Research Interests - Mathematical Logic and Group theory
| My main interests are:
axiomatic set theory (principally independence questions concerning the axiom of choice),
and infinite permutation groups,
though all three are linked.
I have studied in depth some specific infinite permutation groups, mainly the automorphism groups of the random graph, and the rationals as ordered set; also the group of homeomorphisms to itself of the rational numbers as a topological space. I showed for instance that the automorphism group of the random graph is simple, and classified the cycle types of its elements. For the rationals as an ordered set, I proved that it has the 'small index property': any subgroup of index strictly less than 2 to aleph zero contains the pointwise stabilizer of a finite set; a similar result applies to the homeomorphism group of the rationals as a topological space.
Related to the above I defined a notion of 'generic' automorphism, and studied circumstances under which generic, or 'locally generic' automorphisms exist.
In set theory I showed that the boolean prime ideal theorem does not follow from the order-extension principle (joint work with U. Felgner).
In model theory I have studied the elementary theory of quotients of infinite symmetric groups (joint work with S. Shelah). This is related to earlier work of mine on outer automorphisms. Specifically, I showed that the quotient of the group of all order-preserving permutations of the real line by those whose support is bounded above, has no outer automorphisms. This is achieved by an interpretation of enough of the underlying structure in the group to recover its final segments. Similar results apply to certain of the quotients of symmetric groups.
In answer to a question of Azriel Levy, I made an extensive study of 'amorphous' sets, being those which, though infinite, cannot be written as the disjoint union of two infinite subsets. Such sets can only exist if the axiom of choice is false. I provided a 'classification' of many of these. They may be regarded as a set-theoretic analogue of strongly minimal sets in model theory. Recent joint work with Creed treats similar issues for 'o-amorphous' sets (any subset is a finite union of intervals) and 'quasi- amorphous sets' (uncountable sets which cannot be written as the disjoint union of two infinite sets).
My former research student, Richard Warren, made an extensive study of so-called 'cycle-free partial orders', intuitively partial orders in which there is a no cycle, any two consecutive elements of which are comparable (though the correct definition is more complicated than this). His work is published in the Memoirs of the American Mathematical Society volume 614, 1997. Joint work with him, Creed, and Droste, has resulted in a number of papers in which these structures are further investigated. We now have a complete classification of all the ('sufficiently transitive') countable cycle-free partial orders, and know which of the corresponding automorphism groups are simple. The axiomatizability of the class of cycle-free partial orders has been studied, and work currently in progress seeks to determine to what extent these structures and their automorphism groups can be distinguished up to elementary equivalence.
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