PhD Mathematics · University of Cambridge · 1964
office Surrey 14-280
My interests lie in Mathematical Logic. The topics in Model Theory which interest me are: stability theory, homogeneity in the sense of Fraissé, and the connections between model theory and the theory of permutation groups. Of particular interest is the problem of classifying countable homogeneous structures over a finite relational language L. In this context an L-structure M is calledhomogeneous if any isomorphism between finite substructures of M extends to an automorphism of M. A class S of finite L-structures is called an amalgamation class if it is closed under isomorphism and substructures, and for any embeddings f, g of M into P, Q (in S) there exist R in S and embeddings f* of P into R and g* of Q into R such that the composite maps f*f, g*g have the same restriction to M. For fixed L there is a natural 1-1 correspondence between amalgamation classes and the isomorphism types of countable homogeneous structures (the Fraissé correspondence). The crucial question is to what extent the countable homogeneous structures can be classified. The best result to date is that there is a classification of countable simple directed graphs which are homogeneous (due to Cherlin). Much more remains to be done.
In Recursion Theory I am concerned with problems involving recursively enumerable sets and related topics. Of particular interest are: the lattice of recursively enumerable sets and the upper semilattice of enumeration degrees.