This week, Dr Colin Reid from the University of Newcastle in Australia is visiting Simon Smith to continue work on their project to understand the local-to-global behaviour of groups acting on trees that enjoy the Independence Property P described by J. Tit’s.

Colin’s visit is funded by the London Mathematical Society, as part of a Research in Pairs grant that Colin and Simon were awarded in June 2018.

Dr Reid will be giving a talk to the Charlotte Scott Centre for Algebra in Lincoln on Wednesday 16th June at 3pm in INB3305. The title of his talk is: “Locally compact piecewise full groups”.

]]>*Abstract*: An element of a group is said to be right Engel if for every there is a number such that . We prove that if a profinite group admits a coprime automorphism of prime order such that every fixed point of is a right Engel element, then is locally nilpotent.

Abstract: “If G is a finite group, non-isomorphic G-sets X, Y may give rise to isomorphic permutation representations C[X] and C[Y]. These phenomena are called `Brauer relations’ or `linearly equivalent G-sets’, and they turn out to have interesting applications in number theory. I will explain how to classify Brauer relations in all finite groups, the history of the problem and some number-theoretic applications. This is joint work with Alex Bartel.”

]]>Abstract: “I will discuss elliptic curves from the classical number theoretic point of view of trying to solve Diophantine equations. The aim will be both to explain how we think about these creatures and to give an overview of what we can (and sometimes can’t) prove about them, and to illustrate it with explicit examples. I will not try to describe the huge modern technical machine that has been developed to study elliptic curves, so most of the results will come as black boxes.”

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His talk abstract is as follows:

“(Joint with Ben Martin and Lewis Topley). In a paper in the now defunct LMS Journal of Computation I used GAP to compute the Lie-theoretic centralisers in the exceptional groups of elements in their minimal modules in all characteristics, establishing when the centralisers in the groups were smooth. Non-smoothness was only found in very small characteristics, even where there are infinitely many orbits. This led to a question on when all centralisers of elements in a Z-defined representation would be smooth if the characteristic were large enough. With my co-authors we managed to prove this using the Lefschetz principle from model theory applied to Gröbner bases, which gave rise to what we call ‘d-bounded Hopf quadruples’. I’ll explain some of this.”

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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.”

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