PhD scholarships

Applications are now open for scholarships for international PhD students at UWA to commence in the first half of 2019.  Myself and other members of the CMSC are interested in taking on new PhD students so if you are interested then please contact one of us about applying.

The deadline is the 31st of August 2018. Please see here for details on how to apply.

Please note that applications are ranked at our faculty level and in particular there are no scholarships specifically devoted to mathematics students.  In particular the process is highly competitive.

 

Fields Medal

There is lots of excitement at UWA this morning with the news overnight that Akshay Venkatesh was one of the winners of the 2018 Fields medal. He did his undergraduate here in the 90s. I watched the live feed of the opening ceremony to see it. There is a great article at the Quanta magazine about him and his work. Other articles are here, here and here, and a video here.

Postdoc position at UWA

I have just advertised a 9 month postdoc position here in Perth to work on the ARC grant entitled `Symmetries of finite digraphs’.  The deadline is the  28th of March.  More details are available here.

 

Forrest Fellowships

The Forrest Foundation are currently advertising their Forrest Fellowships which are 3 year postdocs at a university in Western Australia.  Of course, The University of Western Australia is one such university.

They are looking for `outstanding researchers of exceptional ability and resourcefulness, having the highest calibre of academic achievements and with the potential to make a difference in the world.’ All research disciplines are eligible to apply. Applicants need to be no more than 2 years post PhD graduation.

More details can be found here. Note that one of the selection criteria involves ability and commitment to mentoring. Applications close at the end of June.

Applicants will need a statement of support from the univeristy that we wish to be at. If you are interested in applying for one of these to join the CMSC then feel free to contact one of us.

8th Slovenian Conference on Graph Theory

Last week I was at the 8th Slovenian Conference on Graph Theory.  This was the latest in what is commonly known as the ‘Bled conference’ but this year was in Kranjska Gora. This meant that the conference excursion was to Lake Bled. It was a very enjoyable conference with lots of interesting talks and it was good to catch up with lots of people. I was one of the plenary speakers and my talk was entitled ‘Bounding the number of automorphisms of a graph’. This surveyed the recent work on the Weiss conjecture and its generalisation the PSV conjecture. It also discussed my recent work with Luke Morgan on the PSV conjecture for semiprimitive groups with a nilpotent regular normal subgroup.  More details can be found on my slides.

Symmetries of Graphs and Networks IV

Last week I was at the Symmetries of Graphs and Networks IV conference at Rogla in Slovenia. The conference webpage is here.  At the same time was the annual  PhD summer school in discrete maths. As usual it was a very enjoyable and well organised conference. It was good to catch up with some of the regulars and meet a few new people as well

I was one of the invited speakers and spoke about some of the work that I have been doing recently with Luke Morgan on graph-restrictive permutation groups. The slides are available here.  The two relevant preprints are on the arxiv here and here.

 

 

 

 

Regular cycles of elements in finite primitive permutation groups

Cheryl Praeger, Pablo Spiga and I have just uploaded to the arxiv our preprint  Finite primitive permutation groups and regular cycles of their elements.   Everyone recalls from their first course in group theory that if you write a permutation in cycle notation, for example (1,2)(3,4,5)(6,7,8,9,10), then its order is the lowest common multiple of it cycle lengths. As I have just demonstrated, not every permutation must have a cycle whose length is the same as the order of the element. We call a cycle of an element g whose length is the order of g, a regular cycle.  A natural question to ask is when can you guarantee that a permutation has a regular cycle? Or in particular, for which permutation groups do all elements have a regular cycle?

Clearly, the full symmetric group contains elements with no regular cycles, but what about other groups?  Siemons and Zalesskii showed that for any group G  between PSL(n,q) and PGL(n,q) other than for (n,q)=(2,2) or (2,3), then in any action of G, every element of G has a regular cycle, except G=PSL(4,2) acting on  8 points. The exceptions are due to isomorphisms with the symmetric or alternating groups.  They also later showed that for any finite simple group G,other than the alternating group, that admits a 2-transitive action, in any action of G every element has a regular cycle. This was later extended by Emmett and Zalesskii to any finite simple classical group not isomorphic to PSL(n,q).

With these results in mind, we started investigating elements in primitive permutation groups. The results of this work are in the preprint. First we prove that for k\leq m/2, every element of Sym(m) in its action on k-sets has a regular cycle if and only if m is less than the sum of the first k+1 primes. After further computation we were then willing to make the following conjecture:

Conjecture: Let G\leqslant Sym(\Omega) be primitive such that some element has no regular cycle. Then there exist integers k\geq1, r\geq1 and m\geq5 such that
G preserves a product structure on \Omega=\Delta^r with |\Delta|=\binom{m}{k}, and Alt(m)^r  \vartriangleleft G\leqslant Sym(m)\textrm{wr} Sym(r), where Sym(m) induces its k-set action on \Delta.

The general philosophy behind why such a result should be true is that most primitive groups are known to be small in terms of a function of the degree.  There is a large body of work bounding the orders of primitive permutation groups with the best results due to Maroti. The actions of Sym(m) on k-sets are the usual exceptions.

Our paper  goes about attempting to prove this conjecture. We make substantial progress and reduced its proof to having to deal with all the primitive actions of classical groups. Note that Emmett and Zalesskii only dealt with simple ones.  One consequence of our work is we showed that every automorphism of a finite simple group has a regular cycle in its action on the simple group.

Pablo and Simon Guest have subsequently gone on to prove the result for all actions of classical groups and so the conjecture is now a theorem.

Pablo gave a great talk about the conjecture and its subsequent proof  at the recent BIRS workshop on Permutation Groups in Banff which you can view here.

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