Chris Ash died in mid-February 1995. He was born on January 5, 1945 in Gorleston in the north of England. In 1963 he went up to St Edmund Hall, Oxford, to read mathematics, choosing mathematics over classics at a late stage of his schooling. After graduating with second class honours - a disappointment which lingered with him - he read for a D.Phil. with John Crossley. He did not like his supervisor's proposed research topic in mathematical logic, so looked for his own and worked on \aleph_{0}-categorical theories. A theory (set of sentences - axioms) is \aleph_{0}-categorical if any two countable models (interpretations) are isomorphic. All the known \aleph_{0}-categorical theories at that time were decidable - there is an algorithm to tell whether a statement is a theorem or not. He found an undecidable one. Glassmire (1971) also found one but Ash's example was simple and supremely elegant (1971).
The thesis, however, was not completed until Ash had followed Crossley to Monash, taking up a Senior Teaching Fellowship in 1970. He stayed at Monash, moving to Lecturer in 1973, Senior Lecturer in 1981 and Reader in 1986. In 1993 he was elected a Fellow of the Australian Academy of Science for his work in mathematical logic, semigroups and universal algebra.
His work had steadily but slowly expanded from the model theory he was doing, to universal algebra and, under the influence of Tom Hall and Gordon Preston at Monash, into semigroups.
In mathematical logic he was a pioneer in recursive model theory - the study of structures in which the basic relations are decidable. For a simple example of such a structure, take the rational numbers with the relation \leq.
Ash was inspired by Anil Nerode, who visited Monash several times from Cornell University. From the Ash-Nerode theorem (1981), Ash's work moved to ever greater heights. The proof of the Ash-Nerode theorem involved a ``priority'' construction, one of the simplest kinds, going just to level two in a certain hierarchy of undecidability. Ash was the first person to develop general theorems on priority constructions, giving conditions (in terms of some abstract relations) which guarantee success, and he did this for constructions at all levels of the hierarchy (finite and transfinite). The details are very technical and we refer the general reader to the annotated bibliography to appear in the Historical Records of Australian Science.
In some aspects of this work he was joined by Julia Knight from Notre Dame University, Indiana and for nine years (starting in 1986) they produced a constantly increasing flow of high-class papers. During this time he also started work on a book entitled ``Computable structures and the hyperarithmetical hierarchy''.
He was a real researcher. He was always interested in other people's questions; indeed, it was easy to get him to attack a problem. One could leave one with him one day and find he had solved it overnight. As for other people's theorems, he was interested to know the theorem but not the proof. That, he would work out for himself - and quickly too. A particular aspect of this approach was that it required a very deep and wide understanding.
The most spectacular demonstration of this came in 1990. He had been aware of the so-called `Rhodes Type II conjecture' since the early '80s. Just before the Monash semigroup conference in honour of Gordon Preston he verified the conjecture. The proof (see (1991a,b)) in this case came about through Chris's consideration of a substantial generalization of the original conjecture. As he said (in his notes on his papers): ``To prove the conjecture, I had to generalize it, and in (1991b) I generalized it even further to labelled directed graphs, for which the Rhodes conjecture becomes the case of a graph with one vertex''.
\midspace{15cm}\caption{ Christopher Ash, FAA }
This work - the verification of the conjecture - but more especially the techniques and depth of vision he brought to bear on it - spawned an industry, to judge by the flood of papers which subsequently appeared.
The Rhodes conjecture had been finally solved in an intense period of three days with sleepness nights. He never spared himself in the pursuit of mathematics.
But the approach was apparent to some of us in earlier work, where he had taken ideas from logic (homogeneous-universal models) and added a new dimension to semigroup theory (see (1979, 1980)).
After 1980 his main interest reverted to recursive model theory and the flow of papers increased. It still continues, thanks to the work of Julia Knight in finishing work they had begun together. The book, too, is now in progress as a joint work.
As a person he was much more difficult to get close to. He had an excessive modesty which tried to conceal his wonderful musical talents. He sang well (at one period with Schola Cantorum Oxoniense ), played piano, violin, viola, cello, recorder, clarinet, bassoon and tenor horn. It was difficult to get him to display these talents but splendid when he did. The last time I heard him he was playing the Schubert Fantasie with J. Alan Robinson, another fine pianist.
He ``couldn't stand the thought of another 15 years of this existence'' (1995), but he certainly left a remarkable amount of work to complete. There is a deep sadness he is not here to work with, to be with.
A fuller obituary will appear in Historical Records of Australian Science.
\item{} {C. J. Ash (1971) Undecidable \aleph_{0}-categorical theories. Notices Amer. Math. Soc. 71T-E10, 423.}
\item{} {C.J. Ash (1979) Uniform labelled semilattices. J. Aust. Math. Soc. (Ser. A) 28 (1979) 385-397.}
\item{} {C.J. Ash (1980) Embedding theorems using amalgamation bases. In Proc. Conference on Semigroups, Monash University, 1980. Eds. T.E. Hall, P.R. Jones, G.B. Preston, Academic Press, New York, 167-176.}
\item{} {C.J. Ash and Anil Nerode (1981) Intrinsically recursive relations. In Proc. Conference on ``Aspects of Effective Algebra'', Monash University. Ed. J. N. Crossley, U.D.A. Book Co., Steel's Creek, Victoria, Australia, 26-41.
\item{} {C.J. Ash (1991a) Inevitable sequences and a Proof of the ``Type II Conjecture''. Monash Conference on Semigroup Theory, 1990. Eds. T. E. Hall, P. R. Jones, J. C. Meakin, World Scientific, Singapore, 1991, 31-42.}
\item{} {C.J. Ash (1991b) Inevitable graphs: A proof of the type II conjecture and some related decision procedures. Int. Journal of Algebra and Computation 1, 127-146.}
\item{} {Christopher J. Ash (1995) Special issue of Monash University Mathematics Department Logic preprint series. }
J. N. Crossley with the assistance of Julia Knight