Plymouth workshops

Plymouth

The title refers not to buildings in the Royal Navy dockyard in Devonport, but to a London Mathematical Society regional meeting here last week. The format of the meeting was, on the first day, an LMS meeting with opportunity for locals to “sign the book” and two big talks introducing the two workshops; and then two days of more specialised talks in the two areas.

The subjects of the two workshops were Combinatorics and Differential Algebra. Quite different subjects, and the organisers had taken quite different views about the organisation. The Differential Algebra workshop was very focussed. Marius van der Put gave an introductory lecture about “Differential equations and finite groups”, and the rest of the workshops provided a second talk by him, two talks by Felix Ulmer, and one by Guy Casale. A linear differential equation has a differential Galois group, which is an algebraic subgroup of GL(n,k) if the equation has degree n over the base field k. The workshop almost entirely concerned the inverse problem: Given such an algebraic group, say over the algebraic closure of the rationals, can you find a differential equatio of which it is the Galois group? The answer is “yes” for all finite subgroups of GL(n,A) except for one outstanding group of order 216. The methods are a real mixture of differential equations and finite groups (and their representations). Casale’s talk was about nonlinear differential equations, where the Galois group has to be replaced by the weaker notion of pseudogroup; he considered three examples, including the first two Painlevé equations.

By contrast, the five speakers in the Combinatorics workshop told us about a wide variety of subjects:

  • Rosemary Bailey told us about the search for Eulerian quasigroups for designing experiments involving neighbour effects: the conjecture is that these quasigroups exist for all orders greater than 4, and this is almost proved. (A quasigroup of order n is Eulerian if the sequence obtained by starting with any two elements and then following a and b by their product in the quasigroup has maximum possible period n2.)
  • Iain Moffatt showed how to axiomatise the notions of deletion and contraction, and use them (via Hopf algebras) to construct “Tutte polynomials” for combinatorial structures other than ordinary graphs and matroids. He illustrated with the Penrose and Bollobás–Riordan polynomials for graphs embedded in surfaces.
  • Robert Brignall talked about well-quasi-ordering of classes of permutation patterns, and permutation graphs, with background on the general notion of well-quasi-ordering for various notions of ordering. The questions for permutation patterns and permutation graphs are subtly different since the graph does not determine the permutation.
  • Karen Gunderson talked about graph bootstrap percolation. Given a graph F (usually a small complete graph), this process consists of taking a graph and adding any edge which completes a copy of F with existing edges; G is said to percolate if the process terminates with the complete graph. She and her four collaborators had examined this for the random graph with edge-probability p, where there are sharp thresholds, not just for percolation, but for the number of steps required. If you want a small example to think about, in the case where F is the triangle, a graph percolates if and only if it is connected, and the time taken is the log to base 2 of the diameter.
  • In the absence of the fourth speaker Tony Nixon, organiser Tom McCourt stepped up and talked about how he and his four collaborators had found the spectrum of orders of distributive Mendelsohn quasigroups. (A Mendelsohn quasigroup is based on a set of points with a collection of cyclic triples so that each pair of distinct points occurs in order in a unique triple, which is their product in the quasigroup; it is distributive if (xy)z = (xz)(yz). The orders of such quasigroups are precisely the Loeschian numbers, those of the form x2+xy+y2.)

We also ate and drank very well, and had the opportunity to take the Cremyll ferry and walk the first Cornish stretch of the South-West Coast Path, coming back via Millbrook Lake where a curlew was sitting on a nest out in the water.

Millbrook Lake

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About Peter Cameron

I count all the things that need to be counted.
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