This is the text and invited lecture given at the 20th Annual Meeting of the Australian Mathematical Society in Perth, May 1976.

In this talk I want to give a survey of some of the areas of research in ordinary differential equations. In view of the vastness of the field, the limited thime available and may own inadequacies I will restrict attention to problems in which I have been involved in some way, although I won't be speaking about my own work. Even with this restriction it will be necessary to be somewhat superficial. However, in spite of all these drawbacks, I thought it would assist communication among mathematicians simply to have an idea of what other people are doing. Since this is my purpose I will not mention the quite mild smoothness hypotheses in the statements of some results.

Necessary and sufficient sonditions are given for the existence of the group and Drazin inverses of bidiagonal and triangular Toeplitz matrices over an arbitrary ring.

A sufficient condition, which may also be necessary, for the regularity of the direct product of two finite regular p-groups is obtained. In an attempt to decide the necessity of the condition, we look at two related questions. Firstly, does the regularity of the direct square of a regular p-group imply the regularity of all the finite direct powers; secondly, is the join of two regular varieties regular? Fainally, the results obtained are used to give some precise answers in special cases.

The set of bounded elements of a unital l.m.c. algebra is characterised as the union of certain naturally defined normed subalgebras, and an analogous characterisation is given for algebras of quotient-bounded operators on a locally convex space. Pseudocomplete l.m.c. algebras are characterised in terms of the completeness of these subalgebras, and the equivalence of this condition with the pseudocompleteness of the quotient-bounded operator algebras established. The scalar multiples of the identity in a unital l.m.c. algebra are characterised in terms of certain boundedness conditions.

It is shown that (n being an integer) any non-trivial finite linear space with n2- 1 points, all of degree at most n+1, is embeddable in a finite projective plane of order n. This generalizes a theorem of Bose and Shrikhande and settles the unsolved case n = 6.

If G is a group, then K(G) is the set of commutators of elements of G. C is the class of groups such that G′ = K(G) is the minimal cardinality of any generating set of dG. We prove: Theorem A. Let G be a nilpotent group of class two such that G' is finite and d(G′) < 4.Then G < G.

Theorm B. Let G be a finite group such that G′ is elementary abelian of order p3. Then G ∈ C.

Theorem C. Let G be a finite group with an elementary abelian Sylow p-subgroup S, of order p2, such that S ⊆ K(G). Then S ⊆K(G).

The symbol IG(X) denotes the inverse semigroup, under composition of functions, of all homeomorphisms between open subsets of a T1 topological space X. The first result is that two such semigroups IG(X) and IG(Y) are isomorphic if and only if the spaces X and Y are homeomorphic. Ideals of IG(X) are next examined and it is shown that for many spaces X the semigroup IG(X) is 0-simple. We also look at congruences on IG(X); one result is that we determine a congruence which in many instances is the largest proper congruence on IG(X).

Congruence-free right simple semigroups with idempotents are simple groups or have order 2. It is shown in this paper that there exist congruence-free simple semigroups without idempotents. Some properties of congruences on right simple semigroups, expressed mainly in terms of their unitary and left unitary subsemigroups, are determined. These are used to provide characterizations of congruence-free right simple semigroups.

In this paper we present a characterization of PSL(2,7) by a condition different from that given in our previous paper.

If ε is an endomorphism of a finitely generated residually finite group onto a subgroup Fε of finite index in F, then there exists a positive integer k such that ε is an isomorphism of Fεk. If K is the kernel of ε, then K is a finite group so that if F is a non trivial free product or if F is torsion free, then ε is an isomorphism on F. If ε is an endomorphism of a finitely generated resedually finite group onto a subgroup Fε (not necessatily of ginite index in F) and if the kernel of ε obeys the minimal condition for subgroups then there exists a positive integer k such that ε is an isomorphism on Fεk.

The Emden-Fowler equation x″(t) + a(t)|x|γ sgn x = 0, t ≧ 0, is said to be in the sublinear or superlinear case according to whether γ < 1 or γ > 1. Conditions on a(t) are given to ensure local uniqueness of solutions in the sublinear case and continuability of solutions in the superlinear case. Boundedness of solutions is also studied.