Day 2 began with Gordon Royle telling us how he had almost completely finished the proof that roots of chromatic polynomials of planar graphs are dense in the interval (3,4). Now, on this topic, only the Birkhoff–Lewis conjecture that there are no chromatic roots of planar graphs in (4,5) remains. Gordon showed us a map with a line joining Perth to Lisbon; I think it was not a rhumb line.
Boris Zilber described his work trying to make sense, from a logician’s point of view, of the limits used by physicists in going from the discrete to the continuous, a problem which (as he pointed out) goes back to Hilbert’s 6th problem in 1900. He has some deep results, not all positive. For example, compact Lie groups are not approximable (in his sense) by finite groups.
Leonard Soicher has written a GAP program, using his GRAPE package, to find the chromatic number of a graph, with the GRAPE philosophy of using symmetry of the graph to cut down the search. Among other things, he can determine, for all primitive groups of degree at most 255, whether or not they are synchronizing; the total time taken is quite small. (There are a few difficult groups where the program needs a hint; we hope that understanding these better will lead to theoretical advances.)
John Meakin, the person I have known longest at this conference (we were undergraduates together in Brisbane from 1964 to 1967), started with the observation that connected covers of a connected topological space are classified by subgroups of the fundamental group. He has defined a gadget which classifies immersions in a similar way. Perhaps not surprisingly, it is an inverse semigroup. He showed us the outline of 15 steps in the argument.
Pedro Silva, with John Rhodes, introduced a class of objects more general than matroids, indeed general enough to include bases of permutation groups (a dream of mine for a long time): these are Boolean representable simplicial complexes, or BRSCs. He gave us a survey of this and mentioned software for computing with these objects. I do hope to say more someday.
A very nice talk from Michael Kinyon followed, about the multiplication group of a loop. A loop is a set with a binary operation having an identity and unique left and right division, in other words, one whose Cayley table is a normalised Latin square; the multiplication group is generated by rows and columns of the Latin square. Many interesting results and conjectures connect properties of the loop with those of its multiplication group; in particular, for non-commutative simple loops the group is primitive. Colva Roney-Dougal (who unfortunately is not here) and Michael Giudici are working on the problem of classifying simple automorphic loops (those for which the stabiliser of the identity in the multiplication group consists of loop automorphisms). The conjecture is that these must be (non-abelian simple) groups. According to Michael, there is light at the end of the tunnel.
Brendan McKay talked about enumeration of three classes of matrices of non-negative integers: rectangular matrices with constant row and column sums; symmetric square matrices with zero diagonal and constant row sums; and adjacency matrices of regular graphs (the previous case with all entries zero and one). He gave exact formulae, asymptotics, and accurate simulations.
Robert Bailey told us about bases and resolving sets for permutation groups, coherent configurations, and metric spaces. We wrote a survey on this in 2011, to try to throw some light on the confusion caused by re-invention of the same concepts in many different fields; this is now his most cited paper, and is on the first page of my papers on Google scholar.
To end the day, Péter Pál Pálfy talked about twisted wreath products, and the role they play in the lattice representation problem. A famous question asks whether every finite lattice is the congruence lattice of a finite universal algebra; an equivalent question is whether every finite lattice is isomorphic to an interval in a subgroup lattice. Twisted wreath products have settled new cases of this conjecture.
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