PREAMBLE:

Mr. Chairman the Vice Chancellor, Colleagues, Staff, Students and Friends of the University of Jos, it is only three months ago that we listened to one inaugural lecture and pertinent remarks on how infrequently inaugurals have held in this University, in the past; Today it is my privilege and pleasure to present another, confident that the next will hold even sooner and confirming that this age old tradition of Universities the world over, is fast taking root in ours.

My lecture is in four short parts; the first, "Mathematics yesterday, today and tomorrow," attempts to reflect on what mathematics is, how it has developed and where we are taking it to. " Complex Analysis in the realm of Mathematics," puts one particular area of mathematics into perspective, and is followed by a focus on specific aspects of this area, effected by providing some instant insights. The epilogue considers the endangered species of the African Mathematician and the challenges that their impending demise poses for Nigeria and the African Continent.

In each part but one, we shall show that the complex is in some sense simple - contradictions? Certainly not! - just excellent paradoxes!

 MATHEMATICS YESTERDAY, TODAY AND TOMORROW.

"Mathematics as an expression of the human mind reflects the active will, the contemplative reason and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction as well as generality and individuality." (Courant and Robbins [28]). Though different traditions may emphasise different aspects of the elements, it is the interplay of these antithetic forces and their synthesis that constitute the life, usefulness and value of the mathematical sciences.

All mathematical development has its psychological roots in practical requirements; but once started under the pressure of applications, it gains momentum in itself and transcends the confines of immediate utility. Mathematics is therefore both an applied and a theoretical science.

It can be visualised as having several concentric layers built on a core which is called pure mathematics; a core hot with ideas, structures and theories, such that ideas from it percolate through the outer layers providing intellectual fuel for the problems of the applied fields, while problems arising from practical requirements simultaneously provide pure mathematics with new concepts, structures and methods. Theory and practice diffuse through ill-defined boundaries between the layers, each layer enriching the other and nourishing the discipline - a necessary and sufficient symbiosis to compensate for the intellect's inability to create all meaningful postulational systems at its whim. Without doubt intrinsic necessity is a catalyst for the free mind to achieve results of scientific value.

Apollonius wrote a classic paper on conic sections in 200 BC and a whole century elapsed before Kepler found an application to parabolic mirrors. Only much later were their relationships with the orbits of the planets discovered.

Cayley's matrix theory developed in 1860, found its first major application in Heisenberg's matrix mechanics in Quantum theory in 1925.

Therefore, it may not surprise you to learn that the mathematical contributions of Liverpool have not yet found applications outside the confines of mathematics. You will be right if you postulate from this that he is a pure mathematician. Generally new mathematics gives immediate applications when it is motivated by an applied problem.

What makes it so hard for mathematicians to transmit the excitement of their current work to the nonspecialist? The discipline is built on structures created and developed not over centuries, but over thousands of years; structures which have remained consistent over all time. For mathematicians, this is a unique asset but for the nonspecialist it may be considered a great handicap as laypersons today do not get exposure to any mathematics less than two thousand (2000) years old. Indeed, if you have taken a first course in Calculus, you have only had a glimpse of eighteenth century mathematics and you are still almost three hundred years away from the frontiers.

In these circumstances, how can the specialist easily transmit the excitement of twenty - first century mathematics to others, resting, if not in slumber, in an eighteenth century mathematical existence? Yet mathematics is inherent in the rational power of every human being. It is part of nature as language, music, art or politics but while in these endeavours, theories come and go, in mathematics theories go on forever!

This activity is yet another celebration of mathematics, bringing it not only to specialists but also to laypersons, the group from which all specialists evolve and to which mathematics truly belongs. Let us hope this activity remains painless for as long as it lasts, as one hour of mathematics presented to the nonspecialist may easily be misconstrued as the infliction of pain; for some, even the infliction of torture.

The intricate interplay between the numerous layers makes it mandatory for young talents to have an extensive and diversified exposure in order to acquire adequate mastery of the science of mathematics and the art of mathematical research if they are to push back the frontiers. New blood African mathematicians therefore, need much more support and encouragement than the near excellent opportunities third world mathematicians had in the last quarter of a century.

In addition, new practitioners are easily deceived, as the interplay between generality and individuality remains poorly documented in standard mathematical literature. While mathematicians spend a lot of time thinking about and analysing particular examples to motivate the development of theory or to reach a deeper understanding of existing theory, this is hardly presented in print.

Results are by tradition presented in a direct way thereby giving the false impression to the uninitiated, that they are arrived at through mysterious means, rather than as a result of hard work and good intuition. The study of the classics, the work of the great mathematicians of each age, is therefore mandatory for a good understanding of some aspects of the philosophy and tradition of the subject. The most useful tool of a mathematician is a mathematician!

Another invention and tool of the mathematician is the all intrusive gadget called the computer; Experimentation and proof in mathematics have been stimulated by its use with the proliferation of results based on intuition and the study of specific cases. Experimental mathematics, as it is currently described, presents the subject today as a living entity, with examples, conjectures and theory, all interacting with and reinforcing one another. Its objective, for now, is to play a role in the discovery of formal proofs but not to replace them. Its range includes diverse fields such as algebraic geometry, cellular automata, dynamical systems, number theory and wavelets.

Although experimental proof has always been around, the introduction of the computer has made a quantitative difference. Programmes such as SnapPea, based on Bill Thurston's work, can quickly compute amazing facts about three dimensional manifolds, including their geometric structure and has had remarkable success in moving forward the understanding of these manifolds. (Epstein and Levy. [29]) Writing such a programme is a spectacular achievement of the constructive approach to mathematics.

There is a feedback loop: computers in mathematics enhance the importance of the constructivist point of view and the constructivists' point of view increases the use of computers in mathematics. But how is one sure that the computer always gives the right answer?

While it is unreasonable to insist that a result which depends on computer calculations must come with a formal proof of correctness, one may require a statement of the algorithm used and a publicly available implementation that can be independently checked.

Another difficulty is that even with a correct programme, the result relies on the correctness of the infrastructure, the absence of computer bugs for example, which is currently impossible to guarantee. However, the fact that despite all these concerns we still use computers to design structures, fly aeroplanes and to play so many other important roles in our day to day life, suggests that we have to learn to assign degrees of reliability to mathematical results that depend on computer calculations, as we do for other mathematical results or for experimental results in other sciences.

There is little doubt that, to date, the role of the computer in suggesting conjectures and enriching our understanding of abstract concepts by means of examples and visualisation is a healthy and welcome development. Far from undermining rigour, the use of computers in mathematics research does enhance it in several ways.

Firstly, mathematicians who write complicated computer programmes soon realise that subjecting lines of code to the usual techniques of mathematical analysis and proof, often reveals faults in the programmes. Programming can thus enhance and extend to a larger population, an appreciation of why mathematicians regard proof as important.

Secondly, the use of computers gives mathematicians another view of mathematical reality and another tool for investigating the correctness of a piece of mathematics, through the examination of examples.

Thirdly, it strengthens the trend towards constructivism helping to keep mathematics on its solid foundation.

Lest you begin to imagine that the advent of the computer poses only non-controversial problems, we shall reflect on "theorems for a price and the computer in mathematics tomorrow." It is said that the most fundamental precept of the mathematical faith is "thou shalt prove everything rigorously." "Yet some mathematicians claim the writing is on the wall. The Silicon Saviour has arrived and a new testament is to be written but there will always be a small group of rigorous old style mathematicians who will insist that the true religion is theirs and that the computer is a false Messiah - hopefully they may be viewed by the mainstream mathematicians of tomorrow, as a fringe sect of harmless eccentrics". (Zeilberger.[33]).

Some mathematicians believe that the computer will do to mathematics what the microscope did to biology. In the future, many mathematicians may be experimentalists and may not care about absolute certainty as many other exciting new facts of new dimensions may be ready for discovery. Will there still be a place for the mathematical mathematicians or will they face extinction? For the mathematical experimentalists of tomorrow mathematical identities may be established by exhibiting a proof certificate. Large parts of mathematics, it is speculated, may be trivialised by reducing mathematical truths to routine, albeit possibly very long and exhorbitantly expensive to check "proof certificates". These certificates would also enable someone plugging in random values, to assert "possible truth" very cheaply.

An era may then begin when one may witness many results for which one would know how to find a proof but one would be unable or unwilling to pay for finding such proofs since "almost certainty" could be bought for so much less. As absolute truth becomes more and more expensive one would, sooner or later come to grips with the reality that only a few non trivial results of this type could be known with "old fashioned" certainty. Most likely one will then wind up abandoning the task of keeping track of the price tag altogether and complete the metamorphosis to non rigorous mathematics.

From the foregoing, one may conclude that mathematics as a discipline is a complex pastime made commonplace by its natural role in every effort made by man to understand his existence and his environment. Mathematics is complex but is still so commonplace - a paradox of the complex!

There are several other dimensions of the impact of the computer on mathematics and scholarship. Mathematical literature has grown exponentially with time, from eight hundred (800) published papers in 1870, to fifty thousand (50,000) annually, today. Typeset in TEX the current output requires 2.5 G.B. of storage, costing less than eighty thousand naira ( N80,000) and this cost would be dramatically reduced using other technologies such as optical disc.(Odlyzko.[32]). As such it is cheaper to electronically store all current mathematical publications than to subscribe to a single established journal. Thus, in the near future personal computers will be able to store not only current publications but all mathematically published work and a mathematician can then call up any paper on screen, with round the clock access and print a copy as needed from the privacy of his study. In such circumstances though the need for a library could still be justified, its role will definitely have to change significantly. I am happily aware that our University Library has already anticipated, and is preparing for, this inevitable change.

Already the presence of electronic technology has presented concerns and opportunities not only for mathematicians, the library and librarians but also for all other scholars, journals and publishers. The electronic media presents promise for an increase in the effectiveness of scholarly work as publication delays disappear and the reliability of the literature increases.

Mathematical electronic journals are already quite commonplace. Electronic preprints have today become the main method in mathematics, for the communication of new results, as departments set up publicly accessible directories from which others can copy, or use preprint servers with all preprints being sent to a central database.

In the long run will all print journals convert to electronic publications as print journal publishers disappear? Your considered judgment, I guess, is as good as mine.

 COMPLEX ANALYSIS IN THE REALM OF MATHEMATICS

The study of mathematics began because it was useful, continues because it is useful and is valuable to man because of the usefulness of its results. As mathematicians, we insist that our discipline should not be studied only for its usefulness but also for its own sake and its beauty as it reveals the very rare combination of power and beauty.

Complex analysis is an old branch of mathematics which is the forerunner of many new ones including homotopy theory and manifolds. It is under the general area of "hard analysis" and underlies a large number of powerful techniques which find their application in other branches of mathematics as well as in Science and Engineering.

Its early applications in applied mathematics derive from dividends of the theory such as the use of residues in the evaluation of integrals and the use of conformal mapping in aerodynamics and potential theory. Some of its current applications in the physics of field theory underpin the understanding of phase transitions in statistical mechanics and string theory.

The evolution of complex analysis began with the introduction of the imaginary unit, i from elementary algebra, in order to derive a solution to the simple quadratic equation, x² + 1 = 0. Using an uncritical approach, i was combined with real numbers to generate complex numbers followed by a justification for the existence of complex numbers using ordered pairs of real numbers and an appropriate algebraic structure.

These numbers are called complex only because they are not real - they are in reality not that complex. The identification of the complex numbers, C with the two dimensional plane, R² and its extension to the extended complex plane, C = C u { }, is effected by stereographic projection using the Riemann Sphere, S. A topology on the extended complex plane, C can then be defined using a bijection :S C which shows that the extended complex plane, C is the one point compactification of the complex plane, C.

Complex analysis is then an extension of the calculus to the extended complex plane, C and further to n dimensional complex space, ⁿ . Differentiation and integration acquire new depth and significance but the range of applicability is radically restricted. As a result, complex functions are in some sense not so complex, being less complex than their real counterparts - a confirmation of the paradox of the complex .

In the geometrically oriented study of analytic functions, conformal mapping plays a dominant role. Existence and Uniqueness theorems allow for the definition of important analytic functions without resorting to analytic expressions; Geometric properties of the domains being mapped, lead to analytic properties of the mapping function. The Riemann mapping theorem deals with the mapping of one simply connected domain onto another and allows for the study of many conformal mapping related problems to be restricted to their study in the unit disc only. This is a gain of overwhelming proportions!

In the last ten years complex analysis has assumed great importance in several areas of contemporary mathematics and science as it has periodically done, during the centuries of its existence. Recent work by Thurston on 3-manifolds show the vital importance of hyperbolic geometry and mobius transformations to this rapidly developing subject.

The concepts of chaos and fractals have become very popular in mathematics and in the non-mathematical sciences as well because the beauty of computer images of complex dynamical systems has attracted the attention of researchers in many disciplines. The more beautiful mathematics behind it all, evolved from the study of iteration of complex functions originating from Fatou [30] and Julia [31] and such studies continue with recent significant contributions by Baker, Beardon, Douady and Sullivan, all major specialists in complex analysis.

The interaction of complex analysis with several other branches of mathematics as well as the other sciences and engineering as illustrated in the foregoing, puts complex analysis in perspective in the realm of mathematics.

INSTANT INSIGHTS INTO COMPLEX ANALYSIS

Let us now make a quick tour of the area of entire and meromorphic functions, a part of complex analysis, with an itinerary calling for stops at the scenic islands of Iteration of analytic functions, [2,4,13,14,17]. Normal families, [7,9,14,20]. Picard sets for entire and meromorphic functions, [1,6,10,11,15,17]. Factorisation of entire functions,[15,19,20]. Value distribution [9,12,21,27]. and the Nevanlinna Theory.[3,7,15,17], all in an attempt to discover its beauty and link it up with some other contributions. [22,23,24,25,26]

Iteration theory includes many interesting results on entire functions with infinite domains of normality. The theory deals with the sequence of natural iterates f_n(z) defined inductively by f₀(z) = z, f_n+1(z) = f(f_n(z) n o. We shall let F= F(f) denote the set of points in the complex plane where the natural iterates, {f_n(z)} do not form a normal family and C(f) its complement. One of the main problems of the global theory is to investigate various possibilities for the structure of C(f) and its complement, F(f).

There are interesting results where assumptions on the structure of F(f) and C(f) are made and the consequences on the generating function f(z) are derived. These consequences relate to growth as expressed in terms of the order of the function ,f(z). (Liverpool[14]).

In the local theory, using permutable power series, formal iterates f (z) are introduced as a generalisation of the natural iterates f_n(z) with arbitrary complex numbers. Baker proved that the values of corresponding to a positive radius of convergence for a normalised family of generalised iterates is either a one or two dimensional discrete lattice or it is the whole complex plane. It was then shown , Liverpool [13], that a 2-dimensional lattice cannot occur in the sense that when it does, the set has to be the whole complex plane.

Normal family studies, extend from their natural occurrence in iteration theory to investigations on the converse of the famous theorems of Schottky and Miranda for a family of analytic functions in a unit disc. Using the normality of the family and the Nevanlinna theory one can obtain, (Liverpool and Nnadi [9]), quantitative versions of the theorems of Schottky and Miranda and then use the results to establish an elementary proof of Picard's theorem. Normal family techniques also give a topological characterisation of a particular solution set, in answer to an open problem. ( American Mathematical Monthly [18]).

Picard sets are defined from considerations of the fundamental theorem of algebra which states that a polynomial of degree n has exactly n zeros in the complex plane, when multiplicity is taken into account. Entire transcendental functions are analytic functions in the complex plane, with an isolated singularity at infinity and can be considered as polynomials of infinitely high degree. It is therefore natural to ask whether such entire transcendental functions have infinitely many zeros in the complex plane.

A meromorphic function is one which is analytic in the complex plane except for a finite number of singular points which must be poles. When such a function has an isolated essential singularity at infinity it is called transcendental meromorphic. The Little Picard theorem states that at most one complex number is absent from the range of a non constant entire function, while at most two are absent from the range of a non constant meromorphic function. The great Picard theorem on the other hand states that given an analytic function f(z) with an essential singularity at z = a, the function, f(z) assumes each complex number with at most one exception, an infinite number of times in each neighbourhood of a.

Immediate corollaries of these results are that for an entire transcendental function, f(z), the function f(z) assumes every complex number with at most one exception an infinite number of times. For transcendental meromorphic functions, all values are taken infinitely often, with at most two exceptions.

From almost all considerations, it appeared that these corollaries reflected the end of the study of a problem of long standing now apparently completely resolved - premature considerations as was shown by Lehto when he defined a Picard set E, as a totally disconnected closed set in the extended complex plane in whose complement, CE, each single valued meromorphic function f(z) with at least one essential singularity in E, takes every value infinitely often except for at most two.

By Picard’s theorem, a transcendental meromorphic function assumes infinitely often all values in the plane except at most two. Hence, given any finite point set, it is automatically a Picard set for meromorphic functions. To find an infinite point set which is a Picard set is therefore an improvement or generalisation of the Picard theorems.

This insightful restatement of an existing result of Picard by Lehto opened the flood gates for simple new questions and complex research activity - the complex deriving from the simplicity of the obvious to an alert mind. It is a classical illustration of one small step for Lehto which became one giant leap for mathematics . Current research mathematics is more often than not, a consequence of very simple questions about complex problems - another paradox of the complex.!

In this area of Picard sets, new proofs of existing theorems as well as improvements and generalisations have led to results which are best possible. The sharpness of these results have been established and appropriate constructions and counter examples generated. Relating Picard sets theory with other aspects of complex analysis is a good example of how apparently unrelated questions in mathematics have been shown to be deeply interconnected. (Liverpool et al [1,16,21]).

THE AFRICAN MATHEMATICIAN - AN ENDANGERED SPECIES ?

Having dwelt at some length and in some depth, on mathematics, let us very briefly turn our searchlight on mathematicians, the creators of mathematics and the impending demise of the African Mathematician as well as the challenges of the present and the future. In this regard, we shall define an African mathematician as a mathematician of African origin resident and working in the continent of Africa.

Mankind today, is at the threshold of a new age, embracing the information revolution and the next millennium with more dependence on Science and Technology than ever before . Africa therefore, cannot afford to maintain her inertia and remain a permanent consumer as costs of goods and services, in the new information age, outstrip our existing economic reality. For mere survival, it is now more critical than it ever was, that we forge ahead. Ironically, with development, the world has become more primitive with a naked recourse to the basic law of the survival of the fittest - a paradox of development and the complex.

Mathematics is the language of Science and Africa cannot produce men of Science and Science, without mathematics. Without Science , we cannot generate wealth as no single nation in the modern world has ignored science and made sustainable growth - indeed every nation that has invested well in science has reaped a bountiful harvest of development and wealth. So, why don’t we invest ?

Reflecting on Science in Africa in the last quarter of a century, one cannot help recalling the bright hopes and bold aspirations of our generation at the dawn of African political independence. Good education was available and free , even though access was limited; academic mentors were role models and becoming a mathematician was an attractive and satisfying option. A university don had a position of character and reasonable economic muscle, as academic links with the rest of the world were strong and the promotion of international collaborative research was the vogue within and without.

It is as if we have decided as a continent to clamour for wealth and not to thirst for knowledge, apparently ignorant of the dependence of wealth on knowledge in this era of an all pervading Science. Our continued existence is thereby being threatened by our own decisions and actions . Dramatic reversals must therefore occur if African academics in African Universities are to make their needed impact on African societies and their due contribution to twenty first century science. For, if the African mathematician becomes an endangered species then Science in Africa would have gone very far down the road to extinction and the African continent can then only hope to survive at the mercy and pleasure of the strong. This reality must constrain us to search for the path of determination and development, identifying our roles in this critical struggle for our very survival by embracing a vision not only for 2010 but for the indefinite future.

As mathematicians , we shall continue to teach and research in mathematics to enhance the discipline by our modest contributions and to train and prepare the next generation of African mathematicians - a sacred duty and responsibility we have to our colleagues, our University and the World.

We shall teach mathematics , stressing understanding so that mastery of techniques is reinforced thus facilitating the creation of, acquisition of and adaptation to, new techniques so that mathematical skills can be an asset to many activities.

As academics we shall make our expertise and experience available to our students and the nation, conscious that they were acquired and nurtured at the expense of the common man.

We shall offer our commitment to the service of Nigeria and Africa as reflected in our continuous service to African higher education over the last twenty five years.

We can ensure that young academics enjoy and can afford to dedicate a life time to academia for the future of our continent is in their hands. We can help them satisfy their curiosity and creative urge and let them experience the thrill of discovery in the same way that we were so lucky and privileged to do.

Colleagues all, ladies and gentlemen, I thank you for listening.

REFERENCES

7. L. S. O. LIVERPOOL."A note on a Paper of Yang Lo". International Journal of Science and Engineering, (1984).

9. L. S. O. LIVERPOOL and J. NNADI. "The Theorems of Schottky and Miranda - a Novel Approach". Journal of Nigerian Mathematical Society, vol.2 (1983)

10. L. S. O. LIVERPOOL and K. B. Yuguda. "Exceptional Sets for Entire Functions". Contemporary Mathematics, American Mathematical Society U.S.A. vol.25 (1983)

11. L. S. O. LIVERPOOL and UMAR UMAR.  "Some Remarks on Point Picard Sets for Entire Functions". Publications de l'Institut Mathematique Tome 32, (1982).

12. I. N. BAKER and L. S. O. LIVERPOOL. "The Value Distribution of Entire Functions of order at most one".  Acta Scientiarum math. Vol.41 (1979).

13. L. S. O. LIVERPOOL. "Fractional Iteration Near a Fix point of Multiplier 1".Journal London Math. Soc. vol.41 (1979).

14. L. S. O. LIVERPOOL. "On Entire Functions with Infinite Domains of Normality". Aequationes Mathematicae vol.10 (1973)

15. I. N. BAKER and L. S. O. LIVERPOOL. "Further Results on Picard Sets for entire Functions". Proceedings of London Math. Soc. vol.26 (1973).

16. L. S. O. LIVERPOOL. "Some Remarks on factorisation of entire functions". Journal of London Math. Soc. (2) 7 (1973).

17. I. N. BAKER and L. S. O. LIVERPOOL. "Picard Sets for Entire Functions".Math. Zeith. Vol. 126 (1972).

21. L. S. O. LIVERPOOL ."Value Distribution and related questions in iteration theory"-in Value Distribution of Meromorphic Functions. Edited by C.C. Yang, Naval Research Laboratories, U.S.A. Marcel Dekker Inc. USA (1982).

33. S. SHIMOMURA. "Entire solutions of a polynomial difference equation". J.Fac. Sci. Univ Tokyo Ser.A1 Math 28 (1981)

34. D. Zeilberger. "Theorems for a price: Tomorrow’s Semi - Rigorous Mathematical culture". Notices of the American Mathematical Society. vol.40 No.8(1995) .