.
Institution: The University of Queensland
Prof. Geoffrey McLachlan and
Dr Deming Wang , The University of Queensland
A/Prof. Padhraic Smyth, University of California, Irvine
|
2000 |
2001 |
2002 |
| Funding |
$51,000 |
$51,000 |
$53,000 |
Title: On Algorithms for the Automatic Analysis and
Segmentation of Correlated Images.
Summary:
We shall develop and evaluate algorithms for the automatic statistical
analysis and segmentation of multidimensional magnetic resonance (MR) images,
which can be used in the diagnosis and treatment of neurogenerative diseases such
as Alzheimer's disease. Even with the recent progress in computer-based image
analysis techniques, automated methods for the segmentation of brain tissue are
still not available for clinical use. Although the algorithms will be developed
and studied in the context of MR images, they will have direct applications to
other multivariate images that are spatially correlated in two or three
dimensions.
Institution: The Flinders University of South Australia
Prof. William Moran
|
2000 |
2001 |
2002 |
| Funding |
$61,000 |
$61,000 |
$61,000 |
Title: Sampling and Interpolation of Signals with
Multi-Band Spectra.
Summary:
The project will solve difficult and important problems in the sampling
and interpolation of signals with multi-band spectra. The solution of such
problems is significant for the compression of signals and images for storage and
transmission. Our aim is to obtain solutions to such problems where the signal
spectrum is constrained to lie in a finite union of intervals. To do this we
shall develop new techniques in three other areas of mathematics: boundary
controllability of distributed parameter systems, Riesz bases of exponentials,
and Wiener-Hopf operators.
Institution: The University of Adelaide
A/Prof. Michael Murray, The University of Adelaide
Dr. Michael Singer, University of Edinburgh
|
2000 |
2001 |
2002 |
| Funding |
$59,000 |
$57,000 |
$62,000 |
Title: Bogomolny monopoles, nullarons and chiral Potts.
Summary:
The aim of this project is to unravel the relationship between Bogomolny
monopoles, solutions of a partial differential equation in three space, and the
solutions of the chiral Potts model in statistical mechanics. This project builds
on previous work with Sir Michael Atiyah FRS and Dr M.A. Singer. The outcome will
be advances in our understanding of the structure of the Bogomolny equations and
their relationship with integrable equations and statistical mechanics.
Institution: The University of Melbourne
A/Prof. Paul Pearce
|
2000 |
2001 |
2002 |
| Funding |
$55,000 |
$53,000 |
$57,000 |
Title: Integrable lattice boundary conditions,
fusion algebras and boundary conformal field theories.
Summary:
Over the last decade sophisticated mathematical techniques, based on the
boundary Yang-Baxter equation, have been developed to study the boundary
properties of integrable lattice statistical models in two dimensions. Recently,
it has been discovered that complete sets of conformal boundary conditions can be
constructed from integrable lattice boundary weights. Importantly, this extends
beyond the usual diagonal theories treated by Cardy and Verlinde and allows a
general and detailed mathematical study of boundary conformal field theories and
their fusion algebras. This will lead to many applications in statistical,
condensed matter and high energy physics.
Institution: La Trobe University
Dr. Reinout Quispel, La Trobe University
Dr. Robert McLachlan, Massey University
Dr. Arieh Iserles, University of Cambridge
|
2000 |
2001 |
2002 |
| Funding |
$56,000 |
$56,000 |
$61,000 |
Title: New Structures in Geometric Numerical Integration.
Summary:
Many scientific phenomena in physics, astronomy and chemistry, are
modelled by ordinary differential equations (ODEs). Often these equations have no
solution in closed form, and one relies on numerical integration. Traditionally
this is done using e.g. Runge-Kutta methods or linear multistep methods. In the
last decade, however, we (and others) have discovered novel classes of so-called
'geometric' numerical integration methods that preserve qualitative features of
certain ODE's exactly (in contrast to traditional methods), leading to crucial
stability improvements. This project will improve, expand and systemise this new
field of geometric integration.
Institution: Macquarie University
Dr. Igor Shparlinski
and
Dr. Bernard Mans
|
2000 |
2001 |
2002 |
| Funding |
$27,000 |
$27,000 |
$27,000 |
Title: Number theoretical methods in graph theory and networks
.
Summary:
The goal of this project is to apply Graph Theory, Number Theory and
Algebra to the theory of Networks. We aim to establish new links between these
areas and solve some well known open questions of Theoretical Computer Science.
In particular, the project will study Circulant Graphs. These apparently innocent
and simple graphs are related to surprisingly many deep questions of graph
theory, number theory and complexity theory.
Institution: The University of New South Wales
Prof. Ian Sloan,
Dr. William McLean and
Dr. Mahadevan Ganesh
|
2000 |
2001 |
2002 |
| Funding |
$70,000 |
$70,000 |
$94,000 |
Title: Numerical Analysis of Evolution Problems in Several Variables.
Summary:
Many phenomena in engineering and science, such as heat transfer,
contaminant diffusion and pattern formation, involve quantities that vary
continuously in space and time. This project will analyse numerical algorithms
used for performing computer simulations of such phenomena. We aim to improve the
efficiency of these algorithms, especially for challenging problems having more
than just a single space dimension. A major focus of the project is a novel time
discretisation that, by means of integral transforms, does away with the
sequential nature of traditional approaches, allowing programmers to exploit more
easily the full potential of parallel computer architectures.
Institution: Macquarie University
Prof. Ross Street
|
2000 |
2001 |
2002 |
| Funding |
$64,000 |
$64,000 |
$64,000 |
Title: Higher-dimensional functorial algebra.
Summary:
The idea to use a symbol to represent an unknown number was an
incredible enabling device for the advance of mathematics. Early in the 20th
Century, the use of arrows directed between mathematical objects had the same
kind of effect; it allowed precise formulations of difficult problems and aided
in organising their solution. Within mathematics and in related sciences, the
applications of structures involving arrows between arrows are being realised.
The process can continue. This is the subject of Higher Category Theory. The
Project asks eight deep questions on this subject and its applications, and
outlines proposals for answering them.
Institution: The Australian National University
Dr. John Urbas
|
2000 |
2001 |
2002 |
| Funding |
$20,000 |
$20,000 |
$20,000 |
Title: Nonlinear Partial Differential Equations.
Summary:
I intend to investigate two closely related classes of nonlinear partial
differential equations: Hessian and curvature equations. These have a wide range
of applications in differential geometry. Monge-Ampere equations are particularly
important subclass having close connections to mass transfer problems, which have
large range of applications in mathematics and in other fields. I will develop a
comprehensive theory of these equations and their applications, encompassing
boundary value problems, extremal solutions, mass transfer problems and curvature
flow problems.
Institution: The Australian National University
Prof. Alan Welsh
|
2000 |
2001 |
2002 |
| Funding |
$53,000 |
$53,000 |
$58,000 |
Title: Analysis and Modelling of Survey and
Other Data Containing Extra Zeros.
Summary:
We will develop and implement improved and more flexible methods for the
analysis of data (including survey data) which contains extra zeros, through the
use of semi-parametric models and models which incorporate dependence. The
methods will impact on the practical analysis of data, enabling us to concentrate
on the substantive problem of interest (rather than nuisance aspects of the
models) and to draw valid conclusions from data containing extra zeros. The
project will result in new statistical methods with associated software and
quality research training.
Institution: The University of New South Wales
Dr. Rodney Weber, Dr. Geoffrey Mercer and Dr. Harvinder Sidhu
|
2000 |
2001 |
2002 |
| Funding |
$51,000 |
$51,000 |
$56,000 |
Title: Unifying the effects of different
geometrical configurations on flammability limits.
Summary:
The determination of the limiting ratios of fuel to oxidant which are
required to sustain vigorous combustion for a single exothermic reaction in three
different, yet related, geometrical configurations is considered. A well stirred
flow system, a propagating flame and a spherical flame ball will each be
systematically studied to determine the limiting ratios. The relationship between
these ratios and the heat losses will be compared and quantified for each of the
configurations. These unified results will then enable us to develop a novel
method of determining flammability limits in different geometrical
configurations.
Institution: The University of Melbourne
Dr. Krzysztof Wysocki, The University of Melbourne
Prof. Helmut Hofer, New York University
Prof. Eduard Zehnder, ETH-Zurich
|
2000 |
2001 |
2002 |
| Funding |
$56,000 |
$56,000 |
$56,000 |
Title: Pseudoholomorphic curves and dynamics on three-manifolds.
Summary:
Each orientable three-manifold possesses a contact from which uniquely
defines the Reeb vector field. This class of vector fields includes, in
particular, Hamiltonian vector fields. The aim of the project is to understand
the dynamics of flows generated by Reeb vector fields. Our understanding is
achieved by constructing a global system of surfaces of section which allows us
to define an area-preserving return which mimics the Reeb dynamics. The
construction of a global system of surfaces of section is based on a novel
approach of pseudoholomorpic curves. A successful outcome of this project would
be a significant step in understanding structural aspects of Reeb flows as well
as Hamiltonian vector fields.
