Below is a sample of some of the PhD research projects offered by staff members in the School of Mathematics and Statistics (collected Oct 2020). Please note that this is not an exhaustive list of all potential projects and supervisors available in the School.
Information about PhD research offerings and potential supervisors can be found in various locations. It's worth browsing the Current Research Students list to see what research our PhD students are currently working on, and with whom.
There is also a Past Research Students list which provides links to the theses of former students and the names of their supervisors.
It's also recommended to browse our Staff Directory, where our staff members' names are linked to their research profiles which provide details about their areas of research and often include the topics they are open to supervising students in.
We host PhD information sessions in the School of Mathematics and Statistics twice a year. Keep an eye on our Events webpage for session information.
Real world problem solving using dynamical systems, stochastic modelling and queueing theory for stochastic transport and signalling in cells.
- Dynamical Systems and Ergodic Theory:
Projects that combine techniques from nonlinear dynamics, ergodic theory, functional analysis, or differential geometry and can range from pure mathematical theory through to numerical techniques and applications (including ocean/atmosphere/blood flow), depending on the student.
Projects are occasionally available in optimisation, mainly using either techniques from mixed integer programming to solve applied problems (e.g. transport, medicine,…) or mathematical problems arising from dynamics.
- Modelling and analysis of ocean biogeochemical cycles including isotope dynamics, inverse modelling of hydrographic data to detect climate-driven circulation changes, and analysis of large-scale ocean transport.
PhD students should be highly motivated, have a strong background in applied mathematics and/or theoretical physics, and will have the opportunity to contribute to shaping their project.
- Data-Driven Multi-stage Robust Optimization:
The aim of this study is to develop mathematical principles for multi-stage robust optimization problems, which can identify true optimal solutions and can readily be validated by common computer algorithms, to design associated data-driven numerical methods to locate these solutions and to provide an advanced optimization framework to solve a wide range of real-life optimization models of multi-stage technical decision-making under evolving uncertainty.
- Semi-algebraic Global Optimization:
The goal of this study is to examine classes of semi-algebraic global optimization problems, where the constraints are defined by polynomial equations and inequalities. These problems have numerous locally best solutions that are not globally best. We develop mathematical principles and numerical methods which can identify and locate the globally best solutions.
- Detection and cloaking of surface water waves created by submerged objects
- Decomposition of ocean currents into wave-like and eddy-like components
- Theory and application of Quasi-Monte Carlo methods:
for high dimensional integration, approximation, and related problems.
Quoc Thong Le Gia
- Computational Mathematics:
with specialised topics in radial basis functions, random fields, uncertainty quantification, partial differential equations on spheres and manifolds, stochastic partial differential equations.
- Discrete Integrable Systems:
These are birational maps with particularly ordered dynamics and their study is a nice motivation for using algebraic geometry, symmetry, ideal theory and number theory in the study of dynamical systems.
- Arithmetic Dynamics:
This field is the study of iterated rational maps over the integers or rationals or over finite fields, rather than the complex or real numbers. I am particularly interested in how the usual structures present in dynamical systems over the continuum manifest themselves over discrete spaces.
- Convex geometry:
Focused on the study of the facial structure of convex sets and the relations between the geometry of convex optimisation problems and performance of numerical methods. The project can be oriented towards convex algebraic geometry, experimental mathematics or classical convex analysis.
- Projection methods and fixed-point theory:
This project is associated with the ARC grant “Geometry in projection methods and fixed-point theory.” The objectives are to capture the geometry of feasible sets affecting the convergence of projection methods, accelerate the convergence of projection and hybrid optimisation-projection algorithms and solve specific problems related to fixed-point iterations of nonexpansive mappings.
Algebraic and Geometric Aspects of Integrable Systems:
The ubiquitous nature of integrable systems is reflected in their (apparent or disguised) presence in a wide range of areas in both mathematics and (mathematical) physics. Projects focus on the algebraic and/or geometric aspects of discrete and/or continuous integrable systems, depending on the individual student's background and preferences.
- Fluid flow in channels with porous walls
- Mathematics education
- Nonlinear differential equations
- Difference equations
- Dynamic equations on time scales
- How many oceans are there? Using novel statistical and machine learning techniques to characterise oceanic zones and provide a blueprint for quantifying the ocean's role in a changing climate.
- How does heat get into the ocean? An investigation of the physical mechanisms that control the ocean's uptake of heat and its effect on climate.
- Making climate models work better: Developing new methods to validate and improve the inner workings of numerical climate models and improve their projections of global warming and its impacts.
- Will it mix? New perspectives on turbulence in rotating fluid flows and how we estimate mixing from observations.
- Graph theory
- Coding theory
- Extremal set theory
- Operator algebras
- Mathematical physics
- Group theory
- Jones subfactor theory
- Quantum field theory
- von Neumann algebras
Connection re: Vaughan Jones between conformal field theory, Thompson groups and knot theory.
- Noncommutative algebra
- Algebraic geometry
- Random graphs
- Asymptotic enumeration
- Randomized combinatorial algorithms
Extremal and probabilistic combinatorics:
Possible subjects therein include Ramsey theory, random graphs, positional games and hypergraphs.
- Unlikely Intersection in Number Theory and Diophantine Geometry:
These are problems of showing that arithmetic “correlations" between specialisations of algebraic functions are rare unless there is some obvious reason why they happen. These “correlations” may refer to common values or to values factored into essentially the same set of prime ideals and similar.
- Arithmetic Dynamics:
This area is concerned with algebraic and arithmetic aspects of iterations of rational functions over domains of number theoretic interest.
- Counting integral and rational solutions to Diophantine equations and congruences.
The goal is to obtain upper bounds on the number of integer solutions to some multivariate equations and congruences in variables from a given interval [M, M+N]. Similarly, for rational solutions one restricts both numerators and denominators to certain intervals.
- Kloostermania: Kloosterman and Salie sums and their applications.
A classical direction in analytic number theory where the goal is to obtain new bounds on bilinear sums of Kloosterman and Salie sums and apply them to various arithmetic problems, such as the Dirichlet divisor problem in progressions.
- Curves and polynomials over finite fields.
Belonging to the area of arithmetic statistics, where one wants to get asymptotic formulas for some algebraic objects, such special curves and polynomials over finite fields, with desired properties.
- Non-commutative functional analysis and its applications to non-commutative geometry, particularly those related to quantised calculus and index theorems.
- Singular (Dixmier) traces and their applications
- Non-commutative integration theory
- Non-commutative probability theory
- Various aspects of Banach space geometry and its applications
- Motivic cohomology and algebraic K-theory - an intersection of algebraic geometry and algebraic topology
- Equivariant algebraic topology
- Extreme Value Analysis:
Projects available on the modelling of the dependence of multivariate and spatial extremes, spatio-temporal modelling, high-dimensional inference. Interests in environmental/climate applications.
- Symbolic Data Analysis:
Projects available on symbol design, distributional symbols and others. Applications in big and complex data analysis.
- Ancient river systems and landscape dynamics with Bayeslands framework
- Bayesian inference and machine learning for reef modelling
- Deep learning for the reconstruction of 3D ore-bodies for mineral exploration
- Bayesian deep learning for protein function detection
- COVID-19 modelling with deep learning
- Variational Bayes for surrogate assisted deep learning
- Bayesian deep learning for incomplete information
- Computational Statistics
- Event sequence data analysis
- Hidden Markov Models and State-Space Models and their inference and applications
- Financial data analysis and modelling
- Point processes and their inference and applications
- Semi- and non-parametric inference
- Nonparametric and semiparametric statistics:
Nonparametric dependence modelling (copulas) and nonparametric functional data analysis.
- Bayesian extension of the Black-Litterman model through copulas, for portfolio optimisation
- Nonparametric spatio-temporal clustering via Dirichlet processes
- Copula factor models
- Bayesian spatial clustering with application in animal behaviour and ecology
- Approximate Bayesian computation: choice of summary statistics, approximation of the likelihood, estimation of long-memory processes, problems of model-choice
- Analysis of high-dimensional genomic datasets: analysis of gene-expression data, study of the direction of the expression, single-cell technologies, GWAS.
- For some examples of my current projects, have a look at my page: https://research.unsw.edu.au/people/dr-clara-grazian
- Social network analysis
- Statistical models for dependent categorical data
- Survey sampling (design and inference), particularly for network data
- Statistical computing, particularly MCMC-based methods
- The development and application of burden of disease methods, especially population attributable fraction.
- Topics include: modelling continuous exposures, repeated measurements, measurement error, competing risks, different study designs, meta-analysis and pooled analysis, future burden projections, comparison of different methods, novel applications.
- Dependence measures
- Complex-valued random variables
- Goodness-of-fit tests
- Machine learning (with potential applications in neuroimaging)
- Time series analysis
- The development of statistical methods for epidemiologic research:
Topics include regression to the mean, interrupted time series, meta-analysis, and population attributable fractions.
- Analysis of capture-recapture data
- Estimation of animal abundance
- Measurement error modelling
- Model selection for multivariate data
- Statistical ecology
- High-dimensional data analysis
- Computational statistics
- Simulation-based inference
- Eco-Stats project ideas