Title of ASC |
Brief description |
Suitable for |
Contact |
| Introduction to Group theory |
Investigate elementary properties of groups, classify abelian groups,
and investigate some applications (such as finite reflection groups and symmetries of Euclidean spaces). |
First year students taking math1115/6, second year |
Bryan Wang |
| Hyperbolic geometry |
Study one of the classic non-Euclidean geometries, two-dimensional hyperbolic space, using a simple model. |
First year with math1116, or second year |
Andrew Hassell |
| Construction of optimal experimental designs |
Experimental designs are used in a wide range of practical situations, for example from field trials through to bridge tournaments. The construction of optimal experimental designs has a long history in combinatorics. More recently progress has been made on unsolved problems with a combination of theory and computing. Balanced Howell designs are used in bridge tournaments and there are a number of optimal designs waiting to be found or disproved. This project is to look at methods for the construction of optimal experimental designs and investigate approaches for unsolved problems. |
First year students taking math 1115/6 |
Emlyn Williams |
| Penrose tilings |
Investigate the basic properties of aperiodic tilings of the plane. |
Could be done by first year students |
Adam Rennie |
| The Bootstrap |
The bootstrap is an important statistical tool which can be viewed as a simulation-based method of constructing statistical inferences. It provides a simple, practical way of making inferences and also throws light on the ideas of statistical inference. Several different projects can be constructed in this area, including applying the bootstrap in different problems and the empirical evaluation of bootstrap methods.
|
First year students doing math 1115/1116 |
Alan Welsh |
| Smoothing |
Smoothing is a basic technique in Statistics which is used to remove variation and noise to show the broad overall structure. There are a variety of ways of approaching smoothing. Projects can involve applying and exploring (theoretically or empirically) one or more of the approaches.
|
First year students doing math 1115/1116 |
Alan Welsh |
| Smoothing and rendering |
Smoothing has a wide variety of applications including medical imaging, data mining and 3D rendering (also see the Advanced Studies Course offered by Allen Welsh). Finding the smoother is computationally expensive, particularly when dealing with large data sets, and applications such as medical imaging require real time results. The project looks at several issues related to the difficulties of calculating the smoother, including efficient linear solvers, different discretisation techniques and high dimensional data sets. Some programming experience is beneficial. |
Prerequisite: math 1116 |
Linda Stals |
| Nash's papers on game theory |
Learn some basic game theory and read some of Nobel prizewinner
John Nash's classic papers on the subject. |
Second year, with or after math2320 |
Andrew Hassell |
| Metric spaces and operators |
This project will study metrics on spaces defined using commutators with
a distinguished `Dirac' operator. For the simplest finite spaces, this project
requires only a knowledge of matrix algebra. Since the mathematics gets more
complex and sophisticated as the space becomes more complicated, this project can be undertaken in any year. |
Any year |
Adam Rennie |
| Fermat's Last Theorem |
Higher arithmetic. The failure of unique factorization in generalized number systems. Integer and rational solutions to algebraic equations. Fermat's Last Theorem. |
Add on to math 2322 or math3345, or a stand-alone ASC |
James Borger |
| Crystallographic groups |
Investigate the relation between group theory and repeating "crystal" patterns in 1, 2, and 3 dimensions |
Add-on to math2322 |
James Borger |
| Spherical data |
Spherical data consist of observations on the surface of a hypersphere which can be treated as unit vectors in space. Thus the sample space is a circle in 2-dimensions, a sphere in 3-dimensions and special methods which take into account the structure of the sample space are needed to model the data. This course is based on topics in modelling spherical data. A good recent reference to the topic is Mardia and Jupp (2000, Directional Statistics).
|
Prerequisite: Stat2001 |
Alan Welsh |
| Topics in Cox Proportional Hazards Model |
Since its introduction in 1972, the Cox proportional hazards model has become the workhorse model for the analysis of censored survival data. This course is based on topics around the Cox proportional hazards model. Depending on the level of background knowledge of the student, the course may begin with introductory topics and then consider more advanced topics such as time dependent variables, dropout, errors in variables, dependence and robustness. An indication of the level of the course is given by the text Therneau and Grambsch (2000, Modelling Survival Data).
|
Prerequisite: Stat2001 |
Alan Welsh |
| Simulation studies |
Simulation is a powerful tool for checking the properties of
statistical models in situations where theoretical results are
unavailable. Additionally, it can be a useful source of insight
in the exploration of theoretical results. Applications that
will be explored include models that allow for dependence
and the effects of variable selection in classification and/or
regression problems. |
Prerequisite: stat2001 |
John Maindonald |
| Review of selected data mining literature |
Investigate the handling of problems of inference
in the data mining literature, and draw comparisons with the
approaches that have been favored by statisticians. |
Prerequisite: stat2001 |
John Maindonald |
| Topics in dynamical systems, stability, and chaos |
In this reading program we will review aspects of the
mathematics of dynamical systems, stability, and chaos within a
historical framework that draws together the two major threads
of its early development: celestial mechanics and control
theory, and focusing on qualitative theory. From this
perspective we will study how concepts of stability enable us
to classify dynamical equations and their solutions and connect
the key issues of nonlinearity, bifurcation, control, and
uncertainty that are common to time-dependent problems in
natural and engineered systems. Building on this foundation, we
may investigate new problems involving stability of complex
networks.
|
Students with first year + some second year maths. A good
background study for MATH3062. |
Rowena Ball |
| Polynomial Hulls of Sets
in Complex Euclidean Space |
Compute the polynomial hull of various sets in complex n-space, n at least 2. |
Prerequisite: complex variables |
Alexander Isaev |
| Galois theory in topology |
Explore the analogy between the Galois theory of fields and the theory of covering spaces in topology |
Add-on to math3345 |
James Borger |
| The statistics of point clouds |
Given a cloud of points in R^n, try to find a pattern, to describe
the shape that they fill out in space. The subject is still very new,
but there are
attempts to compute geometric invariants, ranging from homology to finer
invariants, such as curvature. |
Some background in geometry and/or algebraic topology would be useful. |
Amnon Neeman |
| Elliptic curves |
A cubic equation in the plane has solutions
which satisfy a group law. These are called elliptic curves. At a very basic level one could look at the situation over finite
fields, where everything is very hands-on and computable, and where the subject
plays a key role in cryptography. Fancier versions would involve looking at it
over the integers, where the problems are much harder; they touch on very
modern mathematics, ranging from the proof of Fermat's Last Theorem to the
conjecture of Birch and Swinnerton-Dyer. |
Prerequisite: math2322 |
Amnon Neeman |
| Numerical Simulations of Plasma Flow |
Physicists are interested in studying the behaviour of any instabilities that arise in plasma flows. In order to properly characterise such behaviour numerical simulations must be carried out over long periods of time and this places a large, sometimes impossible, demand on computational resources. The project will explore the theory and implementation of various ODE solvers with the aim of reducing the computation time. Some programming experience is preferred. |
math2305/2405 |
Linda Stals |
| Introduction to Differential Topology |
Obtain topological invariants of spaces such as
the degree of a map and the genus of a surface using geometry and analysis.
Reference: John W. Milnor "Topology from the differential
viewpoint". |
Prerequisite: math2320 |
Bryan Wang |
| Introduction to Dirac operators |
Clifford algebras and Clifford modules; Spin structures and Dirac
operators,
their geometric properties, and some examples, possibly including Witten's proof of the positive mass theorem |
Prereq: math2320, math2322 |
Bryan Wang |
| Differential forms and de Rham cohomology |
Reference: "Differential Forms in Algebraic
Topology" by Bott and Tu. |
Pre/corequisite: math 3344 |
Bryan Wang |
| Exotic smooth structures on spheres |
Investigate why spheres of dimension 7 or more have finitely many,
but more than one smooth structure |
Pre/corequisite: math 3344 |
Bryan Wang |
| From chaos to structure in turbulent plasma and
planetary flows. |
There is a recent upsurge of interest world-wide in the
self-structuring properties of quasi two-dimensional flows,
motivated by the need to control transport in next-step fusion
energy experiments and understand variations in planetary
circulations mediated by climate change. What is the deeper
physics behind the remarkable fact that in such flows ordered
structures and patterns can arise from chaotic or turbulent
fluid motions? Since thermodynamics tells us that disorder or
entropy tends to increase, such behaviour may seem
counterintuitive. In this module we shall study equations of
motion for planetary flows in the geostrophic approximation and
for magnetic fusion plasmas in the electrostatic approximation,
and compare and contrast the physics expressed by each system. |
math2405/2305 plus some physics |
Rowena Ball |
| The Banach-Tarski Paradox |
Use the axiom of choice to slice up and re-glue an apple into two
apples of the same size! |
math 3320 |
Adam Rennie |
| The Riemann zeta function and the distribution of primes |
Dirichlet's theorem on primes in an arithmetic progression, Riemann zeta function, distribution of prime numbers.
|
Second or third year student |
James Borger |
| Graph C*-Algebras |
Study the definition and basic properties of graph C*-algebras. These are
amongst the easiest C*-algebras to visualise and study, and yet are generic
enough to give a good idea how C*-algebras behave in general. |
Prerequisite: math 3320 |
Adam Rennie |
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