I am back after my trip to the annual meeting of the Australian Maths Society in Adelaide. There were 12 plenary speakers this year. Going to one of these general meetings always reveals how big a subject mathematics is and how little of it I know. For this reason I always prefer “big picture” plenaries over more detailed ones. A couple of personal highlights were Terry Tao’s public lecture on `Structure and randomness in the prime numbers’, Jacqui Ramagge’s talk on `Totally disconnected, locally compact groups’ which outlined the theory of the subject developed mainly by George Willis, and Akshay Venkatesh’s talk on his work with Ellenberg and Westerland on `The Cohen-Lenstra heuristics over global fields’.
There were up to 12 parallel sessions on at any given time. I mainly attended the combinatorics session where there were many good talks and I spoke on `3/2-transitive permutation groups’. I have uploaded the slides for my talk. I also ventured to the Topological groups session and enjoyed Daniel Horadam’s talk there on `Automorphism groups of trees’. There were many other talks I would have liked to have gone to but they clashed with other interesting talks or me speaking.
Overall the conference went very well and the organisers should be congraulated on the great job they did. The conference dinner was also very well done: the food and wine was great and there was even a band.
I was lucky enough to be the “keynote” speaker at the special session on Combinatorics so I got a one-hour time slot, which is usually easier to deal with than a shorter one.
The talk itself was a more focussed variant of a talk I gave at the British Combinatorial Conference in July this year, and describes my recent work with Alan Sokal, which is essentially complete, but just needs wrestling into final publishable shape. Working with Alan is a tremendous privilege, but exploring every possible generalization, hypothsis-weakening and related application that his incredibly fertile brain throws up is incredibly time-consuming for a mere mortal!
Anyway the gist of this work is that we want to find a parameter to replace maximum degree in the result that the moduli of chromatic roots are bounded by a function of maximum degree. The parameter maxmaxflow which is defined to be the maximum number of edge disjoint paths between any pair of vertices is a promising alternative, and this talk gives a very general overview of our proof that it works for series-parallel graphs.
One of the things John and I do around here is run activities for groups of school students who are visiting the university. Today we had such a visit from a group from Esperance. The visits often include activities in other disciplines across campus, for example physics demonstrations or building bridges/gyroscopes/electric motors in engineering. This presents an interesting dilemma as to what the maths activity should be. It needs to be very much hands on so that the students get to do something. John and I are also keen for it to be very different to the sort of maths done at school.
Currently, our main activity revolves around the game of Nim. We introduce the rules of the game, give a short demonstration and then call for challengers. We offer a chocolate bar to anyone who can beat us. After a couple of games, we then give all the students some counters and get them to play the game amongst themselves so that they can try and work out a winning strategy and continue to try to beat us. Towards the end of the activity we outline binary numbers, Nim addition and how to use it to win. The activity goes down really well. It draws on their competitive instincts and some students get right into trying to beat us. It has worked with the whole range of high school ages and we once even ran the activity successfully for a group of primary school students (minus the maths behind it though).
We are interested in learning about other suitable activities. It would be good to have a couple up our sleeves as there are sometimes students who attend more than one of these visits. John is looking at developing something involving Conway’s Rational Tangles. He has demonstrated it to a couple of groups of students now and has been very succesful but it is more of a demonstration than an activity. Ideally we should pair it with a couple of other things as we usually have 45-60 minutes. Please let us know of other activities that work well which could be done with or separately to these.
We’re all getting ready here for next week’s Australian Math Society conference in Adelaide, so I’ve been watching a few practice talks. Watching these, along with a few other recent talks, reminded me of a pet peeve.
Nowadays, even in mathematics conferences, most talks are computer presentations, and most of these are prepared using Till Tantau’s beamer package for LaTeX.
Although beamer is amazingly good – or perhaps because beamer is amazingly good – its sheer ubiquity means that so many talks “look the same”.
And one way in which they look the same is that although I have never seen anyone use them in a talk, almost everyone seems to keep the default navigation symbols at the bottom corner of every page, where they often overlap the content and generally clutter the page.
But it is so easy to turn them off – just add
before the beginning of the document and away they go!
Today I had a meeting at our Institute of Advanced Studies to discuss the programme for Akshay Venkatesh‘s visits here as a Professor-at-Large for 2010 and 2011. This will involve a visit each year and the tentative schedule is that over the course of his two visists he will give a public lecture, a masterclass for postgraduate students and teach parts of some honours courses.
Akshay did his undergrad here at UWA before going off to the US for his postgrad. He is now a Professor at Stanford and is an invited speaker at next year’s International Congress of Mathematics. His first visit as a Professor-at-Large will be directly after the congress. He was also here today for the meeting and we were able to reminisce about sharing an office in the period between our honours year and going off to separate parts of the world for our PhD’s.
I have just come across the Newton Institute’s Web seminars. It looks a great resource of various talks given at the Institute including those given as part of their programmes and workshops. They seem to have quite an extensive recording mechanism as there appears to be several cameras involved. The files are a bit big but I can see myself wasting hours watching them as long as my australian broadband can cope. The site includes a talk by Gordon.
I have just finished reading the proofs for my paper `On finite edge-primitive and edge-quasiprimitive graphs’ with Cai Heng Li which has just been accepted in JCTB. This paper has been a long time in the making as Li and I started the work in 2002! Usually when I submit a paper to a journal I put a preprint up on my research web page but I should really start uploading them also to the arxiv so I have just done so and it is now available there.
A graph is called edge-primitive if its automorphism group acts primitively on the set of edges. Many famous graphs such as the Heawood graph, Tutte-Coxeter graph, Hoffman-Singleton graph and Higman-Sims graph are edge-primitive. The paper begins a systematic study of edge-primitive and edge-quasiprimitive graphs by analysing the possible actions on vertices and edges.
It is basically a dual version of the old problem of characterising graphs whose chromatic polynomials have integer roots. Chordal graphs, which are built up from a single vertex by repeatedly adding a new vertex adjacent to a clique, are one such class of graphs. But there are lots of non-chordal examples, and it seems hopeless to try and classify them all.
We considered the dual problem of the graphs whose flow polynomials have integer roots. Again there is an obvious class of examples, namely the planar duals of chordal graphs, but in this case there are no other examples. So the punchline of our paper is that “the obvious examples are the only examples”.
In matroid terms, the result says that a cographic matroid with integer characteristic roots is necessarily supersolvable.
We needed the new version because a student at MIT, Maria Monks, spotted a (fixable) error in the first version. This was despite the paper having been refereed and accepted already! So the power of the arxiv has saved Joe and I some red faces.
You may have heard about the open problem on Moore graphs which is one of the most famous problems in Algebraic Graph Theory. Here is a brief synopsis…
A Moore graph is a regular graph which is extremal in the sense that its number of vertices is large compared to its degree and diameter . Equivalently, a Moore graph is just a regular graph with girth equal to . There is much one can say about Moore graphs, but I just want to concentrate on the diameter 2 case; these are exactly the strongly regular graphs with and . A nice algebraic graph theory argument leads to the Hoffman-Singleton Theorem:
Theorem: Any Moore graph with diameter 2 must have degree 2, 3, 7, or 57.
The graphs with degree 2, 3, and 7 are the pentagon, Petersen graph, and Hoffman–Singleton graph, respectively, and these are the unique such diameter 2 graphs with these degrees. The last case, degree 57, is unsolved; we still do not yet know whether such a graph exists!
What we do know is that a Moore graph of diameter 2 and degree 57 is vertex-intransitive and the automorphism group has order not divisible by 4. In fact, an involutary automorphism must fix 56 points. Not much more could be said until very recently, and I want to make special mention of a recent preprint of Martin Mačaj and Jozef Širáň. They prove that the automorphism group must have size at most 375.
So what do you think? Does a Moore graph of diameter 2 and degree 57 exist?
Well here is my first post and the first post for our blog. Welcome. Over the last year I have become an avid reader of various maths blogs but haven’t come across one covering research in my corner of the mathematical world. After much discussion and dreaming, John, Gordon and I have finally taken the plunge and decided to start one ourselves. Hopefully between the three of us we will have something interesting to say.
To get the ball rolling I should start with a bit about my research. My main area of interest is permutation groups. This includes various areas of combinatorics, in particular questions involving automorphism groups of structures. So far these have mainly been graphs but it is starting to involve a bit of finite/incidence geometry. I guess I’m interested in anything with a decent group involved.