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THE AUSTRALIAN NATIONAL UNIVERSITY

SCHOOL OF GENERAL STUDIES

FACULTY OF ARTS

DEPARTMENT OF PURE MATHEMATICS

ANN"VAL REPORT 1966

Staff:

Full time Professor Hanna Neumann, D.Phil., D.Sc.(Oxon). Senior Lecturer M.F. Newman, M.Sc.(Syd), Ph.D.(Manc). Lecturers A.R. Jones, M.A., Ph.D.(Melb).

D.H. H. Held, Dr.phil (Frankfurt) (resigned 7 August 1966) M.A. Fard, B.Sc(Melb), B.A., Ph.D. (A.N.U) (since 6 January 1966) W.L. Steiger, S.M. (M.I.T.) (since ll:. January 1960) C. Christensen, M.Sc., Ph.D.(Manc) (since 15 September 1966)

Temporary Lecturer N.D. Gupta, B.A.(Kashmir), Ll.B., M.A.(Alig) Ph.D. (A. N. U. )

Senior Tutor Patricia. J. Gaudry, B.Sc.(Q 1 ld) (resigned 12 August 1966) Temporary Tutor J.H. Coates, B.Sc. (A.N.U.) (1 January 1966 - S August 1966) Tutor N.D. Porter, B.Sc.(Melb) (since 15 August 1966) Secretary Mrs. F. Wickland.

Part-time Mr. M. S. Brooks Mr. R.A. Bryce Mr. C.K. Cheong Mr. P.J. Cossey (until 6 August Mr. T.M. Gagen (until 6 August

1966) 1960)

Mr. Mr. Mr. Mr. Mr.

A.G.R. Stewart P. W. Aitchison A. Shafaat (since 5 September 1966) P. Pentony (since 5 September 1966) I.W. Wright (since 5 September 1960)

As the list of staff shot·JS, we have at no time during the year been understaffed; but the serious lack of balance between senior and junior appointments causes a heavy load of responsibility and work to fall on the few senior members of the department. Owing to the world shortage of mathematicians, this situation is likely to continue and we must be grateful if, as in this past year, we succeed in attracting well qualified young men and are able to enlist useful temporary help.

Under these conditions staff changes are bound to be frequent. Dr. Held left for Monash to continue his \Jork with Professor Janko; Dr. Hard joined the department upon completing his Ph.D., Mr. Steiger came from the United States, and Dr. Christensen from the Manchester College of Science and Technology. Mrs. Gaudry left for Paris whereher husband holds a post-doctoral fellowship and Mr. Porter took her place as tutor. Our own graduate of last year, John Coates, joined us as a tutor for two terms until he, too, left for Paris, to work under Professor Serre at the Sorbonne.

Teaching In the light of our experience of new courses in 1965, we revised

to some extent the second and third year units for the pass degree before the start of the year. We nou believe to have achieved a much better approximation to our aim, a pass course of substantial mathematical content, yet within the capabilities of as many students

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as we can hope to attract to taking a full major in pure mathematics. Several of these students have availed themselves of the possibility to take part in some of the uork provided for Honours students; they will now either attempt to join the honours class by means of further additional work or to enrol for a qualifying course towards a Master's degree. In the honours classes we had again a number of quite outstanding students, and again we are grateful that these students were able to benefit from visiting appointments to the Department of Mathematics in the I.A.S: Dr. Smithies (Cambridge, U.K.), Dr. Mary Rees (Reading, U.K.), and Professor E. Hille (University of California), gave courses of lectures, in successive terms, that were attended by many of our students. Moreover two fourth year students and one second year student took an active part, and held their own successfully, i n a quite advanced study group for research scholars in the Institute. The benefit our best students derive from being welcome to many of the activities in our sister department in the I.A.S. is immeasurable.

Table of enrolments and results . Examinations

i Enrolments Total No. Total No. CR D RD entered of passes

Gen. Math. 66 4.1 35 5 - -Pure Math. I Pass ll2 92 72 20 1 -Pure Math. I Hons. 27 25 25 7 ll 6

Pure Math. II Pass 62 46 39 5 - -Pure Math. II Hons. 28 16 16 3 5 6

Pure Math. III Pas s 24 22 21 2 3 -Pure Math. III Hons. 8 6 6 1 2 3

Pure Math. IV 6 IIAtl I:5

All our fourth year student s this year specialized in Pure Mathematics. One student has completed successfully a qualifying course for the M.Sc., Mr. Porter was a full-time student for the M.Sc . , and there are three part-time students working for the M.Sc. During the year four students were awarded Master's degrees, two in the Faculty of Science (A.G.R. Stewart and Patricia Gaudry), t wo in the Faculty of Arts (N . Hu Bong and A. Majeed).

Research and Research Supervision Group theory remains ~he most strongly represented research interest;

of the new members of the department both Dr. Ward and Dr. Christensen are group theorists, and lively cooperati on with the group theorists in the I.A.S. continues .

All the more welcome was the addition to the department of an analyst: Mr. Steiger works on the Moment Problem, and related topics, but he also developed during the year a keen interest in axiomatic geometry . Dr. Jones' work on Hamiltonian differential equations is just being submitted for publication. He, Dr. Newman and myself were again kept busy with the supervision of research students, both Masters students and Ph .D. students enrolled in the Institute, while by way of reciprocation, Professor Mahler supervised both Mrs. Gaudry and Mr. Bong and Dr. Edwards helped with the supervision of Mr. Porter. We are most grateful to both of them for their assistance.

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Three of our fourth year students spent the major part of the long vacation preceding their final honours year working in the department, either as vacation scholars or as undergraduate assistants. Keith Hutchison did valuable work on generalized functions under Dr. Newman's supervision, Andrew Struik tackled a problem in group theory under my supervision, and George Maxwell -nominally under my supervision - prepared a course of lectures that I had asked him to offer us , the staff, in order to prepare us for the substance of his fourth year essay. All three students' achievements amply justified our belief in the wisdom and the value of giving time to keen students during their last vacation.

Other activities In May, Drs. Gupta, Newman, Hard and myself attended the

Annual Meeting of the Australian Mathematical Society in Perth, ~.A. I am grateful to the A.N.U. for making it possible for me to

attend the International Congress of Mathematicians in Moscow during the August vacation. The chance to report to normally far distant colleagues on work going on here, and to establish personal contact with colleagues long known to us through correspondence made my attendance at the Conference useful and fruitful. Immediately after this Dr. Ne\·Man and I attended the Inaugural Conference of the Australian Association of Mathematics Teachers in Melbourne, Dr. Newman as one of Canberra 1 s two delegates to the Coun­cil and I as one of the Vice-Presidents of the new Association.

The meeting r.;as strenuous, but worth while and there is hope that the work of the new association will be beneficial to the teaching of mathematics in Australia.

Throughout the year Dr. Newman was again available to teachers for help and advice with the fifth and sixth year syllabuses fo~ schools. Besides doing similar work in the early part of the year, I acted as External Examiner in Pure Mathematics for the University of Newcastle.

List of Publications

Coates, J.H.

Gupta, N.D.

On the algebraic approximation of functions. I, II, II.I Koninkl. Nederl. Akad. Wetensch Proc,A 69(1966), l(.21-l:.3l:., 435-l:.43, 4l:.9-li,61.

0 On Commutation Semigroups of a group. J. Austral. Math. Soc. ~(1966) 36-45.

0 Some group-laws equivalent to the commutative law. Arch. Math. lZ.(1966) 97-102.

0 Groups with Engel-like conditions. Arch. Math. lZ.(l~ J 6) 193 ··199.

Gupta, N.D. and Newman, M.F. On metabelian groups. J. Austral. Math. Soc. ~(1966), 362-368

Jones, A.R. On symmetry in periodic solutions of Hamiltonian systems. J. Aust. Math. Soc. i(l965), 463-468 .

Neumann, Hanna. Probability, Pamphlets for Teachers No.2, Puhl. by the Canberra Math. Ass. 1966.

r/J Based on work done while a member of the Department of Mathematics, Institute of Advanced Studies.

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Newman, M.F. Another non-Hopf group. J. London Math. Soc. 41(1966) 292. /

_K_o_v_a_c_s..,, __ L_._G_._; __ a.-n_d_""N_e_v.m.-_a_n ..... ,_M_._._F. Minimal verbal subgroups. Proc. Cambridge Philos. Soc. 62(1966) 3li.7-350.

Cross varieties of groups. Proc. Roy. Soc. London . Ser A 292(1966) 530-536.

On cri tical groups. J. Aust . Math. Soc. 6(1966) 237-250.

Gupta, N.D. and Newman, M.F. see under Gupta, N.D.

Stewart, A.G.R.

Ward, M.A.

On the class of certain nilpotent groups. Proc. Roy. Soc . London. Ser A 292(1966) 37l~ -379.

0 Remarks on a Theorem of Fuchs. Acta Mathematica Acad. Sci. Hung . , l.Z.(1966) 5-8.

Hanna Neumann.

0 Based on work done while a member of the Department of Mathematics, Institute of Advanced Studies.

1 A member of the Department of Mathematics, Institute of Advanced Studies.