School of Mathematics & Physics, University of Lincoln
On Wednesday the 28th of November 2018, Dr Nick Gill (University of South Wales) visited the Charlotte Scott Centre for Algebra in Lincoln and gave a talk “Cherlin’s conjecture for finite binary permutation groups”.
Abstract: “A mathematical object C is called HOMOGENEOUS if any local symmetry can be extended to a symmetry of C itself. The category of vector spaces, for instance, is replete with homogeneous objects: if U_1 and U_2 are vector subspaces of V that are symmetric, i.e. there is an invertible linear transformation T between them, then we know that we can extend T to an invertible linear transformation V -> V.
In other categories, though, homogeneous objects are hard to find — for instance, if one considers the category of graphs, a classical theorem of Sheehan/ Gardiner tells us that there are only a couple of infinite families, plus a couple of sporadic examples. Our interest lies in understanding Gardiner’s theorem as a special case of a general theory concerning HOMOGENEOUS RELATIONAL STRUCTURES. This wider perspective allows us to (a) generalize Gardiner’s result; (b) understand the presence of sporadic examples in Gardiner’s result; (c) understand relational homogeneity for any finite permutation group.
This is joint work with Francesca Dalla Volta, Francis Hunt, Martin Liebeck and Pablo Spiga.”