Number 88 August 2003 NEWSLETTER OF THE NEW ZEALAND MATHEMATICAL SOCIETY (INC.) Contents PUBLISHER’S
NOTICE ISSN 0110-0025 THE NATIONAL COMMITTEE FOR MATHEMATICS (1967-1996) Author's note: The rapid approach of my retirement has prompted me to delve into some of the more dust-covered corners of my office. In one of these I rediscovered the archives of the National Committee for Mathematics, which I chaired at the time of its demise in 1996. Since many newer members of the NZMS may not have heard of this body, which was among other things involved in the founding of the NZMS in 1974, I thought a few historical notes might be of interest. In terms of documentation, the story of the New Zealand National Committee for Mathematics begins with a letter dated 7 February 1966 from the secretary of the International Mathematical Union (IMU) to Professor John Kalman of the University of Auckland, in response to a request for information about the possibility of New Zealand joining the IMU. It was suggested that New Zealand should apply for Group I membership (the same group as Australia), thereby paying the lowest possible annual subscription (Member countries of the IMU fall into five membership groups; the higher the membership group number, the more votes a member country has and the higher its annual subscription is. A country may apply for membership in the group of its choice; approval or rejection of the application is in the hands of the member countries.). It was also explained that adherence to the IMU would be through a committee of the Royal Society of New Zealand (RSNZ), to be called the National Committee for Mathematics, the composition of which would have to be reported to and recognized by the IMU before New Zealand's membership of the IMU could become effective. The matter was discussed at the first two New Zealand Mathematics Colloquia (in 1966 and 1967), and it was decided to recommend to the RSNZ that New Zealand should proceed with an application to join the IMU. Consequently the RSNZ appointed a National Committee for Mathematics (NCM) consisting of Professors Simon Bernau (Otago), John Butcher (Auckland), Brian Hayman (Massey) and Gordon Petersen (Canterbury), Mr Doug Harvie (Victoria), Dr Hamish Thompson (DSIR) and the International Secretary of the RSNZ (ex officio). The first meeting of the committee took place in Wellington on 15 December 1967. Hamish Thompson advised members of the date and time by telegram! At this first meeting Simon Bernau was elected as chairman, and a constitution was agreed which included among the objects of the NCM the following:
The constitution also laid down that the NCM should comprise 9 appointed members in addition to the International Secretary of the RSNZ. This constitution was approved by the Council of the RSNZ on 15 February 1968, and subsequently three more members were appointed: Professors Desmond Sawyer (Waikato) and Cecil Segedin (Auckland), and Mr David Goldsmith of the Department of Education, Christchurch. The first matter which the RSNZ brought to the attention of the NCM was a shortage of money! The NCM had asked that funds be made available for its members to travel to the May Colloquium, at which a meeting of the NCM would be held. The response from the RSNZ was that "... the Society will have to live through the next financial year on a smaller budget owing to the restrictions in Government finance. ... as your Mathematics Colloquium is primarily a University activity, Council would expect that the travel expenses of Members of the National Committee would be met from University sources ... I therefore think that the answer in this case is negative'' (G W Markham (Executive Officer, RSNZ) to S J Bernau, 26 February 1968.) At last in July 1968 the NCM was given something to attend to other than its own internal workings. Following an initiative by the National Committee for Geological Sciences, all National Committees were asked to "examine the situation in New Zealand'' and report on any gaps there might be in University teaching in their particular fields (Memorandum from RSNZ to National Committees, 15 July 1968.) Lack of travel funds meant that the NCM could not meet to discuss this, so members were asked to send comments to the chairman, who reported these views to the RSNZ in a letter which is worth quoting in full: (S J Bernau to RSNZ, 7 November 1968.)
Meanwhile New Zealand's application for membership of the IMU was proceeding through the necessarily formalities, and was finally approved on 15 January 1969. The Secretary of the IMU concluded his letter of notification with the words "... I have received many letters warmly accepting New Zealand as a new member of IMU, and no rejection. Therefore, your application is accepted, and I have the honour of welcoming New Zealand as the 42nd member of IMU, in Group I' (Secretary, IMU to Executive Officer, RSNZ, 17 January 1969.) At the NCM meeting held on 15 May 1969, it was decided that the NCM would circulate a list of research topics and supervisors available in mathematics in New Zealand (This function was later taken over by the NZMS.) Simon Bernau resigned as chairman, as he was leaving New Zealand to take up an appointment at the University of Texas. Brian Hayman took over the chair, and immediately faced the task of gathering data on the names of New Zealand mathematicians to be included in a new (fourth) edition of the World Directory of Mathematicians (the final list comprised 27 names, compared to 11 in the previous edition). Efforts to find a delegate to represent New Zealand at the 1970 General Assembly of the IMU in France seemed likely to fail because the inadequacy of the financial support available from the RSNZ, until John Kalman (who would in any case be attending the ICM in Nice) agreed to represent us. The NCM found little to do over the next few years, apart from recording changes in membership. During 1970 David Goldsmith resigned, and the two vacancies on the NCM were filled by Miss Margaret Ryan (Department of Education, Wellington) and Professor Teddy Zulauf (Waikato). The following year Brian Hayman stood down as chairman and Doug Harvie took over that role. Then in 1972 a constitutional requirement for one-third of the members to retire came into operation; those chosen (by ballot) were John Butcher, Doug Harvie and Desmond Sawyer. They were replaced by Professors John Kalman (Auckland), David Vere-Jones (Victoria) and Bill Davidson (Otago). Things livened up when the RSNZ received and forwarded to the NCM a letter from Professor C J Seelye, Head of the Mathematics Department at Victoria, conveying this recommendation from a meeting of staff of the Department: "That recognising the time might be appropriate for the formation of a New Zealand Mathematics Society this meeting requests the National Committee for Mathematics to submit to the 1973 Mathematics Colloquium General Meeting a detailed proposal for the formation of such a society'' (C J Seelye to RSNZ, 4 September 1972.) At the request of Cecil Segedin, who was then acting chairman of the NCM (For health reasons Professor Segedin did not in the end accept nomination as chairman and Gordon Petersen was appointed instead.), David Vere-Jones presented the NCM with a very detailed proposal including a draft constitution for a New Zealand Mathematical Society. The reaction was very mixed, several members of the NCM being quite strongly opposed to the formation of such a society, largely on the grounds that the Colloquium was working very well and there were dangers in trying to incorporate it into a larger organisation which was unlikely to bring any additional benefits. As it happened, the NCM meeting held during the May 1973 Colloquium did not muster a quorum, and so the matter was passed on to the Colloquium AGM without any recommendation from the NCM. That meeting, as we all know, voted to support the formation of a New Zealand Mathematical Society, which held its inaugural meeting the following year. In 1973, too, came a request from the IMU for data for the next edition of the World Directory of Mathematicians (the New Zealand listing increased to 51). Cecil Segedin resigned from the NCM and was replaced by Mr Steve Kuzmicich (Department of Statistics, Wellington). At its May 1974 meeting the NCM expressed its concern that the current list of Fellows of the RSNZ did not do justice to the international standing of leading New Zealand mathematicians. Following correspondence with the RSNZ, it was suggested that the best solution would be for the NZMS to become a Member Body of the RSNZ; this would give it the right to nominate candidates for election as Fellows. This was duly done; the application from the NZMS to become a Member Body was approved in October 1975. In 1975, largely on Gordon Petersen's initiative, the NCM took up the matter of New Zealand becoming a member of the International Committee on Mathematical Instruction (ICMI). Our application was successful and New Zealand became a member in 1976. A Mathematics Education Subcommittee of the RSNZ was set up to liaise between the ICMI and relevant groups in New Zealand. At this time the archives of the NCM become somewhat sketchy, but it seems that somewhere between the 1975 and 1976 meetings the Committee lost Brian Hayman and Hamish Thompson and gained Professor Graham Tate (Massey) and Dr Mark Schroeder (Waikato). Then the RSNZ decided to reorganise its National Committees and reduce their size to six members. The NCM effectively dissolved in 1977 and did not meet that year; then it reappeared in 1978 with a reduced membership comprising Dr Robert Davies (DSIR), Professor John Kalman (Auckland, chairman), Mr Gordon Knight (Massey), Mr Steve Kuzmicich (Department of Statistics, Wellington), Dr Mark Schroeder (Waikato) and Professor Brian Woods (Canterbury); Steve Kuzmicich retired from the NCM in 1979 and was replaced by Dr Gillian Thornley (Wellington Polytechnic). The new Committee immediately had to deal with another updating of the World Directory of Mathematicians. The exponential growth of the New Zealand mathematical community continued, there now being 109 New Zealand mathematicians listed. Dr Graeme Wake, who had represented New Zealand at the General Assembly of the IMU held in Helsinki in 1978, reported back with a recommendation that New Zealand apply to move from Group I to Group II, as Group I was more for developing countries and our mathematical standing made Group II more appropriate for us. At its 1979 meeting the NCM resolved that this be recommended to the RSNZ and that a case be prepared jointly with the NZMS. This was done and a detailed case was ready early in 1980. However, the RSNZ, at its March 1980 Council Meeting, declined to allow this application to proceed because its limited financial resources meant that it could not afford the increased subscription. During 1980 Robert Davies, John Kalman and Mark Schroeder all retired from the NCM and were replaced by Professor John Deely (Canterbury), Dr David Gauld (Auckland) and Mr Ken Jury (Ruakura); Brian Woods took over as chairman. In 1981 the World Directory of Mathematicians again became due for a new (seventh) edition. The New Zealand list continued to grow, though more slowly, reaching 128 this time. Since then the rapid growth in numbers of the New Zealand mathematical community has ceased, and in fact the last edition of the World Directory which the NCM was involved with (the tenth edition, which appeared in 1994) contained only 92 New Zealand names. From 1982 to 1986 there is a gap in the NCM archives. The brief annual reports which appeared in the Proceedings of the RSNZ for those years show that in fact very little was going on other than the preparation of the World Directory listing. Members came and went; the chair moved from Brian Woods to David Gauld to John Butcher to Gloria Olive. When the NCM awoke from its gentle slumber towards the end of 1986, it consisted of Dr Gloria Olive (Otago, chair), Dr Mike Carter (Massey), Dr Rod Downey (Victoria), Dr Murray Jorgensen (Waikato), Professor Roy Kerr (Canterbury) and Dr Wilf Malcolm (Vice-Chancellor, Waikato). Gloria Olive celebrated her accession to the NCM chair by writing a brief article about the NCM in the August 1986 Newsletter of the NZMS. The same issue of the Newsletter carried an article by Professor Derek Holton (Otago) urging the sending of a New Zealand team to the 1988 International Mathematical Olympiad to be held in Canberra. Derek discussed the activity that was already taking place and concluded by saying that moves were afoot "via the National Mathematics Committee'' to give some sort of formal status to the group of people involved. In fact Professor Gordon Hookings (Auckland), another enthusiast for the Olympiad, had written to Gloria Olive on 9 May 1986 asking the NCM to give its approval to the establishment of a New Zealand Mathematical Olympiad Committee. Gloria responded conveying the whole-hearted support of the NCM, and the Olympiad Committee was duly formed. The outcome of the first meeting of the Olympiad Committee was a letter to the NCM asking it to "consider a proposal that the New Zealand Mathematical Olympiad Committee become a sub-committee of the National Committee for Mathematics''\footnote{G A Hookings to G Olive, 21 October 1986.}. The NCM duly contacted the RSNZ to find out what they thought of this idea. But the RSNZ felt that the connection between the Olympiad and the IMU was tenuous, and proposed instead that the Olympiad Committee should be affiliated to the NZMS\footnote{S M Usher (Assistant Executive Officer, RSNZ) to G Olive, 3 August 1987.}. The Council of the NZMS in due course (May 1990) agreed to this. Meanwhile our team attended the 1988 Olympiad, being placed 34th out of 49 and gaining one individual silver medal—the first of many successful Olympiad entries. A unique event in 1988 was the only joint meeting of the Australian and New Zealand National Committees for Mathematics, which took place during the 1988 Mathematical Sciences Congress held in Canberra. A report of this meeting appeared in the NZMS Newsletter, December 1988. There was much goodwill and extensive discussion on the desirability of fostering closer cooperation between New Zealand and Australian mathematics-related societies, but nothing much seems to have come of all this. However, some members of the New Zealand NCM were persuaded by the example of the Australian body that the New Zealand NCM should take a more active role in New Zealand mathematics instead of largely confining itself to IMU matters. In mid-1989 Gloria Olive retired from the NCM and Mike Carter took on the role of chairman. Other members to retire then or a little later were Wilf Malcolm, Murray Jorgensen and Roy Kerr; the new faces on the NCM were Dr Peter Lorimer (Auckland), Dr Graham Weir (DSIR), Dr Charles Little (Massey) and Professor Derek Holton (Otago). In June 1989 the NCM received a letter from David Gauld advising that the meeting of Heads of Mathematical Sciences Departments held during the May Colloquium had decided to set up a working party on the funding of the mathematical sciences within New Zealand universities. The views of the NCM were sought, and if it saw fit it was asked to nominate two representatives to this working party. The NCM was supportive, and nominated Mike Carter and Wilf Malcolm to represent it on a body which also included representatives of the NZMS, Heads of University Mathematical Departments, the Operational Research Society of New Zealand and the New Zealand Statistical Association. After much discussion the working party presented its report in 1990. From the minutes of the 28th Council Meeting of the NZMS (23 November 1990; see NZMS Newsletter, April 1991) we learn that the report had been circulated to all Heads of Departments, and that "few responses had been received, but were unanimously negative, observing that the report appeared unfocussed, especially on providing reasons for extra funding, and had a whinging tone to it''. The working party report seems not to have got much further, but subsequent NZMS Council minutes show that the issue itself certainly did not (and probably never will) go away. The major initiative of the NCM during 1990 was the push to create a New Zealand subcommission of the ICMI. The functions of such a subcommission would be to maintain liaison with the ICMI and to promote mathematics education within New Zealand. The NCM consulted with the NZMS, the NZAMT and the NZSA and received support from them all. The RSNZ was also supportive and the subcommission was duly established towards the end of 1991 with Megan Clark (Victoria) as convenor. In 1994 the Secretary of the IMU requested New Zealand to consider upgrading its IMU membership to Group II on the grounds of a substantial jump in the quality of mathematical activity in this country; but once again financial considerations obliged the RSNZ to reject the suggestion. The RSNZ decided in 1995 that the work of the National Committees should be incorporated within its Standing Committees. It established a Standing Committee on Mathematics and Information Sciences, within which Marston Conder took over responsibility for liaison with the IMU. The NCM accordingly ceased to operate in June 1995 and was eventually officially disbanded as from December 1996. Over the years the NCM attracted some criticism for being unrepresentative and largely self-electing. It was perhaps not entirely clear, either to itself or to anyone else, what its precise role within New Zealand mathematics was supposed to be. But its formation was an essential step in the incorporation of New Zealand mathematics within the international mathematical community, and it was involved in some important initiatives over the years. As well, it quietly continued with routine matters such as the updating of the New Zealand listing in the World Directory of Mathematicians, the appointment of delegates to represent New Zealand at general assemblies of the IMU and the ICMI, and the nomination of New Zealanders for executive positions within these organisations (though our nominations were never successful). At the end it had indeed outlived its usefulness and the decision to disband it was a sensible one accepted as such by all concerned, but its contribution to New Zealand mathematics over the years should not be forgotten. David Benney
David J. Benney was born in New Zealand and spent most of his early years in Wellington. He was educated at Wellington Boys' College where he excelled in languages! His father, Cecil Benney, was Undersecretary of Mines for many years. Dave Benney graduated from Victoria University of Wellington with a Bachelors in Science (1950) with first class honours in Mathematics and a Master's in Science in (1951). Subsequently he went to Emmanuel College, Cambridge University where he took the Mathematical Tripos. He returned to New Zealand where he taught at Canterbury University College in 1955-56 and then went to the Massachusetts Institute of Technology (1957-59) where he earned a Ph.D. in Mathematics, studying with C. C. Lin. He has been at MIT ever since, holding posts of Instructor, Assistant Professor, Associate Professor, Professor (since 1966) and Head of the Department (1989-1999). Dave met his wife Elizabeth at Canterbury when she was finishing her degree. They were on a group ski trip to Temple Basin in 1955 and were married in 1959. Elizabeth Benney is well known for her remarkable accomplishments in the equestrian world and as a published author. They purchased an abandoned 77 acre farm in Upton, Massachusetts and brought it back to life. They built stables, a house, restored fields, installed fences etc. and made it into a beautiful horse farm. While this was in progress they lived in the original hundred year old farm house that was in very rough shape! Dave and Liz have three children and two grandchildren. They frequently visit New Zealand. There's a lovely story which illustrates Dave Benney's understated character. Dave worked in the garden at the N.Z. Government House during the summer holidays. Sometimes the Governor General, Sir Bernard Freyberg, would walk around and talk about the tomatoes, strawberries etc. with Dave. In the meantime, Dave was nominated for a prestigious scholarship and Freyberg was to be the interviewer. On the day of the interview, Dave worked in the garden and ran home at lunchtime to change into a suit. When the interview started, Freyberg said: "Haven't I met you? You look familiar!" DJB never let on he was his gardener! In 2002 volume 108 of {\it Studies in Applied Mathematics} was dedicated as a tribute to Benney in recognition for his groundbreaking research contributions to applied mathematics and for being Managing Editor of "Studies" for over 30 years. The volume consists of an expository article by C.C. Lin and eight papers by colleagues and former students of David Benney. Benney's work covers a range of issues of wide interest in physical applied mathematics: basic problems in fluid dynamics including flow stability and transition to turbulence, fundamental phenomena in nonlinear wave motion and asymptotic analysis. A preface article "Research Contributions of David J. Benney" by M. J. Ablowitz, T.R. Akylas and C.C. Lin surveys many of his important research contributions. The present article summarizes some of the earlier review and puts into perspective some of Benney's influential work. The reader is encouraged to consult the above volume, articles and references for more information. His research work was often carried out in collaboration with his graduate students; he directed 18 Ph.D students. Such collaborations helped bring up younger generations many of whom made significant contributions of their own, thus amplifying the impact of his own effort. He is well-known at MIT as one of the best classroom lecturers at both the graduate and undergraduate levels. In collaboration with his colleague Harvey Greenspan, he co-authored an important calculus textbook which in addition to fundamentals encompasses both mathematical principles and the processes of mathematical modeling. This book combines the 'spirit' of both pure and applied mathematics. The past half century has been a period of rapid progress in understanding nonlinear phenomena, and the work of David Benney is central to this remarkable success. While his papers deal primarily with classical problems in fluid dynamics and nonlinear waves, Benney's work has had lasting impact in various other fields as well, including meteorology, oceanography, optics and plasma physics. Benney recognized that valuable insights associated with nonlinear phenomena can be gained by working with relatively simple generic evolution equations that capture the underlying physics in certain asymptotic regimes relevant in a variety of physical contexts. This approach to nonlinear problems has proved particularly fruitful, and several of Benney's papers have become true classics. Benney's work on hydrodynamic stability and transition deals with essentially parallel flows such as that in the boundary layer over a flat plate. Beginning with his doctoral thesis, "On the Secondary Motion Induced by Oscillation in a Shear Flow'', Benney made decisive contributions to the theory of transition to turbulence through his studies of the nonlinear interactions of linear instability modes. Benney understood that interacting wavetrains have much in common with coupled nonlinear oscillators, and that the asymptotic procedures (multiple scales, averaging, Poincare frequency shifts etc.) developed for treating oscillator problems could be adapted to handle interacting waves. The mathematical analysis used to study nonlinear interactions of unstable modes in shear flows was the basis of Benney's seminal work on nonlinear resonant interactions of gravity water waves. The spurious 'secular' growth of certain resonant wave components that had been found in earlier works on water waves was shown to be a manifestation of significant energy exchanges among waves that are members of resonant 'quartets'. This resonance mechanism has wide applicability in nonlinear dispersive wave systems. In 1964 Benney derived a system of equations approximating the dynamics of three-dimensional weakly nonlinear shallow-water waves and obtained a class of interacting solitary-wave solutions. Solitary waves and their interaction properties have been a topic of keen interest to researchers. Shallow-water waves are disturbances whose lengthscales are long compared to the depth, and different aspects of long-wave phenomena is a subject to which Benney returned periodically. The 1964 work, in particular, made clear why certain classical equations derived by mathematicians in the late 1800's by Boussinesq and by Korteweg and de Vries were actually generic approximations of weakly nonlinear-weakly dispersive wave phenomena. Hence, these equations applied in many different contexts. In subsequent work on weakly nonlinear modulated wavepackets in 1967 the celebrated nonlinear Schrödinger (NLS) equation was proposed as the canonical evolution equation governing the propagation of a slowly varying wavepacket envelope. Interestingly enough, the broad validity of the NLS equation was not fully appreciated at first—thirty five years ago, when the NLS equation was viewed as one of several proposed asymptotic approaches for studying the then recently discovered modulational (Benjamin-Feir) instability of nonlinear periodic water wavetrains. However, the NLS equation is now generally understood to be the proper evolution equation governing the envelope of weakly nonlinear wave pulses and, among other applications, has been at the centre of major recent technological advances in fibre optics. An important extension of the NLS equation to account for three-dimensional modulations of water wavepackets in finite depth was found in 1969. This work pointed attention to the coupling of the envelope with an induced mean-flow component which plays an important part in oblique instabilities of periodic wavetrains in water of finite depth. A coupled system of equations which governs the evolution of the wave envelope and the induced mean flow was derived. A special case of this system which is obtained in the limit of small fluid depth—sometimes also referred to as the Davey–Stewartson equations—has received considerable attention since it is an integrable 2+1 dimensional system and is a prototype model of nonlinear interactions between short and long waves. Benney made further significant contributions to the theory of long-crested nonlinear waves in 1966. He devised an ingenious perturbation procedure to establish that the Korteweg–de Vries (KdV) equation is the appropriate evolution equation for weakly nonlinear long waves in a variety of flow systems—in the presence of density stratification, shear and rotation. This work went far beyond the classical problem of irrotational waves in shallow water and the result has been of central importance to other areas of fluid dynamics. He employed a similar perturbation approach to analyze waves on thin films, a problem of importance in coating and other manufacturing processes. In this instance, nonlinear, dispersive as well as instability effects are present; the resulting evolution equation has been studied extensively and has brought out a number of interesting physical phenomena. Benney was also the first to draw attention to nonlinear critical layers in parallel shear flows. In 1969 a class of finite-amplitude neutrally stable shear-flow modes was obtained in which nonlinearity dominates viscosity at the critical layer. These disturbances are governed by an eigenvalue problem quite different from that furnished by the classical linear viscous theory. This interesting possibility is now known to be relevant in various instances, and further studies which include stratification and rotation, as well as unsteady and wavepacket effects have been carried out by many others, including Benney. In 1970 Benney derived the asymptotic equations governing slowly varying multiperiodic wave trains in a class of nonlinear systems. This was the first time modulations of fully nonlinear multiphase wave systems were obtained, extending earlier work from modulations of singly periodic to quasi-periodic waves. Similar asymptotic equations and their solutions, which involve algebro-geometric constructions, have been actively studied during the past 30 years and are still a topic of considerable interest. In his seminal 1973 work on strongly nonlinear surface waves in shallow water Benney derived a novel system of nonlinear equations and showed that they admitted an infinite number of conservation laws. This system, which is referred to as the "Benney Equations", have been intensively studied. While its complete solution is still not known, the properties already uncovered are so intriguing that undoubtedly the Benney equations will continue to be carefully investigated for years to come. During the eighties and the nineties, in parallel with his duties as Department Head of Mathematics, Benney continued his work on nonlinear wave interactions and flow instability. Among other contributions, he pointed out that instabilities with algebraic, rather than exponential, growth can become important under conditions of 'direct resonance' owing to generalized eigenfunctions corresponding to degenerate eigenvalues in non-conservative systems. He also proposed a theory for shear-flow instability that provides an explanation for the significant three-dimensional distortion of the mean flow which has been observed experimentally and which cannot be accounted for by classical weakly nonlinear theories. I wish to thank Liz Benney for relating the many interesting personal items to me. Dr Hannah Bartholomew Dr Hannah Bartholomew joined the mathematics education unit in the Department of Mathematics at The University of Auckland in April. She has a PhD in mathematics education from the University of London, and worked as a Research Associate at King's College London for four years before moving to New Zealand. Prior to that she obtained a BSc in mathematics from the University of Bristol, and an MSc in Pure Mathematics from the University of Manchester. Her research interests include gender issues in mathematics education; the formation of students' identities as learners of mathematics; the impact of grouping students by ability; and the ways in which these issues intersect with the types of understandings that students develop in and about mathematics. Dr James Goodman Dr James Goodman joined the Department of Computer Science at The University of Auckland in January 2003 as a Professor. He has moved from the University of Wisconsin-Madison, where he was a member of the Computer Sciences and Electrical and Computer Engineering departments for 23 years. He served as Chair (HoD) of the Computer Sciences department from 1996 to 1999. Jim's research focus is on computer architecture, and specifically memory systems, multiprocessors, and synchronization. He has contributed particularly in the art of snooping caches and other extensions of shared-memory multiprocessing. Recent work has focussed on efficient implementations of transactional memory, providing a clean programming model for multiprogramming. After he had accepted the position at The University of Auckland, Jim discovered that William Steadman Aldis, the first Professor of Mathematics and Mathematical Physics at Auckland University College (from 1884 to 1893), was the brother of his great-grandfather. This came as a great surprise, since he had not previously known of any relatives in New Zealand, nor of any relatives in the academic community any place in the world.
Dr Tim Stokes Dr Tim Stokes joined the Department of Mathematics at The University of Waikato on the first of April (despite fears it was all a practical joke!), having previously worked at Murdoch University in Perth, Western Australia for nine years. His first job was as a Post-doctoral Fellow at the University of Tasmania in Hobart, from 1991 to 1993 (where he also completed his BSc and PhD, and indeed was born and raised). His interests at that time were automated theorem proving and algebra, mainly ring theory and universal algebra. On moving to Perth as a contract Associate Lecturer he continued to work on these areas, until his contract ran out at the end of 1998. He then worked as a Research Associate on several projects at Murdoch, and picked up new interests in such diverse areas as free surface problems in fluid mechanics and modal and temporal logic and their applications in computer science. He kept up his other interests however, and is looking forward to pursuing them more vigorously, whilst maintaining the newfound interests, in the years to come. He is also very much looking forward to exploring all this beautiful country has to offer. SPRINGER-VERLAG PUBLICATIONS Information has been received about the following publications. Anyone interested in reviewing any of these books should contact David Alcorn Arveson W, Noncommutative dynamics and E-semigroups. (Springer Monographs
in Mathematics) 434pp. Geometric Numerical Integration The relatively new subject of geometric integration has arisen in response to perceived shortcomings of traditional methods of solving evolutionary problems numerically. The conventional approach is to seek approximations in a single time-step which keep the norm of the error committed small. Assuming that appropriate stability conditions hold, this will mean that the overall error at a fixed output point, after many, but increasingly small, time steps, will converge to zero in some specific manner. For example, the error at the fixed output point can typically be estimated in terms of a specific power of the stepsize. This approach is too general and heavy-handed in situations in which accuracy in some directions is regarded as more important than in others. In the modelling of a scientific problem for which the solution is known to lie on some manifold, error components normal to the manifold are usually regarded as more serious than errors tangential to it. Thus, even though we can never eliminate computational error entirely, we can often at least preserve geometric or structural aspects of the approximation. The authors of the present book all work at the forefront of this rapidly developing subject and are also skilled expositors. Accordingly, it is an outstanding contribution to the rapidly growing literature on the subject of geometric integration. The first chapter introduces some of the questions fundamental to geometric integration by surveying some keys problem areas from the Lotka-Volterra problem in population dynamics, through the Kepler problem and molecular dynamics, to highly oscillatory problems. The second chapter introduces the basic classes of numerical methods that are already known to be useful in geometric integration. These include Runge-Kutta methods and their extensions to second order systems through to composition methods and splitting methods. The third chapter explores order conditions from the classical tree-based approach through to the Baker-Campbell-Hausdorff formula and some of its ramifications. Chapter four is devoted to the study of solutions which evolve on manifolds. On this theoretical basis, the Crouch-Grossman and the Munthe-Kaas generalizations of classical Runge-Kutta methods are introduced. Chapter five makes a special study of numerical methods which are symmetric under time reversibility. Chapter six deals with the important special case of Hamiltonian systems and numerical methods which preserve the symplectic property. Besides being important in terms of applications and the maturity of the mathematical knowledge associated with this problem, this topic is important historically. Amongst the eight further chapters I will make special mention of chapter 13 which deals with highly oscillatory problems. The study of oscillatory systems of this type is of crucial importance because of the need to model such systems in scientific applications with due attention paid to long term behaviour on the one hand and computational efficiency on the other. Important examples are applications to the Schrödinger equation and to transmission in fibre-optic cables. As I have indicated, each of the authors of this impressive work is, in his own right, at the leading edge of research in geometric integration. Together they are a superb team. They bring mathematical knowledge, scientific insight and computational experience together in a way that makes for a most successful exposition of this rapidly developing and vital subject. Intuitive Combinatorial Topology This is a translation of the Russian original which was published in 1982 and was itself a major revision of the authors' short survey which appeared in instalments from 1957. Since its beginnings in the middle of the 19th century topology (The word 'topology' was apparently first used by Listing in 1847.) has developed to a vast extent. Initially essentially a curiosity (think of the Königsberg bridges which, I concede, were sorted out earlier than the mid-19th century) it soon became a very respectable branch of Mathematics with its deep and challenging problems. Perhaps the most honourable of these problems is the Poincaré Conjecture, first formulated in 1900, reformulated by Poincaré himself in 1904 when he published a counterexample to his original conjecture, and remaining unsolved 99 years later. (Many 'proofs' have appeared in the intervening years, often with lots of accompanying publicity even in the daily press but all, except possibly the latest by Grigori Perelman which was publicised in the New York Times on 15 April, 2003, seem to have foundered. Ironically, a natural generalisation of the Poincaré Conjecture to higher dimensions has for long been completely solved, gaining their solvers Fields Medals.) As the 19th century faded and the 20th raced by topology grew, with a number of distinct subfields developing. One of these is combinatorial topology, which is based on figures made up of closed line segments, closed triangles and their higher dimensional analogues. In a sense it is sufficient to understand combinatorial topology to grapple with the Poincaré Conjecture. Because of the simplicity of the objects studied much progress can be made yet in the sense alluded to in the previous sentence one can understand all topology of the space we supposedly live in from just studying combinatorial objects. Not surprisingly many introductory books on the subject have appeared over the years. My library is pretty small but even there I find about 10 books on the subject starting with a translation as Elementary Concepts of Topology of Alexandroff's 1932 book through to Huggett and Jordan's 2001 book A Topological Aperitif. The first chapter, Topology of Curves, introduces the topic, discusses some simple topological invariants, including the Euler characteristic of a graph and gives a proof of the classic Jordan Curve Theorem for polygonal curves. (Actually, despite some shaky starts to the proof of the Jordan Curve Theorem (Jordan himself gave a faulty proof) it is relatively simple to prove the result in general. Some of my Auckland colleagues may remember a seminar in which, in the space of an hour, I gave a complete proof from scratch, by which I mean that I assumed only elementary facts about metric spaces and their maps.) The authors then ask the question 'What is a curve?' and this leads them to interesting (not really combinatorial) objects such as the lakes of Wada, Sierpinski carpets, Menger sponges and Peano space-filling curves. Chapter 2, Topology of Surfaces, extends the Euler characteristic then classifies surfaces, with a proof in the case of orientable surfaces. The discussion of vector fields on surfaces includes Poincaré's link to the Euler characteristic and hence indices. There is a nice discussion of the four colour problem/theorem, including the dilemma of the computer-based proof and also the simplicity of the corresponding problem for other orientable surfaces. The hardly fair but nevertheless true proof of the corresponding result for other orientable surfaces is given. As was the case with Chapter 1 there is an excursion into the non-polygonal, in this case Antoine's necklace and a construction of a version of Alexander's Horned Sphere from it. The rest of the chapter is devoted to a quick look at knots with the emphasis being on spanning surfaces. At this point I became a little frustrated: I am used to calling these surfaces Seifert surfaces but in this book not necessarily orientable versions are called Frankl surfaces and even when orientable versions are given there is no mention of Seifert, but nor is there a specific reference to Frankl's work to enable one easily to compare the temporal and other connections between these two authors' work on the subject. The final chapter is titled Homotopy and Homology. The fundamental group is introduced and a heuristic argument is given to identify the fundamental group of the circle. This is followed by a discussion of covering spaces, including a discussion of the fact that the plane is a universal cover of every surface except the sphere and the projective plane. Applications to proving the fundamental theorem of algebra and knot groups are given. There is a good description of H_{1}(X) and a brief (what else is needed in this context?!) discussion of H_{0}(X) for a space X, followed by a very brief nod to H_{r}(X) for x2. This, of course, leads again to the Euler characteristic. The chapter ends with brief discussions of fibre bundles and Morse theory (the Euler characteristic again!!). All of what I have described above is a fairly standard collection of topics which give an enjoyable introduction to low-dimensional topology with hints of the more general. The discussion is clear and it is all superbly illustrated by 200 figures. For me, what sets this book apart from others is the ten page appendix, Topological objects in nematic liquid crystals. Nematic liquid crystals are long molecules whose interactions tend to arrange them in parallel and hence determine vector fields except that the molecules are directionless. This leads naturally to a map to the projective plane and the appendix analyses from a topological point of view the interaction of discontinuities in the map. Overall I am happy to recommend this book as giving a good introduction to topology of the space we live in through combinatorial topology with some interesting excursions to some of the wild places. I noticed only one misprint (the title of section 1.3 in the table of contents). Perhaps the index could have been more comprehensive to reflect more precisely the wide range of interesting topics, especially those that keep reappearing. ICIAM2003 ICIAM2003 was a real success for the Applied Mathematicians fortunate to attend this Sydney gathering between July 7 and 11. This was claimed to be the largest mathematical meeting to be held in the Southern Hemisphere, with over 1600 registered attendees. This figure should be a significant underestimation of attendees though, because while it was obvious that many attendees had left by the last day of the meeting, after having delivered their talks and succumbed to the attractions of home or downtown Sydney, the head count at the closing ceremony was 1700. A large attendance at ICIAM2003 was important because the NZ and Australian Mathematical Societies had underwritten the meeting financially. The concerns raised recently by September 11, Afghanistan, Iraq, SAARS, etc have made international travel more uncertain, and the organisers did face many late cancellations. However, as in show business, things came right on the night, and the Treasurer, Bill Summerfield, was very pleased with the large numbers attending. I have attended engineering meetings larger than ICIAM2003 before. Specifically, an annual petroleum industry meeting had about 10,000 attendees, and I regularly attend the annual American meeting of chemical engineers which has typically over 2000 attendees. These meetings are very well run, being planned down to the last minute. ICIAM2003 was nothing like this, with a free spirit moving throughout the meeting, giving it a charm and style of its own. I arrived too late for the opening addresses, and decided to attend a talk from one of the following 43 parallel sessions. The opening speaker (who was also the Session Chair) had trouble with his powerpoint links, and did not start until seven minutes into the talk. That was no problem though, because one of the following speakers had cancelled, and so each of the three speakers in that session had been allocated 40 minutes for their talk by the Chair. While intrigued at this initiative by the Chair, I decided to chill out, and note what developed. None of the audience seemed to be concerned by the time issue, and indeed, they listened very attentively, and were really interested in the new idea being presented, and were keen to hear how it compared with other methods. After this opening talk, I decided to switch sessions. This is something one should not do at these meetings, because I arrived at the end of the 3rd talk in that session, although the time to switch between the two sessions was only a few minutes. This was a disappointment to me, as I was looking forward to hearing that talk, but I had learned my lesson that one should avoid switching sessions, if at all possible. A special feature of the first two days of parallel sessions were the student talks. This was a real inspiration to those attending and no doubt to those presenting. The student talks I attended were very well prepared and rehearsed, and a real testament to the students' supervisors. It is clear that ICIAM2003 captured the hearts of many maths students in Australia, and no doubt many from NZ. I met one undergraduate maths student from Adelaide who had paid the attendance costs herself. I am sure that ICIAM2003 will be a pivotal moment in the lives of many of the young (and not so young) applied maths students fortunate to attend. The morning and afternoon talks began with Invited Speakers. These talks were on the whole really impressive, outlining the state of play in the latest computational methods, the impact of the net, operations research, multi-scaling, fluid mechanics, pdes, elasto-plasticity, geometry, finance, education, random matrices, biomathematics, meteorology, statistics, ergodicity, turbulence, imaging, communications, stability, and so on. It was clearly impossible to attend all of these invited sessions, and delegates were frequently faced with difficult choices. In addition to the invited and parallel streams, nine other special sessions were held. There were five embedded meetings: the 6th Australian-NZ Maths Convention; 11th Computational Techniques and Applications Conference; 6th Engineering Mathematics and Applications Conference; 17th National Conference of the Australian Society for Operations Research; and the 2nd National Symposium on Financial Mathematics. As well, there were Industry Day, Education Day, Community Day and a session by the IMU. Then of course the parallel sessions were subdivided into contributed talks, mini symposia and submeetings. ICIAM2003 was too large to hold on one site, and a 750m walk was needed to switch between venues. Nevertheless, there were excellent opportunities for networking. Morning and afternoon teas were held in the same room as the booths and posters, and so foot traffic to these displays was excellent. A full social program was available, including icebreakers, reception, drama, opera and a dinner cruise. Looking back, I am sure attendees will rate ICIAM2003 as a real success. There were some quirks, but I am still thinking on some of the ideas which were stimulated by this meeting, and my colleagues are as well. I believe the planning and support in NZ and Australia for ICIAM2003 over the last four years or so has now been justified. ICIAM2003 will certainly have been a major success for mathematics in 2003. REPORT FROM THE NZAMT8 CONFERENCE The NZ Association of Mathematics Teachers (NZAMT) held their Eighth Biennial Conference in Hamilton from July 8th to 11th, 2003, with the theme "Celebrating the Magic of Maths''. With eight plenary speakers and many workshop presenters—14–16 workshops to choose from at every session—it was a very full programme, with plenty to interest teachers at every level. One day there was a special emphasis on primary teachers with appropriate workshops. The trade displays were well visited, especially the stall organised by the Department of Engineering Science of The University of Auckland, which showed some of the opportunities in Mathematics-intensive tertiary study. The weather was kind, and the venue at St Pauls Collegiate School was first-class. The highlight of the social programme was the conference dinner, featuring a Harry Potter theme, held at The University of Waikato Centre of the Performing Arts. REPORT ON ANODE 2003 John Butcher's 70th birthday year was celebrated with the last in the Anode series of conferences. Anode, which stands for Auckland Numerical Ordinary Differential Equations (not Auckland nerds overdoing differential equations, as my son suggested), attracts a significant number of overseas participants who are keen to work with John. This year the overseas invited speakers, and their lecture series, were: \noindent Hermann Brunner, Memorial University of Newfoundland, Canada, on Collocation methods for Volterra functional integral and integro-differential equations. Chris Budd, University of Bath, UK, on Geometric integration of ODEs and PDEs, and its applications. Mari Paz Calvo, University of Valladolid, Spain, on Some aspects of the time integration with Runge-Kutta type methods of evolutionary partial differential equations. Roswitha März, Humboldt University, Germany, on Differential algebraic equations with properly stated leading term. In a new departure this year, New Zealand applied mathematicians from various areas were also invited to speak. These were: Rick Beatson, The University of Canterbury, on Fast computation with radial basis functions for applications to image reconstruction and geophysics. Ian Coope, The University of Canterbury, on Grids and frames in computational optimisation. Peter Hunter, The University of Auckland, on Computational modelling in biology: integrating physiological function from cell to intact organs. Stephen Joe, The University of Waikato, on Construction of good quasi-Monte Carlo rules for functions in weighted spaces. Robert McLachlan, Massey University, on Runge-Kutta-Nyström methods and the entropy of classical mechanics. David Ryan, The University of Auckland, on Scheduling problems and the set partitioning model. Philip Sharp, The University of Auckland, on Realistic test problems for N-body simulations of the solar system. David Wall, The University of Canterbury, on Computational methods and inverse problems. Not only did these talks provide a showcase for the wide variety of applied mathematics being studied in New Zealand, but also established some contacts between the speakers and conference participants. An excursion to the Waitakeres took place on the wettest day of the conference, as per usual, but the rain did hold off long enough for a short bush walk to see several large kauri trees. At the conference dinner, the tributes made to John Butcher emphasised the esteem in which he held in the numerical ODE community worldwide. During the week after the conference, a follow-up workshop was held at the Tamaki Campus. This gave an opportunity for those participants who stayed on and the Auckland group to meet in a more informal way, and to understand the work of others in a deeper way. Thanks are due to the Marsden Fund, for paying the expenses of overseas visitors for the conference, and to New Zealand Institute of Mathematics and its Applications for supporting the post-conference workshop. MODELLING CELLULAR FUNCTION, ON WAIHEKE A conference on "Modelling Cellular Function'' was held on Waiheke Island from the 14th to the 18th of July. It was funded by the NZIMA and (theoretically) organised by Nic Smith, Peter Hunter (both from Bioengineering, Auckland) and James Sneyd (Maths, Auckland). In actual fact James Sneyd and Peter Hunter did nothing at all, while all the real work was done by Nic Smith and Catherine Lloyd (Bioengineering, Auckland). That being the case, the organisation was superb. One suspects that the conference had a lucky escape. Nic and Catherine had arranged for lots of sun with very little rain, and the Northern Hemisphere visitors (particularly, let me add maliciously, those from Chicago) were impressed by our "typical'' Auckland winter weather. We didn't enlighten them. There were a number of eminent international speakers. James Keener (Utah) spoke about the topology of cardiac defibrillation, Philip Maini (Oxford) enlightened us on tumours, Andrew McCulloch (UC San Diego) gave a wonderful talk on heart models, Peter Deuflhard (Free University of Berlin) talked about designer drugs and how to model them, and Philip Kuchel (Sydney) spoke about a mathematical model of the red blood cell. The programme was filled out with a number of talks from other international and local speakers, and a poster session at which each participant had to give a three minute blurb about their poster. It was an excellent way to learn about a wide variety of mathematical work in biology, and rarely have I attended a conference that has held my interest so consistently. For some less strenuous entertainment, Pepe, the magician/physiologist from Chicago, showed off his magic tricks at the final dinner. He was persuaded not to saw anybody in half, but did a sort of thingy with blocks and balloons and scarfs and cards that had the assembled multitudes going oooooh aaaaah. He was ably assisted by Catherine (among others) and less ably assisted by someone who wishes to remain anonymous. Just as a side note, it was the considered opinion of the conference that if that same anonymous person is to be asked, firstly, to give after-dinner speeches, and secondly, to assist with magic tricks, he should be made to pay for his own wine. Nevertheless, despite those occasional moments, the conference overall was a great success, demonstrating to our visitors the strength of mathematical biology in New Zealand, and giving local students and faculty a valued opportunity to meet some of the most famous people in the field. KOREAN INDUSTRIAL MATHEMATICS INITIATIVE Supported in part by the RSNZ, a conference to initiate mathematically based interactions with Industry was held in Daejeon, South Korea 1st-3rd July 2003. This support was provided under the Memorandum of Understanding between South Korea and New Zealand for scientific cooperation. About 65 persons attended including many graduate students in applied mathematics. This conference was co-directed by Professor Graeme Wake of Massey and Canterbury Universities, who is also Director of the ANZIAM Mathematics in Industry Study Group 2004–5. During the Initiative, case study problems were used to demonstrate the power of mathematically based problem-solving. A plenary discussion at the conclusion of the conference canvassed options for the development of further initiatives in Industrial Applied Mathematics in Korea. In addition to Professor Wake, three other New Zealanders contributed. These were: Associate Professor Mark McGuinness (VUW), Professor Robert McKibbin and Dr Bruce van-Brunt (both from Massey University—Albany and Palmerston North respectively). The front row is, from left to right: Conferences in 2003 November 23–27 (Queenstown) Delta '03 Conferences in 2004 January 3–11 (Nelson) NZMRI meeting on Computational Algebra, Number
Theory and Geometry January 12–16 (Nelson) NZIMA meeting on Logic and Computation January 18–22 (Dunedin) Australasian Computer Science Week February 9–14 (Wellington) VIC 2004 February 16–20 (Whakapapa) Annual New Zealand Phylogenetics Meeting
MINUTES OF THE 29TH ANNUAL GENERAL MEETING Present. Charles Pierce, Shaun Hendy, Graeme Wake (Chair), Kathirgamanathan Padmanathan, Ulises Carcamo, Robert McLachlan, Tammy Smith, Rick Beatson, Nicoleen Cloete, Mark McGuiness, Stephen Joe, Alona Ben Tal, Peter Renaud, Aroon Parshotam Apologies. Robert McKibbin, Marston Conder, David Gauld, Rod Downey, Graham Weir, John Burnell Shaun Hendy opened the meeting. It was moved (Hendy and Smith) that in the absence of the President and vice-President, Graeme Wake as former President would chair the meeting. The motion was carried.
The meeting closed at 6-15pm. The AGM was ratified by the NZMS Council on July 22, 2003. NZMS RESEARCH AWARD 2003 The New Zealand Mathematical Society Research Award for 2003 is made to Rod Gover from the University of Auckland for his highly original contributions in conformal differential geometry that have led to the solution of some outstanding and difficult problems. AITKEN PRIZE 2003 The Aitken Prize for the best student talk at the Colloquium is awarded to Cynthia Wang (Massey University – Albany) for her talk "Modelling a plate of arbitrary shape in infinitely deep water using a higher order method.'' The judging panel had great difficulty separating the top three talks and have awarded Highly Commended to Jonathan Marshall (Massey University – Palmerston North) for his talk "Analyticity of solutions to functional differential equations'' and to David Byatt (University of Canterbury) for his talk "Performance of various BFGS implementations with limited precision second-order information.'' POSITION ANNOUNCEMENT The Mathematical Biology Group is seeking a research assistant who can help with
and a range of associated tasks. This position would suit a recent graduate (or near graduate) who has a strong degree with numerate-oriented emphasis. As most of the work involves models of biological processes it would be advantageous if the applicant also had some background in the life sciences. The position would be based at the Ruakura campus of AgResearch, a very pleasant site near The University of Waikato in Hamilton. Please bring this to the attention of any suitable recent graduates or final year students. Any further enquiries to: Tony Pleasants, Leader, AgResearch, Ruakura Research Centre, GRANTEE REPORTS I would like to thank the New Zealand Mathematical Society for providing the summer support of $500 at The University of Auckland. I finished the reading course successfully on time. The Mathematics Department had been pleased to offer me a tution scholarship and $3000 allowance to continue with my PhD research. Currently I am a graduate student in Mathematics Department at The University of Auckland. Thank you. REPORT ON ICIAM 2003 The fifth International Congress on Industrial and Applied Mathematics (ICIAM 2003) was hosted by Australia and New Zealand Industrial and Applied Mathematics (ANZIAM) during the week 7–11 July 2003. The conference was held at the Sydney Convention and Exhibition Centre at Darling Harbour, adjacent to the central business district of Sydney. Parallel sessions took place at the nearby Haymarket campus of the University of Technology Sydney. The conference covered the full spectrum of industrial and applied mathematics focusing on the strong links between applied mathematics, industry and finance. A number of other meetings were embedded within ICIAM 2003, including: 6th Australia-New Zealand Mathematics Convention, 17th National Conference of the Australian Society for Operations Research, 11th Computational Techniques and Applications Conference, 6th Engineering Mathematics and Applications Conference, 2nd National Symposium on Financial Mathematics, the 2003 meeting of ANZIAM and the annual general meeting of the New Zealand Mathematical Society. There were 27 invited speakers, more than 40 parallel sessions spread over the two sites, minisymposia, an exhibition, a career development workshop (6 July) and three special days: Industry Day (8 July), Education Day (9 July) and Community Day (10 July). These special days provided the opportunity for the mathematical community to meet and engage with others. Over 1500 delegates from around the world attended the conference. Contributed talks were held every 30 minutes with speakers given 20 minutes to present their material, followed by five minutes of questions and five minutes to get to the next talk. The book of abstracts is over 400 pages long and the conference program is over 200 pages. As well as the academic programme there was a full social schedule providing plenty of opportunity for people to interact. The optional social programme included: a reception at the Powerhouse Museum, Sydney Harbour Bridge climb, Proof (a play at the Sydney Opera House), a night at the opera, the Down-under Experience, Sydney harbour cruise and Sydney sightseeing tours. The next ICIAM meeting is in four years time in Zurich 16–20 July 2007. I would like to thank NZMS for providing financial assistance to attend ICIAM 2003. REPORT ON ICIAM 2003 I attended ICIAM 2003 in Sydney, July 7–11, along with Massey (Albany) colleagues Mike Meylan, Winston Sweatman, Robert McKibbin and Graeme Wake. The attendance was close to 1700, with seemingly every Australian mathematician in attendance, along with representatives from many other countries. Almost everyone attending gave a presentation, with the result that several times a day one had to choose from a selection of 43 parallel sessions! I had been asked by SIAM to organise a mini-symposium on "Pattern formation in neural systems'', and I arranged for Jack Cowan (Chicago), Carson Chow (Pittsburgh) and Steve Coombes (Loughborough) to speak, along with myself. I was also invited to speak in a minisymposium on "Neuronal and biological dynamical systems''. Overall the conference was a great success, with many opportunities to meet colleagues and make new contacts. I particularly enjoyed some of the plenary sessions, as they often gave insight into other areas of applied maths that I knew little about. I thank the New Zealand Mathematical Society and Massey University (IIMS) for providing financial support for my trip to Sydney. R gave me a DAE underneath the Linden tree Most of my life has been concerned with understanding how to solve ordinary differential equations numerically. Just when I think I have got somewhere, I find that some new aspect of the problem has become important and I have to start at the beginning again. So it was when "stiff" differential equations were recognised as constituting a distinct class of numerical problems. I thought I knew something about how to solve the easier non-stiff problems; all one needed was a stable and consistent numerical approximation and this can be turned into an effective algorithm. Most of the traditional numerical schemes are generalizations of the famous Euler method in which the solution at a time value x_{n} is found by adding to the approximation at x_{n-1} the value of an approximation to the derivative, evaluated at x_{n-1}, and multiplied by x_{n}-x_{n-1}. If the differential equation is y'(x) = f(x,y(x)), and y_{n} is the approximation to ,y(x_{n}), then we can write, for the Euler method, y_{n} = y_{n-1} + h f(x_{n-1}, y_{n-1}), where h = x_{n }- x_{n-1}. In the special case of the linear problem, y'(x) = ly(x), the recursion for the approximations would be y_{n} = (1 + lh) y_{n-1}. Sometimes, buried in a complicated differential equation system, are components which behave in this fashion. Sometimes, when l is a (possibly complex) number with negative real part, we are supposed to be approximating a negative exponential so that the effect of these components should be only transitory. However, if l is outside a disc with centre -1 and radius 1, powers of 1 + hl are unbounded and the computed result will be spoilt by the effect of these terms. In such a situation, the problem will be stiff and it needs to be solved by alternative methods, such as the implicit Euler method y_{n} = y_{n-1} + h f(x_{n-1}, y_{n}). For this method, the factor 1 + lh, which was the source of trouble for the forward Euler method, is now replaced by $(1 - hl)^{-1}, and no harm is done by l being negative and of large magnitude. I do not remember when I first became aware of differential algebraic equations, but I know when this crucial event occurred in the life of my colleague and friend Roswitha März of the Humboldt University in Berlin. I was attending a conference in the former West Germany in 1981 when someone asked me if I would like to visit East Berlin, because this person could arrange it for me. With some trepidation, I agreed to fit in with the arrangements that had to be made, and a few days later I first made the acquaintance of Roswitha. I had in my hand a number of reprints of papers by various people and some of these were on the relatively new subject of numerical methods for differential algebraic equations. I gave these to Roswitha in case she could make use of them and she certainly could. She soon became the leader of one of the most productive and insightful research groups working on this subject and I have always been proud of my small role in her introduction to this subject. Over the last 6 or 7 years we have held a series of "ANODE" (Auckland Numerical Ordinary Differential Equations) workshops and what will almost certainly been the last of these has just finished. It was a delight to welcome Roswitha März, as an invited speaker, at this meeting. In fact it was a triple benefit because Roswitha came with two colleagues, René Lamour and Caren Tischendorf, who are outstanding research workers in their own right. As an easy introduction to the subject of numerical differential algebraic equations, I will quote an example problem presented by Caren. This consists of the coupled system This is a differential algebraic equation because it contains a differential equation together with an algebraic constraint. It is said to be of "index 1" because a single differentiation of (2) and a rearrangement converts it to the differential equation system If the problem in its original formulation is solved by a natural extension of the implicit Euler method it is found that the z values satisfy the recursion z_{n} = (1 +lh) z_{n-1}. It is unfortunate that this is the same recursion that would apply to (4) being solved by the explicit, rather than the implicit Euler method, and is disadvantageous if l negative with a large magnitude. One of the main thrusts of the Humboldt group led by Roswitha März, is that this sort of anomalous behaviour is avoided if the problem is formulated differently. What they call "numerically qualified" would cast this example problem in the form The crucial detail concerning the two matrices and , is that im(D) is constant and that ker(A)Åim(D)= . An implementation of the implicit Euler method using this formulation would propagate only values of , and use the algebraic constraint to evaluate the individual components y_{n} and z_{n}. During an eight month visit to Auckland, Steffen Schulz, a postgraduate student at Humboldt, wrote the Auckland Mathematics report number 497: "Four lectures on Differential-Algebraic Equations". This is a good introduction to the subject and to some of its literature. John Butcher, butcher@math.auckland.ac.nz
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