Honours projects in Pure Mathematics

Below you will find some descriptions of Honours projects or areas of staff interest. If you find a staff member listed below, but no associated information, you should consider talking to them to find out more about the opportunities for carrying out an Honours project with them.

• Dr Sue Barwick

  I have supervised honours projects in the area of Projective Geometry, and in the area of Applications of Projective Geometry to Information Security (in particular to Secret Sharing Schemes). Projects in these areas generally involve a literature review. That is, the students use the library to find research papers applicable to the project topic, read the papers and their project consists of a survey of known results about the topic. A prerequisite to work on a project in these areas is the third year subject Fields and Geometry III. Following are brief descriptions of two potential projects in these areas. I am also able to supervise projects on topics in Coding and Cryptography. Students are encouraged to come and see me to discuss possible project topics.

  • k-arcs in projective planes

    A k-arc in a projective plane is a set K of k points such that no three points in K are collinear. A non-degenerate conic is an example of a k-arc. There is a rich and interesting literature on k-arcs, and generalisations of k-arcs. Some of the questions to be answered are: what is the largest k for which a k-arc exists? what are some constructions of k-arcs? and can we classify k-arcs for certain values of k? There are several different paths that can be followed in this topic, and the student can choose the one they are most interested in.

  • Geometric secret sharing schemes

    Suppose we have a secret, a group of people, and a list of subgroups of people (called an access structure) who are authorised to obtain the secret. For example, the secret may be the ability to open a bank safe and the security policy of the bank requires a minimum of 3 tellers to be present to open the safe. We can solve this problem by issuing physical keys, but some access structures can mean that everyone has to carry a large number of keys. A more practical and secure solution is to use mathematics, and it turns out that solutions involving projective geometry are very effective. A project in this area would involve studying how projective geometry can be used to construct secret sharing schemes, as well as looking at ways secret sharing schemes can be generalised.

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• Dr Nicholas Buchdahl

  Please note: I am not able to supervise any Honours students from the middle of 2006 to the middle of 2007.

  I am happy to supervise Honours projects in the following areas, not arranged in any particular order:

  • Complex analysis in one variable;
  • Complex analysis in several variables;
  • Differential geometry;
  • Problems in analysis;
  • Partial differential equations;
  • Gauge theory;
  • Algebraic topology;
  • Differential topology.

  I am also happy to consider supervising in other areas, provided I feel sufficiently competent and sufficiently interested.

  In the past I have supervised Honours/Summer projects on

  • The theorems of H. Cartan on Stein manifolds (Several Complex Variables);
  • Topics in Several Complex Variables;
  • The Riemann-Roch theorem (1 complex variable/partial differential equations);
  • Donaldson's theorem on definite 4-manifolds (Gauge theory);
  • Constructing constant curvature metrics on surfaces (Differential geometry/Analysis);
  • Topics in advanced complex analysis (one variable).

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• Professor Michael Eastwood

  • The Mathematics of Wallpaper Design

    There are 17 essentially different wallpaperings of the Euclidean plane, all of which occur in Moorish art. The project should present a complete proof of this with variations if time permits (e.g. how to wallpaper the sphere or the hyperbolic plane). Historical note: this is one of two mathematical ingredients in William Lawrence Bragg's winning of the Nobel prize in 1915 (at age 25) after studying Honours Mathematics at the University of Adelaide.

  • The Geometry of Straight Lines in Three-space

    How best to parameterise the space of straight lines in Euclidean 3-space? The Pluecker embedding realises this space as a quadric in 5 dimensions. Alternatively, one can view it as the tangent bundle to the 2-sphere. The project should discuss the various possibilities and their hidden relations with complex analysis. If time permits, applications to medical imaging can be included.

  • Consequences of the Triangle Inequality

    Suppose a,b,c are the side lengths of a triangle. Which homogeneous symmetric polynomials in the 3 variables a,b,c are non-negative as a consequence of a,b,c being the sides of a triangle? How about polynomials in 6 variables a,b,c,d,e,f being sides of a tetrahedron? Warning: this is an open-ended project and might be quite hard! There are potential applications to image recognition problems.

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• Dr Finnur Lárusson

  I am happy to supervise honours projects in complex analysis and the many areas of mathematics that interact with it. An honours topic that I particularly like is the theory of compact Riemann surfaces. It provides an opportunity to meet and apply important ideas from functional analysis, homological algebra, manifold theory, partial differential equations, and sheaf theory in an accessible geometric context. The first goal of a project in this area would be to get to the central theorem on compact Riemann surfaces, the Riemann-Roch Theorem. After that, the project could continue in various different directions.

  There are many other options, including several complex variables, topos theory, logic, and homotopy theory. In 2006, I am supervising two honours projects. One is on wavelets, a newly developed area within real and functional analysis with important applications to image compression. The other is about the correspondence between planar trees and so-called Shabat polynomials; this project involves combinatorics, covering space theory, and complex analysis. I am glad to meet with prospective honours students to discuss possible topics.

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• Professor Michael Murray

  I am happy to supervise Honours projects in any of the following areas.

  • differential geometry
  • mathematical physics
  • algebraic topology
  • differential geometry and statistics
  • Lie groups and Lie algebras
  • gauge theories
  • Bogomolny monopoles
  • twistor theory
  • bundle gerbes
  • computer vision

  Please contact me for details on any of the above.

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• Dr David Parrott

  I am happy to supervise students in the following areas, with some recent Honours topics listed as examples.

  • Finite group theory
    • Constructions of the Leech Lattice
    • The Conway groups
    • Extended Sylow Theorems
  • Lie Algebra
    • The classification of the semi-simple complex Lie Algebras
  • Logic/Computability
    • Computability and Decidability
    • Computability, Turing Macines and Complexity Classes
  • History of Mathematics
    • The History and Development of Number Systems
    • The Development of Calculus
  • Mathematics Education
    • Secondary School Mathematics in South Australia

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• Associate Professor Mathai Varghese

  I am happy to supervise Honours projects in the following areas:

  • Atiyah-Singer index theory of elliptic operators and generalizations,
  • Noncommutative geometry,
  • Positive scalar curvature metrics - obstructions and existence,
  • Spectral theory of elliptic operators and their discrete analogs,
  • L2 methods in geometry and topology,
  • K-theory and twisted analogues,
  • Applications to the fractional quantum Hall effect and string theory...

  In the past I have supervised Honours projects on:

  • The proof of the Atiyah-Singer Index theorem;
  • Cyclic cohomology.

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• Dr Alison Wolff

  I am prepared to supervise Honours projects in number theory, combinatorics, and applications of these to other areas.

  Descriptions of some past projects I have supervised:

  • Numerical Integration with respect to Fibonacci lattice rules

    This project was done by a student doing joint Honours in Pure Maths and Computer Science. The project was on an application of number-theoretical ideas to numerical analysis, thus combining the student's favourite areas of study from his undergraduate courses.

  • On Diophantine Approximation

    This was a straight number-theoretical project, in which the student researched in greater depth a topic studied in Number Theory III. In particular, he investigated a fascinating sequence of numbers called the Markoff Spectrum, which is concerned with how well certain types of irrational numbers can be approximated by rationals.

  • Solving Large Sparse Singular Homogeneous Systems over GF(2)

    Another joint Pure Maths/Computer Science project, using number theory and linear algebra, and with applications to factoring large numbers, a topic which is of interest to cryptographers.

  • On the Period Length of Pseudorandom Number Sequences

    The 'random' numbers used in many applications are in fact almost always elements of periodic sequences produced deterministically by algorithms known as pseodorandom number generators. Clearly a long period length is desirable, and this project was a study of the period lengths of several common types of generator. Algebraic and number-theoretic methods were used.

  • Path Counting and Applications

    This project dealt with the combinatorial problem of counting certain restricted types of lattice paths between two points in the coordinate plane (A lattice path is one composed of steps along the integer grid lines x=n, y=m where m,n are integers.) Applications to such diverse areas as seismography and vote-counting were investigated.

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