The Bulletin will expand to accomodate, in each issue, a mathematical or historical article of interest to the mathematical community, as well as a generalist section on `What's new in Mathematics', adapted from the AMS section with the same title and reprinted here with the kind permission of the AMS.
We hope the Bulletin may be a contribution to the main purpose of CIM to act as a forum for the Portuguese mathematical research community.
Gracinda M. S. Gomes (University of Lisbon, Portugal), Jean-Eric Pin (University of Paris VII, France) and Pedro V. Silva (University of Porto, Portugal).
Date: May - July, 2001.
The Term is designed to make Coimbra the gathering point of researchers in the subjects of semigroup theory and automata theory during the months of May, June and July 2001. Besides providing a basepoint for the development of joint research projects, the Term includes multiple activities such as specialized schools and workhops on relevant specific subjects. Postgraduate students will be particularly welcome.
Each school consists of several 5 hour courses held by prominent researchers. The workshops include 50 minute invited lectures and a limited number of 20 minute talks on the specific topics of the workshop, proposed by the participants. Anyone wishing to present such a communication is invited to submit a 1 to 2 page long abstract before February 28 to the e-mail address term2001@cii.fc.ul.pt.
The programme of events is the following:
2-11 May: School on Algorithmic Aspects of the Theory of Semigroups and its Applications
Invited lecturers: J. Almeida (Porto), C. Choffrut (Paris VII), J. Fountain (York), S. Margolis (Bar-Ilan), L. Ribes (Carleton), M. Sapir (Vanderbilt), M. Volkov (Ekaterinburg), T. Wilke (Kiel).
4-8 June: School on Automata and Languages
Invited lecturers: M. Branco (Lisbon), V. Bruy ere (Mons), O. Carton (Marne-la-Vallée), A. Restivo (Palermo).
11-13 June: Workshop on Model Theory, Profinite Topology and Semigroups
Invited lecturers: J. Almeida (Porto), T. Coulbois (Paris VII), H. Straubing (Boston College), P. Trotter (Tasmania), P. Weil (Bordeaux).
2-6 July: School on Semigroups and Applications
Invited lecturers: K. Auinger (Vienna), M. Lawson (Bangor), W. D. Munn (Glasgow), A. Pereira do Lago (São Paulo).
9-11 July: Workshop on Presentations and Geometry
Invited lecturers: R. Gilman (Stevens Inst. of Tech.), D. McAlister (DeKalb), J. Meakin (Lincoln), S. Pride (Glasgow), N. Ruskuc (St. Andrews), B. Steinberg (Porto).
The venue for all events is the Observatório da Universidade de Coimbra, in the peaceful setting of Mount Santa Clara.
Registration fees
May school
Before February 28: Euro 100
After February 28: Euro 120
June events
Before February 28: Euro 100
After February 28: Euro 120
July events
Before February 28: Euro 100
After February 28: Euro 120
Full term
Before February 28: Euro 200
After February 28: Euro 240
In return for their fees, the participants are entitled to receive school/workshop documentation and to participate freely in the social activities, including the corresponding Term dinners, to be held on May 10, June 12 and July 10.
Accompanying persons wishing to join the social pro- gramme will pay 75% of the normal fee. Early payments can be made by international cheque addressed to "Centro Internacional de Matemática" (CIM). The cheques should be sent to:
Patrícia Paraíba,Detailed information on accommodation and travel arrangements will be supplied in the second announcement, to be issued soon.
C.A.U.L., Av. Prof. Gama Pinto 2,
1649-003 Lisboa, Portugal.
A limited number of scholarships will provide funding for those participants in need of financial support, particularly postgraduate students. Anyone wishing to apply for support is invited to do so before December 31, mentioning the amount of funding required and justifying such a request.
In order to receive further information on the Thematic Term, the potential participant should fill out the at- tached preregistration form and return it by e-mail to the address term2001@cii.fc.ul.pt. The information contained in the form will be assumed to be provisional.
For more information on these events and registration forms, please visit the site
http://alf1.cii.fc.ul.pt/term2001/The Thematic Term has the support of Fundação Calouste Gulbenkian, Fundação para a Ciência e Tecnologia, Centro de Matemática da Universidade do Porto and Centro de Álgebra da Universidade de Lisboa.
Estelita Vaz (University of Minho, Portugal), J. Maia (University of Minho, Portugal) and K. Walters (University of Wales Aberystwyth, United Kingdom).
Date: 25-29 June, 2001.
Aims
Despite its relevance to a wide number of industries, Rheology and Non-Newtonian Fluid Mechanics are subjects that are often viewed as being of prohibitive complexity to newcomers to the field and have often not been used to the fullest possible extent. The aim of the School is, therefore, to interest young researchers into the field by helping to bridge the gap between the available theoretical tools and existing problems of a mathematical nature in industry and academia.
The School will be held in the Guimarães Campus of the University of Minho, Portugal.
Lecturers
K. Walters, University of Wales
A. R. Davies, University of Wales
M. H. Wagner, Technical University of Berlin
G. Marrucci, University of Naples
R. Keunings, Catholic University of Louvain
F. P. T. Baaijens, University of Eindhoven
Registration and fee
Registration can be performed by e-mail, fax or letter directly to the School Secretariat at:
Summer School SecretariatThe registration letter must specify the participants' name, institution, address, telephone, fax and e-mail.
Ms. Elisabete Santos
School of Sciences, University of Minho
4800-058 Guimarães, Portugal
Phone: 351 253 510 159
Fax: 351 253 510 153
e-mail: s.school@ecum.uminho.pt
The following School Fees will apply:
Before March 31, 2001: 350 Euro.
After March 31, 2001: 450 Euro.
The fee includes attendance at the lectures, the support materials (programme, book, etc.), five lunches (monday to friday), a visit to the Port Wine cellars and admission to the school dinner. The fee can be paid either by cheque payable to Summer School 2001 or by bank account transfer. The banking details are:
Bank: Caixa Geral de DepósitosDeadlines
Branch: University of Minho - Azurém
NIB (account number): 0035 0130 00001457300 06
The absolute deadline for the receipt of registrations is April 30, 2001. The number of participants is limited to 80 and, therefore, early registration is advised.
For more information on this event, to be held in Guimarães, please visit the site
http://www.dep.uminho.pt/SummerSchool/
Filomena Dias d'Almeida (Engineering Faculty, Univ. of Porto, Portugal) and Paulo Beleza de Vasconcelos (Economics Faculty, Univ. of Porto, Portugal).
Date: 3-6 July, 2001.
Aim of the School
The aims of this advanced school are: to present the state-of-the-art methods and tools to solve large scale linear problems, namely large linear systems and large eigenvalue problems, to bring together specialist researchers on computational mathematics and to encourage the interchange of new ideas, to create a suitable environment for the participants to get acquainted and involved in today's computational mathematics research problems.
Topics
Claude Brezinski, Univ. Lille, France (to be confirmed)
Jack Dongarra, Univ. of Tenesse and Oak Ridge Nat.
Lab., USA
Iain Duff, CERFACS, France and RAL, UK
Joaquim Júdice, Mathematics Dep., Coimbra Univ.,
Portugal
Osni Marques, LBNL, USA
Francisco Moura, Informatics Dep., Minho Univ., Portugal
Orlando Oliveira, Physics Dep., Coimbra Univ., Portugal
Rui Ralha, Mathematics Dep., Minho Univ., Portugal
Program and registration fee
A detailed version of the program will be available soon. Registration fee: Euro 100. It includes the school documentation and coffee.
Travel information and accommodation
Soon we will provide information about travel information and accommodation.
Financial support
CIM - Centro Internacional de Matemática
CMAUP - Centro de Matemática Aplicada da Universidade do Porto
(applied for)
FCT - Fundação para a Ciência e Tecnologia (applied for)
FEP - Faculdade Economia do Porto
FEUP - Faculdade Engenharia da Univ. Porto
For the registration form and more information on this event, to be held
in Porto, please visit the site
http://www.fep.up.pt/docentes/pjv/LSC.html
F. Miguel Dionísio (IST, Technical University of Lisbon, Portugal), José Carlos Teixeira (University of Coimbra, Portugal) and Bernd Wegner (Technische Universität Berlin, Germany).
Date: 13-15 September, 2001.
Aims
The workshop will provide an open forum for the exchange of information and presentations on electronic media in Mathematics for mathematicians and people using mathematics in applications. Three main subject areas are to be covered:
A. B. Cruzeiro (University of Lisbon, Portugal) and L. Streit (University of Bielefeld, Germany).
Date: 18-22 September 2001.
Aims
The need for the development of infinite dimensional Analysis on spaces of continuous paths or of less regular objects such as distributions has become evident mainly by physical motivations (e.g. Quantum Mechanics and Quantum Field Theory). These spaces are endowed with probability measures, one of the more regular cases being the law of Brownian motion. In this case the It o calculus provides the underlying techniques to manipulate irregular functionals of the paths and the corresponding infinite dimensional Analysis has developed intensively in the past recent decades giving rise to important results in Mathematics, but also applications outside the initial framework (e.g., Filtering and Control Theory, Financial Mathematics). More recently, special attention has been given to the geometry of (curved) spaces. The goal of the workshop is to bring together various approaches to infinite dimensional Analysis.
Information about some of the events not included in the Thematic Term is given here, in complement to the announcements published in previous issues of the Bulletin.
In the year 2000, professors António Ornelas (Évora), J. C. David Vieira (Aveiro), Joaquim Madeira (Coimbra) and J. C. Tiago de Oliveira (Évora) taught courses at the Instituto Superior de Educação of Cabo Verde, in the city of Praia. These missions received support from Instituto de Cooperação de Portugal and Instituto Superior de Educação of Cabo Verde.
The new chairman of the Cooperation Committee for 2001 is Prof. J. C. David Vieira (Aveiro).
These facilities are located at Complexo do Observatório Astronómico in Coimbra and include: office space, computing facilities, and some secretarial support; access to the library of the Department of Mathematics of the University of Coimbra (30 minutes away by bus); lodging: a two room flat.
At least one of the researchers should be affiliated with an associate of CIM, or a participant in a CIM event.
Applicants should fill in the electronic application form
http://www.cim.pt/cim.www/cim_app/application.htmalso acessible from the CIM web page (see below).
http://www.cim.ptThis is mirrored at
http://at.yorku.ca/cim.www/
Alvarus Thomas (fl. 1509), a Portuguese master at the University of Paris at the beginning of the sixteenth century, is still a poorly known historical figure. In his Liber de triplici motu he presented a comprehensive and sophisticated analysis of the theory of proportions and of the science of motion of his time, in the characteristic form of the Calculatory tradition. Besides some interesting criticism of contemporary physical theories, this work is also relevant from the point of view of mathematics since Thomas achieves some surprising results in the study of infinite series.In this paper I summarize the present knowledge on Alvarus Thomas. I collect all biographical information currently available and present very briefly the contents of the Liber de triplici motu adding some observations on its historical context and influence.
Front page of the Liber de triplici motu
Sirvan estas notas sobre el Liber de triplici motu de incentivo para que alguien lo analice, tan minuciosamente como merece, y para que alguna corporación ibérica emprenda su traducción. De los eruditos portugueses esperamos que indaguen en sus archivos datos bastantes para trazar la biografia de este sutil ingenio, digno precursor de Pedro Núñez 2.These words were written in 1926 but unfortunately such a programme of study on this "digno precursor de Pedro Nunes" remains to be accomplished.
Rey Pastor was not the first scholar to mention the work of Alvarus Thomas. In 1913, the great historian of science Pierre Duhem, in his Études sur Léonard de Vinci presented the first modern analysis of Thomas' work 3. Duhem noted Thomas' erudition and brilliance, and commented that
Les problèmes que ces ma itres et régents s'acharnent à résoudre, dont ils entrevoient parfois la solution, en dépit de leurs connaissances rudimentaires en Mathématiques, ce sont les deux grands probl emes de l'intégration des fonctions et de la sommation des séries. Et l'on se demande alors quels résultats ces hommes n'eussent point obtenus, quelle promotion ils n'eussent point imprimée aux Mathématiques s'il leur e ut été donné de lire Archimède 4.Following these pioneer studies, other historians of science, while investigating the contributions of sixteenth century authors to the development of physics and mathematics, pointed to the important role played by Alvarus Thomas. Of special interest is the work by the eminent historian of mathematics Heinrich Wieleitner, in which Thomas' techniques for the summing of infinite series are analysed 5. In more recent years, scholars such as Marshal Clagett, William Wallace, Edward Grant and Edith Sylla devoted some attention to the work of Alvarus Thomas. Their studies on the significance of Thomas' contributions will be used throughout this paper. At this point it is sufficient to quote the evaluation made by William Wallace:
At Paris [...] there can be little doubt that Thomaz was the calculator par excellence at the beginning of the sixteenth century, and the principal stimulus for the revival of interest there in the Mertonian approach to mathematical physics. 6Strangely, the impact of these authoritative voices in the Portuguese community has been practically nil. If we except the three brief pages that Gomes Teixeira devoted to Thomas in the História das Matemáticas em Portugal 7, and a short, but correct, notice in the most important Portuguese encyclopedia 8, there are no more detailed or reliable references to Thomas' scientific work, much less a careful analysis of his book. Garção Stockler, Rodolfo Guimarães and Pedro José da Cunha in their classic studies9 do not mention Alvarus Thomas, and later historians of Portuguese science follow essentially the same pattern10. The most apt characterization of this state of affairs was perhaps Joaquim de Carvalho's assertion that Thomas was "[...] uma das figuras mais lamentavelmente esquecidas da nossa história científica."11 Further evidence of the oblivion into which Alvarus Thomas has fallen in his own country is that references to his biography or work often contain innacuracies: caveat lector. The objective of this paper is to provide a very brief introduction to the life and work of Alvarus Thomas. I do not claim to present here any new findings related to this Portuguese scholar, nor a detailed explanation of his work. However, the neglect into which this Portuguese master has fallen among his countrymen and the importance of his work justify that even such a modest project should be undertaken 12.
The facts of Thomas' life that can be ascertained on a documentary basis are very few and cover a time span of only about ten years 13. The first piece of evidence we have is his book - as far as is known, his only work - published in Paris in 1509 (or 151014). The complete title is: Liber de triplici motu proportionibus annexis magistri Alvari Thomae Ulixbonensis philosophicas Suiseth calculationes ex parte declarans, a translation of which runs as follows: "Book on the Three [kinds of] Movement, with Ratios Added, by Master Alvarus Thomas of Lisbon, Explaining in Part Swineshead's Philosophical [i.e. Physical] Calculations".
The "explicit" of the Liber de triplici motu states that it was "compositus per Magistrum Alvarum Thomam ulixbonensem. Regentem Parrhisibus in Collegio Cocquereti." Another document confirms that in 1513 Thomas was still teaching Arts, i.e. Natural Philosophy, in the same college.
Thus, Alvarus Thomas was born in Lisbon and acted as Master of Arts and "regens" in the Collège de Cocqueret in Paris from, at least, 1510 to 1513. This Collège had been established in 1439, and although it never rose to the distinction of the Collège de Montaigu or the Collège de Saint-Barbe, it had among its teachers and students some leading intellectuals of the time 15.
The "regens" was generally a student of one of the higher Faculties (Theology, Law or Medicine) who payed for his studies by teaching Arts in a college. Indeed, it is known that Thomas enrolled at the Faculty of Medicine in 1513 and it is very likely that he studied there while teaching Arts at Cocqueret. He completed his licentia examinations in Medicine two years later and obtained his degree of doctor in 1518. In that same year he was appointed professor at the Faculty of Medicine. After 1521 his signature no longer appears in the University archives. What happened afterwards is not known.
The fact that he was studying Medicine in 1513 and that he received a doctorate in 1518 makes us suppose that he was not yet a middle aged man at that time. On the other hand, the sound command of an impressive range of sources that he shows in his book and his teaching position in 1510 at Cocqueret are hard to imagine (but not altogether impossible) in a man in his early twenties. Comparing with the academic career of other Portuguese scholars in Paris at the same period, it is plausible to suppose that Thomas had arrived in Paris by 1500, as a young man of around 16-18 years of age, and that he wrote the Liber de triplici motu a few years after having finished his studies in Arts and before embarking on the study of Medicine. This would mean that he would have been born in Lisbon around 1480-85.
The presence of Alvarus Thomas in Paris is quite natural. After a period of lesser prominence during the fifteenth century, by the turn of the century the University of Paris had recovered the glory of past ages. It had established itself as the most reputable university in Europe, attracting students from everywhere except Italy where the local Universities disputed this leading position. It is known that Portuguese students had been sent to the University of Paris since as early as 1192. In the period 1500-1550 around 300 Portuguese attended the University of Paris16.
In the first decades of the sixteenth century a remarkable group of Portuguese students was at Paris. Besides humanists, philosophers, and theologians, among the contemporaries of Thomas one finds men who will greatly contribute to the scientific history of Portugal. Such is the case of Pedro Margalho (1471?1556), Fran- cisco de Melo (14901536) and João Ribeiro, for example. Also outstanding was the group of Spaniards. Among others, Gaspar Lax (14871560), Pedro Ciruelo (14701554), Juan Martínez Silíceo (14861557), Juan de Celaya (14901558), were contemporaries of Thomas in Paris17. These Iberians would play an important role in the History of Science, which prompted a modern historian to comment that
Among the many foreigners at Paris at the turn of the sixteenth century, no group is more interesting than that of the Spaniards and the Portuguese18.The history of the intellectual relations between these men is, to a great extent, still to be made. The study of their influence in the Iberian Peninsula is also a desideratum. To a greater or lesser extent these men seem to have been influenced by the Scottish nominalist John Major (14671550) who was, at the beginning of the sixteenth century, the leading intellectual figure in Paris. Major pontificated in what was perhaps the most important of the colleges of the University of Paris at the time, the Collège de Montaigu, but his pupils would eventually occupy chairs in all other colleges, thus extending his influence to the whole of the University of Paris. There is no evidence of Thomas being directly associated with Major or of having been his direct disciple, but no doubt he benefited from the intellectual environment around the Scottish master.
Even surrounded by men of great intellectual prestige, Thomas seems to have been a leading figure. One of his contemporaries considered him to be superior to Pierre d'Ailly19 and modern historians confirm Thomas' intellectual position among his peers20.
By mid fourteenth century the study of motion - a central and always problematic question in the corpus of aristotelian physics - was radically changed due to the contributions of a group of men at Merton College, in Oxford. In a period of about twenty years, the successive appearance of a number of texts on proportions and ratios, motion, and logical rules applied to physical questions, heralded a new approach to problems of natural philosophy. Of these, the most important were: Thomas Bradwardine, De proportionibus velocitatum (1328), William Heytesbury, Regulae solvendi sophismata (1335), John Dumbleton, Summa logicae et philosophiae naturalis (1349), Richard Swineshead, Liber calculationum (ca. 1350). Instead of pursuing an analysis of motion in the traditional categories of act and potency, these men adopted a formal and highly speculative analytical approach which considered motion essentially as a ratio. Their analysis of motion included detailed discussions on the possible types of motions (uniformiter, uniformiter difformis, difformiter difformis, etc.), the description of each of these different types of motion, and an inspection of the origin of each motion. These studies were abstract, without reference to any natural event or artifact, and made extensive use of logical techniques originally developed in other intellectual pursuits such as the study of language.
The original context of these discussions was the much debated question of the "intensio and remissio formarum", which is, basically, the question of how qualities varied in intensity. To the Oxford "Calculators" - such was the designation by which they came to be known, and Swineshead "the Calculator" - variations of velocity, that is, local motion, were treated as variations in the intensity of a quality, in the same way as color changes its hue or a body becomes warmer. But the problems they addressed had a much broader context than merely the question of understanding local motion (motus localis); calculatory techniques were also used in medicine and theology, for example. From the perspective of the history of mechanics, the contributions of the Merton school have been summarized thus by one of the most competent historians of medieval science:
From the discussions of these four men at Merton emerged some very important contributions to the growth of mechanics: (1) A clear-cut distinction between dynamics and kinematics, expressed as a distinction between the causes of movement and the spatial-temporal effects of movement. (2) A new approach to speed or velocity, where the idea of an instantaneous velocity came under consideration, perhaps for the first time, and with a more precise idea of `functionality'. (3) The definition of a uniformly accelerated movement as one in which equal increments of velocity are acquired in equal periods of time. (4) The statement and proof of the fundamental kinematic theorem [...]22These are no small intellectual accomplishments. Although to a modern reader the texts that these men produced are certainly prolix, confused and difficult to follow - a critique that some contemporaries also made - underneath this complexity lies an exceptional ability to seize upon and extract the mathematical features of the problem of motion. What is perhaps their greatest feat - sometimes considered the most outstanding medieval contribution to physics - was the statement and demonstration of the so-called "Mean Speed Theorem" for uniformly accelerated motion. In modern terms, this theorem asserts that a body in uniformly varied motion during a certain interval of time will traverse the same distance as a body with a uniform velocity equal to the instantaneous velocity at the middle instant, in the same interval of time. The power of this theorem lies in equating, for the purpose of calculating the distance traversed, an accelerated motion with a uniform motion. This theorem was proved by means of many different geometrical and numerical arguments and became the cornerstone of the studies of motion by the Calculatores.
The Calculatory tradition evolved significantly when it arrived on the Continent. In Paris, the ideas and techniques of the Merton approach were incorporated into the more realistic framework which had been worked out by fourteenth century thinkers such as Jean Buridan and Nicole Oresme. A salient feature of the Parisian achievements was the introduction of the notion of impetus in the analysis of motion.
The influence of Swineshead's Liber Calculationum is clear, but Thomas' exposition is more systematic and better organized. The first impression any reader has is the extent of Thomas' knowledge. His sources for mathematics range from the older Nicomachus or Boethius to the very recent edition of Euclid by Bartholomeus Zambertus (Venice, 1505). He is at ease with the Englishmen Swineshead, Bradwardine and Heytesbury, but also with Parisians such as Oresme, and with the Italians (Paul of Venice, James of Forli, etc.). The Portuguese master is in the exceptional position of knowing both the formal techniques of the Merton approach, the conceptual tradition of the Parisian school, and the Italian contributions24.
But the contribution of Alvarus Thomas would not be correctly described by mentioning simply his role as the catalyst of the Merton tradition in Paris. From the perspective of his mathematical accomplishments the Liber de triplici motu contains remarkable results. Since it is impossible to even review its contents, I will simply comment on some aspects that relate to summing infinite series.
Thomas follows strategies typical of the Calculatory tradition, and by ingenious and complex use of the Mean Speed Theorem he manages to establish some suprising results. The approach used by Thomas can be better understood by using present day terminology. The reader can imagine that one is considering a modern graph with velocity represented in the vertical axis and time in the abscissas25. In such a depiction, a motion at constant velocity is represented by a horizontal line, and a uniformly accelerated motion by a straight line with some finite slope. A generic motion will be represented by some curve. In all cases, the total space traversed by the mobile is given by the area below the curve. Alvarus Thomas considers different types of motion. The question he tries to answer is inspired by the Mean Speed Theorem, but now for these much more complicated motions: Given a certain complex motion, what should the uniform velocity be such that a body moving with this constant velocity traverses, in the same time, the same distance as the body following the more complex motion?
Thomas cannot address the problem in the most general terms, but he considers complex motions that correspond to a division of the time axis in a geometrical progression. In each interval the velocity is assumed to be constant or uniformly accelerated. By this judicious construction and using the Mean Speed Theorem, Thomas is then able to calculate the total space traversed by the mobile and the corresponding uniform velocity which would make it traverse the same distance in the same time. It is not difficult to realize that, from a mathematical point of view, Alvarus Thomas is calculating the sum of an infinite series.
One of the motions considered by the Portuguese master corresponds to the series
1 + 2x + 3x2 + 4x3 + ...Thomas is able to show that the sum of this series is equal to the square of the sum of the series:
1 + x + x2 + x3 + ...In a typical Calculatory spirit, Thomas will stretch his techniques to the limit, considering motions progressively more complex. With this he is able to obtain remarkable mathematical results. For example, he shows that the series
1 + (2/1)x + (3/2)x2 + (4/3)x3 + ...is bounded above by
1 + x + x2 + x3 + ... = 1/(1-x)and bounded below by
1 + 2x + 3x2 + 4x3 + ... = 1/(1-x)2.Rey Pastor observed that several of the series analysed by Thomas would cause many difficulties even to presentday students. He is able to sum series such as,
1 + x + ax2 + bx3 + a2x4 + b2x5 + ...or even,
1 + (3/2)(1/2) + (5/4)(1/22) + (9/8)(1/23) + ...Naturally, there is no detailed inspection of the criteria of convergence of the series studied, nor an attempt at rigorous definitions. Nevertheless, Thomas is aware that while some of the series he proposed can be summed, others cannot, either because it is technically very difficult (or impossible) or because the partial sums of terms increase very rapidly.
In what concerns the study of infinite series, Alvarus Thomas is the high point of an intellectual tradition which had reached its limits. The discursive approach to these mathematical problems would soon be abandoned, and forgotten, with the development of the much more powerful algebraic approaches of the seventeenth century. Jakob Bernoulli's Tractatus de seriebus inifinitis (1689) ushers in a new world in the study of infinite series. But in the study of local motion, the book of Alvarus Thomas was perhaps of much more relevance. It has been plausibly argued that the Liber de Triplici Motu may have been influential in the scientific formation of Domingo de Soto (14951560), either directly or via Soto's teacher in Paris, Juan de Celaya. A possible intellectual connection between Thomas and Domingo de Soto is of the utmost historical significance since it is known today that Soto was the first author to have argued that the free fall of bodies is a motion uniformiter difformis with respect to time. That is, in modern terms, that in free fall the body traverses spaces in direct proportion to the squares of the times of fall28.
After the investigations of William Wallace, it is today agreed that Galileo was well acquainted with the results of the tradition of the Calculatores after Domingo de Soto. It is very likely that it was from this knowledge that Galileo first noticed the correct law for the free fall of bodies, which he presented at the beginning of the seventeenth century29. In this sense, the role of Alvarus Thomas, as a leading figure in the Calculatory tradidion that ultimately led to Galileo's outstanding contributions, needs to be noted. But there is more to interest Portuguese readers in this fascinating story. Galileo's knowledge of the ideas and techniques of the Calculatores was drawn from his study of the lecture notes of the Jesuit professors at the Roman College. In the efficient Jesuit network of colleges the Calculatory tradition underwent a diffusion originating in the Iberian Peninsula - a direct consequence of the return to the Peninsula of former students at Paris, and in particular, of the teaching of Domingo de Soto. In fact, a substantial number of manuscripts from Jesuit colleges confirms that in the last decades of the sixteenth century the terminology and notions of the Calculatores were being used in the analysis of the nature of motion in Portugal30.
Instead of elaborating on these outstanding contributions, which are both far from the area that I have specialized in, I would prefer to reflect on what has impressed me from a more personal point of view, restricting myself to my own area of research.
We have witnessed enormous growth in Differential Geometry in the last quarter of a century. Gromov and Yau are clearly in the center of this development. Yau's solution of the Calabi conjecture is maybe the single most important result with important consequences in other areas like Algebraic Geometry. Gromov's most influential contribution might be his paper on pseudo-holomorphic curves which started a line of research in Symplectic Topology that recently culminated with a solution of the Arnold conjecture. One should also mention his highly original theory of hyperbolic groups and his numerous contributions to Riemannian Geometry.
If I look at the whole century and not only the few years that I have been able to witness personally, the theory of semi-simple Lie algebras and Lie groups and their rep- resentations comes to my mind. This is a theory that started in the last decades of the nineteenth century with the work of Killing and Cartan and took its final shape in the twentieth century with contributions of Cartan and Weyl and many others. First of all the beauty and intricate structure of this theory is fascinating. Then it is also central in so many areas of Mathematics that it certainly deserves to be considered as one of the the truly important contributions to Mathematics in the twentieth century.
Gudlaugur Thorbergsson was born in Melgraseyri, Iceland, having graduated from the University of Iceland. His postgraduate studies were done at the University of Bonn and he held positions at Bonn, IMPA - R. J. and University of Notre Dame. He is currently at the University of Köln. An important part of Professor Thorbergsson's research is in submanifold geometry. Recently he contributed a survey on isoparametric hypersurfaces and their generalizations to the Handbook of differential geometry, Vol. I, published by North-Holland.
However, if I was asked to suggest some development of lasting importance and greatest impact then I might propose the work of Alan Turing on Mathematical Logic in the 1930s.
His conceptual development of what is now popularly known as a universal Turing machine anticipated the modern computer. His description of an abstract machine which can be made to carry out complicated tasks through a combination of simple instructions inserted on a tape is easily recognised now as a description of a programmable computer (with the tape as the programme).
Turing's motivation came from one of the most abstract of ideas in mathematics: The problem of "decidability" (Hilbert's second problem) which relates to whether, for a given well formulated mathematical problem, a solution necessarily exists or not. In the context of the Turing machine, given a finite set of instructions, it may be impossible to decide whether the machine would continue forever, or stop in some finite time.
The first practical application of Turing's ideas was during the second world war. Turing worked for the British Government Code and Cypher School on decoding military transmissions encoded by the german "Enigma" machines, developing practical decoding machines based on his original abstract ideas. The second important application of his ideas was the construction of the first programmable computer in Manchester under the aegis of Max Newman, in the late 1940s.
Mark Pollicott has held positions at the universities of Edinburgh and Warwick, as well as visiting positions at IHES, MSRI and IAS (Princeton). He was an Investigador Auxiliar of INIC from 1988-92, whereafter he took up a Royal Society University Fellowship at Warwick. He presently holds the Fielden Professorship in Pure Mathematics at Manchester University, England.
Originally published by the American Mathematical Society in What's New in Mathematics, a section of e-MATH, in
http://www.ams.org/new-in-mathReprinted with permission.
Professor Zeeman, at 7 you were fascinated when your mother showed you how to solve a problem using the unknown x. I'm sure that during your mathematical career some of the results you proved must have given you a similar feeling.
Which were the peaks of your research?
When my mother showed me at the age of 7 how to use x for an unknown it was a revelation to me. However, I think the feeling of revelation that you get when someone reveals something to you is different from the feeling of exhilaration that you get when you discover something for yourself. Revelation can be wonderful, but exhilaration can be even better!
I can distinctly remember a few revelations such as un- derstanding limits rigorously for the first time (and hence calculus), or understanding the complex numbers as the algebraic closure of the reals, or using groups and fields to show the insolubility of the quintic, or proving the knottedness of knots, or understanding Newton's proof of elliptic orbits, and much later realising that Newton's equations are contained in the symplectic structure of a cotangent bundle, or understanding Mather's proof of Thom's theorem on elementary catastrophes.
When I began proving my own theorems each one seemed the best at the time, but in retrospect I suppose I am particularly fond of having unknotted spheres in 5-dimensions, of spinning lovely examples of knots in 4-dimensions, of proving Poincaré's Conjecture in 5-dimensions, of showing that special relativity can be based solely on the notion of causality, and of classifying dynamical systems by using the Focke-Plank equation. And amongst my applications of catastrophe theory I particularly liked buckling, capsizing, embryology, evolution, psychology, anorexia, animal behaviour, ideologies, committee behaviour, economics and drama.
When he introduced you as the 1992/93 Johann Bernoulli Lecturer, Floris Takens mentioned that you served as a flying officer in R. A. F. during World War II. I presume it must have been between high school and university.
What are your recollections of that experience and how did it affect your mathematical path?
I served in the Royal Air Force during the war from 1943 to 1947 (between the ages 18 - 22). I was a navigator on bombers, trained for the Japanese theatre, but that was cancelled because they dropped the atomic bomb a week before we were due to fly out. Since the death rate was 60% in that theatre it probably saved my life, but at the time I was disappointed not to see action, although relieved not to have to bomb Japan, the land of my birth.
The air force was a rewarding experience, a breath of freedom that allowed my self-esteem to recover from the prison of boarding school. It enabled me to realise that I loved mathematics, and wanted to do that more than any other career. I was unashamedly happier my first day back as a student than my last day as an officer in the air force. Of course by then I had forgotten all my mathematics, and so it set me back 5 years in my mathematical career, but then who cares now that I am 75 and still at it. I am grateful to the air force for providing an opportunity for personal development, and for enabling me to laugh at myself slightly as an academic ever since.
And yet "They (the problems one sets about solving) are rarely solved", I quote from the interview mentioned at the beginning.
Were there problems of which the solution eluded you? Do you still think about them from time to time?
Of course the solutions to many problems have eluded me, and I still think about them from time to time. A good mathematician probably has 25 failures to each success. The important thing is that new ideas keep coming.
One of my favourite failures is the 3-dimensional Poincaré Conjecture, which I spent the first year of my research thinking about, and which is still unsolved today. Another little hobby is to try and rediscover Fermat's own original proof of his last theorem, at least for n = 3, without using complex numbers (which he is unlikely to have used). I have done half of it.
At the moment I am busy trying to unfold some difference equations in higher dimensions using alien techniques from dynamical systems, algebraic geometry and number theory. Last month I managed to prove a theorem that I conjectured 25 years ago about Eudoxus' theory of proportion. I suspect that Eudoxus was able to take ratios of ratios, which Euclid was not able to do in Book 5 (nor in Book 6, in spite of Definition 5, which is a later blemish added by other writers) because he had fouled up Eudoxus' beautiful abstract approach by, ironically, introducing the Euclidean algorithm too soon.
"Among students the good ones are automatically good and it is not possible to improve the bad ones' performance". You are talking about Maths students. I agree with you and it brings to my mind the following question.
What do you think of Mathematical Education as a scientific discipline?
There are two different meanings to the word "discipline". The first meaning is my definition of an academic discipline as a corpus of works of genius that a student can study without the interference of the lecturer. In this sense mathematics is a discipline, as are also physics, chemistry, biology, literature, etc. But mathematical education is not.
This became sharply clear to me once at Warwick. Each year the Mathematics Institute there runs a year-long symposium, with some 80 long-term visitors, in topics like topology, groups, dynamical systems, algebraic geometry, etc. One year we debated whether to run a symposium on mathematical education, and tried out a pilot week to examine the potential, but it transpired that there was not enough material: it was not an academic discipline.
On the other hand vocational apprenticeship to the profession of mathematical teaching needs discipline if the student is to master the necessary techniques. And such discipline needs to be taught, needs specialists to teach it, and needs to be supported by research on curriculum reform and the analysis of learning techniques.
"Deep down I am a geometer and geometry is very clear. ... Proofs are rigorous and very satisfying from the aesthetic viewpoint". This is something you said. On the other hand René Thom is well known for statements such as "I do not think that a mathematician's vocation is to prove theorems" or "One can always find imbeciles to prove theorems". Such different approaches to Mathematics and however parts of your works overlap very significantly.
How did that come about? It would be unthinkable to interview you and not bring up Catastrophe Theory ...
Thom is quite witty, and he occasionally talks rubbish because he loves being provocative. At the same time he is the greatest genius I have had the privilege to know well. He is the fountainhead of many wonderful ideas. Sometimes he does not bother to be rigorous, nor to get down to the nitty-gritty of proofs, whereas I do. I like to rework and repolish a proof until it is in its simplest rigorous form.
Thom occupies a position halfway between mathematics and philosophy. He was reluctant to get his hands dirty predicting experiments, lest the potential failure of those predictions detracted from the purity of his theory. He quoted the unfortunate example of D'Arcy Thompson who got all his theoretical ideas right but all his experimental predictions wrong, and said he did not want to be caught the same way.
I, on the other hand, occupied a position halfway between mathematics and science. I wanted to get my hands dirty, and make predictions, and get the experimentalists to test them, because I knew that the scientific community would never take a theory seriously unless it was capable of being tested experimentally. And I was gratified that several of my predictions were confirmed. Some were refuted, and others remain to be tested.
Since we occupied different positions Thom and I complemented each other. We met over the mathematics and the theory in between, and our collaboration turned out to be very fruitful.
In the 60's when you were in your early forties you founded the Mathematics Institute and Research Centre at Warwick. It must have meant a lot of paper and administrative work and surely it must have affected your mathematical output.
Do you have any regrets?
It is true that the founding of the Mathematics Institute and Research Centre at Warwick was a big administrative load, that prevented me from doing much research in topology during the first 5 years 1964 - 69 (while I was 39 - 44). But I certainly had no regrets, because founding Warwick was one of my best and most reward- ing achievements. And it made me into a much broader mathematician. During 1968/9 I learnt all about dy- namical systems by running a symposium on it for a year, with many of the world leaders including Smale and Thom coming for long periods. Then in 1969/70 I had the good fortune to spend a sabbatical year with the latter at the IHES in Paris, where I learnt all about catastrophe theory. So I was very fortunate to get in on the ground floor of such beautiful new subjects.
Your mathematical career has been showered with lots of awards and other forms of recognition: A knighthood, an F. R. S. fellowship, the Senior Whitehead prize, a Forder lectureship, book dedications ...
Was it important for you to have achieved such a recognition?
Of course I was very pleased to receive such recognitions, although I never set out to achieve them - I merely did what I liked best in teaching and research. The awards proved useful in that they enabled me to go ahead and do further things.
I was elected a Fellow of the Royal Society primarily for my work in geometric topology, which helped to resuscitate that subject in the 60's, and partly for my work in dynamical systems and catastrophe theory. The Whitehead Prize and Forder Lectureship were for both research and teaching. I attach great importance to teaching, and at Warwick I insisted that it should be given as much importance as research, which is one of the reasons why the Warwick Mathematics Institute remains so robust today. I was given the Royal Society Faraday Medal for my contributions to the public understanding of science, in particular for giving the Royal Institution Christmas Lectures in 1978, out of which grew the Mathematics Masterclasses for 13-year-olds (which have now been flourishing for 20 years and have spread to 50 centres around the country). My knighthood was probably for four things: my research, founding Warwick, creating masterclasses, and heading an Oxford College.
Your wife is a jeweller (I think she even coined the term "bracelet" in "bracelet umbilic"), you are now in Portugal to give a talk in connection with a video of which the title is "Geometry and Perspective" ...
Are you interested in Art? Do you have a favourite painter, a favourite sculptor? I'm tempted to mention Barbara Hepworth or Henri Moore but that is a bit too obvious perhaps ...
My wife Rosemary is indeed a jeweller and makes beautiful very feminine enamelled jewelry. Although she has never been a mathematician, yet she loved geometry at school, and so I try to explain geometrical things to her from time to time.
I coined the term "umbilic bracelet" when I tried to explain to her the natural stratification of the 4-dimensional space of real cubic forms in two variables. The elliptic and hyperbolic umbilics form the two open strata, and are separated by the parabolic umbilics, which form a codimension-1 stratum, which is a cone on the bracelet; the bracelet itself being a bundle over S1 with fibre a triangular hypocycloid and group Z3.
Yes, I am very interested in art. My favourite painters are from the Renaissance: Masaccio, Giovanni Bellini, Piero della Francesca, Botticelli, Leonardo, Filippino Lippi and Raphael; and (later) Vermeer, Ingres, Velasquez and Turner. Favourite sculptors include the Pisanos, Donatello, Michelangelo and Rodin, as well as individual pieces of sculpture like Djhutmose's unfinished quartzite head of Queen Nefertiti from Armana (the Cairo one rather than the Berlin one), the Egyptian wooden harp head from the Louvre, Myron's Diskobolos and Greek wrestlers from the 5th century BC, the Winged Victory of Samothrace, and (more modern) Boccioni's "Unique forms of continuity in space", Duchamp-Villon's "The great horse", Teddy Hutton's "Pregnant Woman", and Makonde sculptures from Tanzania and Mozambique. Modern painters I like include Rodolfo de Sanctis, Gordon Onslow-Ford, Edith Smith, Joe Brotherton, Peter Edwards and Picasso (although some of his work is junk). Your suggestions of Barbara Hepworth and Henry Moore have topological appeal but they do not make my spine tingle or move me to tears, as do the sculptures listed above.
(Questions and picture by F. J. Craveiro de Carvalho)
Sir Erik Christopher Zeeman is one of the great XXth century mathematicians. His university studies were at Christ's College, Cambridge and he also received his PhD from Cambridge.Professor Zeeman spent most of his career in Cambridge, Warwick (where he founded the Mathematics Department and Research Centre) and Oxford.
His election to the Royal Society of London in 1975, the Senior Whitehead Prize in 1982, the first Forder Lectureship of the London Mathematical Society in 1987 and the Royal Society's Faraday Medal in 1988 are some of the honours he received. He was also knighted in 1991.
Professor Zeeman is the author of the video Geometry and Perspective based on the Royal Institution Christmas Lectures he gave in 1978.
We will be talking about Professor Bento Caraça the man, who, from his birth in 1901, fought for survival and was only saved by a miracle. A life which only lasted 47 years, but which was enough to enrich his era and to bequeath to us a cultural and ethical legacy of the highest and incomparable value.
Bento de Jesus Caraça
Of the many great personalities who marked national life over the last century, Bento Caraça was particularly noted for the greatness and universality of his messages and for his courage, even his spirit of sacrifice, in defending them.
He was made to pay dearly for this defiance, respectable though it was. Bento Caraça was mercilessly persecuted by the police under the dictatorship: he was imprisoned at Aljube, he lost his professor's chair where he was a teacher like no other, and he suffered much economic hardship, whilst his health was at risk.
But the ultimate shock for the Professor was his expulsion from his university teaching post, in 1946, when he was professor at the Instituto Superior de Ciências Económicas e Financeiras, an institute for which he had so much affection and which owed so much to him.
On his own merit and as an exceptional measure, Bento Caraça was appointed 2nd Assistente of the 1st Group of Chairs of Mathematics at the ISCEF at the early age of 19, and when he was only 23 he was appointed Professor Extraordinário. Five years later, in 1928, he was appointed Professor Catedrático.
As a result of his training, he was particularly interested in economic issues and introduced methods of Econometrics in Portugal. In 1938, with his fellow professors Mira Fernandes and Beirão da Veiga, he founded the Centro de Estudos da Matemática Aplicados a Economia, of which he was President and immediately afterwards, with other mathematicians, he launched the "Gazeta de Matemática".
Following these efforts to provide and innovate economic knowledge in Portugal, Bento Caraça, in the final period of his life, encouraged a group of young economists, all of whom were his ex-students, to launch a specialised publication, in a country in which information and knowledge were notoriously scarce and mishandled.
Hence the appearance of "Revista de Economia" in 1948, in which the opening article in the first issue "Sobre o Espaço de Capitalização" was written by Bento Caraça.
A cultured and very sensitive man, the author of a book as up-to-date and inspired as "A Cultura Integral do Indivíduo", he lived the problems which affected Portuguese society as if they were his very own. The fact is that this society, which cultivated obscurantism and anti-democratic ideas which he deplored, was the same society which was at the basis of his own humble origins, as the son of poor farm workers from the Alentejo region.
A feverish worker - as if he foresaw his early demise - he faced all manner of adversities and disenchantment, without ever wavering, because reason was on his side, together with the love and satisfaction at having fulfilled his duty.
This position as citizen, master and friend, lover of Nature and all that is beautiful, combining reason and heart in an exceptional manner, was a constant in the life of the Professor.
In his modest life, rich in moral and cultural concerns, teaching and mathematical research occupied a special place. In his classes, which he gave in a unique style and which were revolutionary in educational terms, he captivated his students through his fascinating way of presenting subjects. This soon transformed Professor Caraça into a great idol, beloved not only among his students but among the whole academic community.
This general feeling can be observed, for example, in the commentaries of Professor Sebastião e Silva, another great mathematician, on his book "Lições de Álgebra e Análise": "For the first time, mathematics has been presented by someone who lives the profession with the soul of an apostle and of an artist."
As a writer, communicator and polemist, he favoured biographies of great, universal names, of inspiring examples and acts, such as Romain Rolland, Rabindranath Tagore, Evariste Galois, Leonardo da Vinci, Galileo Galilei and others. He also maintained a notable polemic with António Sérgio, another great name of the 20th century, in the magazine "Vértice", conducted by both with utmost elegance.
On another level of his activities, involving cultural, civic and political institutions and undertakings, Bento Caraça was unable to remain indifferent the existing socio-political situation, marked by odious dictatorship. The overt politics of Professor Caraça in this context were, as we have seen, focused mainly on the culturalisation of the individual, on teaching and on the defence of major democratic values.
To promote this political and cultural process, there were social and artistic meetings, conferences and debates, most of which took place at the "Voz do Operário" and, in particular, at the "Universidade Popular Portuguesa", which was a meeting place for the city "intelligentsia" at the time, and of which Bento Caraça was the President for many years.
On a similar level, another prodigious activity, due to the responsibility it demanded, was his commitment to the project "Biblioteca Cosmos", undoubtedly one of the finest and most significant cultural achievements of this century and which was conceived and organised by Bento Caraça.
Over a period of less than eight years, this publisher brought out over 114 titles, of great cultural interest and unique in Portugal, agitating and mobilising the best collaborators in Portugal.
Bento de Jesus Caraça died on 18 June 1948. It was astonishing to see the crowds of people of all social classes who joined together spontaneously in the streets of Lisbon to pay their last heartfelt respects to the Master, to the citizen, to the great Friend.
Ulpiano Nascimento
Economista
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