Research projects

One of the things I most like about my job is supervising students in research projects. From AMSI Vacation Scholarship students to honours students to PhD students, I enjoy collaborating with a student that is seeing a dedicated research problem for the first time, and then becoming a master of their subject.

For PhD supervision, I require that the student know me first: too often do I receive an email from someone who wants to do a PhD in, say, fluid mechanics with me; a subject I have very little idea about! I also require proof that you are capable of research at a high level. This gruelling test consists of asking the student to write a difficult proof in exquisitely written english on a result of my choosing. So if you are interested in doing a PhD with me, send me an informal chatty email (formal emails will be deleted), and the areas of my research that you are interested in.

Prospective honours students, I have outlined what I think you need to know in making an informed decision on whether you want to work under my supervision. The styles of projects can be roughly categorised as follows:

• A scholarly investigation that demands a lot of learning on a subject that is not taught in undergraduate. This is not an `instant-starter’ and would mean the student does a lot of reading on a subject before embarking on a research problem. In most instances, the research problem will not be known from the beginning but will evolve organically from a deep understanding of the subject. The main objective is to produce a dissertation that gives a summary of the research area that allows someone else of a similar background entry into the area.
• An instant-starter means that I have up my sleeve a problem in combinatorics, number theory, or group theory that requires little background and the student can start thinking about straight away. However, this rarely happens! Ask me if I have such a problem; most years I won’t.
• Computational enumeration problems are excellent problems for honours students as they learn a subject (like projective geometry) by a needs basis. This style of project requires the student to have an affinity with computer programming and at least feel that they are “good” at programming. The project would usually consist of searching for examples, exhaustive search of examples, and analysing the results with the mathematics they have learned.

As for research areas, here is how I describe them for a student:

Group theory

I am mainly interested in finite permutation groups, but I sometimes offer a project where my curiosity has taken me beyond the finite realm. The project is likely to be an abstract investigation and be co-supervised with another member of our research group; the Centre for the Mathematics of Symmetry and Computation are world-renown for their strengths in finite group theory.

Geometry

My research has mostly been in finite projective or polar geometry, generalised polygons, and related structures. Recently I have been interested in the foundations of geometry and the Bachmann school of metric geometry. I have plenty of nice problems in incidence geometry, and this would be my comfort zone in terms of supervision.

Algebraic Combinatorics

The use of association schemes has recently been very successful in the existence and non-existence questions of substructures in finite geometry. This currently a hot topic of mine that I have many side-projects on the fly. There are interesting avenues for computational investigation in this area.

Elementary number theory

First a disclaimer: I am not a number theorist, and I cannot pretend to be at the cutting edge of modern number theory. A lot of students are easily swayed by the beauty of this subject, however, the modern day number theory is quite a deep and complicated profession. The research projects I would offer under this heading would not be cutting edge research problems, only curiosities that have come up in my day-to-day research. If you would like a problem that trains you to be at the forefront of modern research in number theory, there is an excellent group at Macuqarie University and ANU that I would suggest you contact. However, if you like the “style” of mathematics of number theory, then you might also like “combinatorics” too, and we can certainly offer projects under that broad heading.

Students I have supervised

I have been enormously lucky to have supervised some remarkable students. Here is a list of the students and their projects.

Current students

• Vishnu Arumugam, PhD, Groups of Lie Type Acting on Generalised Quadrangles, co-supervised with Michael Giudici, 2022 –
• Jacob Smith, PhD, TBA, co-supervised with Michael Giudici, 2023 –

Past students

PhD Students

1. Jesse Lansdown, PhD, Designs in finite geometry, co-supervised with Gordon Royle and Alice Niemeyer (cotutelle with Aachen), 2015 – 2020
2. Mark Ioppolo, PhD, Codes in Johnson graphs associated with quadratic forms over $\mathbb{F}_2$, co-supervised with Cheryl Praeger and Alice Devillers, 2012 – 2020.
3. Jon Xu, PhD, Chevalley groups, Schubert varieties, and finite geometry, co-supervised with Arun Ram, 2012 – 2017.
4. Sayeed Hassan Alavi, PhD, Triple factorisations of groups, co-supervised with Cheryl Praeger, 2008 – 2011.
5. Frédéric Vanhove, PhD, Incidence geometry from an algebraic graph theory point of view, co-supervised with Frank De Clerck, 2007 – 2011.
6. Geoffrey Pearce, PhD, Transitive factorisations of graphs, co-supervised with Cheryl Praeger, 2004 – 2007.

Masters/MPhil Students

1. Saul Freedman, MPhil, p-groups related to exceptional groups of Lie type, co-supervised with Luke Morgan, 2017 – 2018.
2. Melissa Lee, MPhil, The m-covers and m-ovoids of generalised quadrangles and related structures, co-supervised with Michael Giudici, 2015/2016.
3. Sylvia Morris, MPhil, Symplectic translation planes, pseudo-ovals, and maximal 4-arcs, co-supervised with Michael Giudici, 2011 – 2013.
4. Guang Rao, Masters, Symmetric translation planes of square order, 2009/2010.
5. Cedric Raemdonck and Els Quintelier, Masters, Databanken en websites voor wiskundig onderzoek, co-supervised with Leo Storme, Michel Lavrauw, Geert Vernaeve, 2008.
6. Frédéric Vanhove, Masters, Een studie van m-systemen en aanverwante incidentiestructuren, co-supervised with Frank De Clerck, 2006/2007.

Honours Students

1. Jolyon Joyce, The incidence-based foundation of hyperbolic planes over arbitrary and Euclidean ordered fields, 2024.
2. Jeff Saunders, Affine geometric algebra, 2024.
3. Huxley Berry, Finite abstract ovals, co-supervised with Luke Morgan, 2024.
4. Jacob Smith, Non-regular 2-closed permutation groups that are not automorphism groups of digraphs, co-supervised with Michael Giudici, 2022.
5. Darryl Teo, On intersection multiplicity of plane curves, 2021.
6. Ann Linehan, Relationships Between Geometric Propositions which Characterise Projective Planes, co-supervised with Jesse Lansdown, 2021.
7. Hefu Yu, The quest to classify finite flag-transitive projective planes, 2020 – 2021
8. Dominique Douglas-Smith, Skew Projection, co-supervised with Jesse Lansdown, 2020
9. James Evans, Generalised polygons and their symmetries, co-supervised with Emilio Pierro, 2019.
10. Jacob Morhall, Eigenvalue bounds on cliques and cocliques, co-supervised with Gordon Royle, 2018.
11. Timothy Harris, Abstract ovals, 2017.
12. Calin Borceanu, Breaking symmetry in backtrack algorithms, 2016.
13. Saul Freedman, Maximally symmetric p-groups, co-supervised with Luke Morgan, 2016.
14. Jesse Lansdown, Metric coset geometries, co-supervised with Joanna Fawcett, 2014/2015.
15. Melissa Lee, Relative hemisystems on the Hermitian Surface, co-supervised with Eric Swartrz, 2014.
16. Blake Segler, Fault-tolerant quantum computation, co-supervised with Jingbo Wang, 2013.
17. Murray Smith, A generalised clique-coclique bound, 2012.
18. Mark Ioppolo, A classification of rational 2-tangles via coset spaces, 2011-2012.
19. David Raithel, The 24-cell, co-supervised with Cheryl Praeger, 2011-2012.
20. Daniel Skates, Sets of type (m,n) in projective planes, co-supervised with Gordon, 2011.
21. Adrian Dudek, Expander graphs, co-supervised with Gordon Royle, 2010.
22. Andrew Cannon, Graphs of the Johnson schemes, co-supervised with Cheryl Praeger, 2009 – 2010.
23. Claire Collier, The norm of a group, co-supervised with Maska Law, 2006.
24. Michael Pauley, Flag-transitive linear spaces and spreads of projective space, 2005/2006.
25. Jonathan Cohen, Small base groups, large base groups and the case of the giants, co-supervised with Alice Niemeyer, 2004.
26. Geoffrey Pearce, Products of groups and graphs, co-supervised with Cheryl Praeger, 2003.