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Research Interests |
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| In group theory, the notion of
the index of a subgroup (given by counting the number of cosets of
that subgroup) is of fundamental importance. In particular, there
are many theorems showing how similar groups are to their
subgroups of finite index, in terms of the algebraic, combinatorial and
computational properties that they posses. One of the central aims of
my current project "Finiteness
Conditions and Index in Semigroups and Monoids" (funded by the EPSRC)
is to develop a correspondingly powerful theory of index in the more
general context of semigroup theory, which will have
applications in the study of finiteness conditions of semigroups.
Ultimately this research will strengthen connections between group theory and semigroup theory.
Currently in this area I am working on projects with A. Malheiro (Lisbon), N. Ruskuc (St Andrews), M. Kambites (Manchester), S. J. Pride (Glasgow), and C. Carvalho (Lisbon).
Before taking up my fellowship I was a Research Fellow at the University of Leeds, working with J. Truss and D. MacPherson, on an EPSRC funded project entitled "Homogeneous structures, bipartite graphs, and partial orders". The main aim of this project was to classify various families of sufficiently symmetric relational structures: usually graphs, digraphs or posets. The main interest is in infinite structures, especially those that are countable. In practice a level of symmetry is achieved by imposing a transitivity condition on the group of automorphisms of the structure. There are various strengths of symmetry one may impose. The strongest of these is the notion of ultrahomogeneity which is a concept that was introduced by Fraissé (1954). A structure is ultrahomogeneous if any isomorphism between finite substructures extends to an automorphism. This property is very strong and a number of important classifications of ultrahomogeneous families are known including: countable graphs (Lachlan and Woodrow (1980)), countable partial orders (Schmerl (1979)), countable homogeneous tournaments (Lachlan (1984)), and countable digraphs (Cherlin (1998)). Other important less restrictive symmetry conditions include: k-arc-transitivity, distance-transitivity, k-set-transitivity, and k-CS-transitivity. In this area, and am currently working on projects with J. Truss and D. MacPherson (Leeds), and also R. Moller (University of Iceland), C. E. Praeger (University of Western Australia) and G Royle (University of Western Australia). |
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| [ University of St Andrews ] [ Mathematics ] [ Agebra and Combinatorics Group] [CIRCA] | ||
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