Part 1

(Academic work)

STUDIES

I obtained my B.Sc. (1973) and M.Sc.(1974) from Imperial College, London University, and my Ph.D. (1977) under the supervision of J. A. Erdos at King’s College, London University. The title of my thesis was On some reflexive lattices and related algebras, and the topic was Operator Theory.

EMPLOYMENT AND TEACHING

My first position was at the University of Crete, when it had just opened, in 1978. This was a fascinating experience because we had to set up a Department from scratch: there was no library, the courses had not been designed, student handouts were unavailable etc. Everything had to be done ab initio. Although this took a disproportionate amount of effort, the fascination and the experience gained made it worthwhile.

In 1985 I was promoted to an Assistant Professor and in 1991 to an Associate, a position that I still hold. I spent my 1985/86, 1991/92 and 1998/99 sabbaticals at, respectively, the University of Alabama and King’s College London (twice). In the summer of 1988, 1990 and 1992 I was at the University of Western Australia, with a research grant, to teach and collaborate with my colleagues there.

In the various places I worked, I have taught a plethora of courses at the undergraduate or the graduate level, such as Calculus, Real Analysis, Topology, Measure Theory, Functional Analysis, Operator Theory, Analytic Number Theory, Set Theory, Functional and Theoretical Numerical Analysis, ODE’s, PDE’s, Euclidean Geometry, History of Mathematics, History of Calculus and others.

Finally I videotaped the course Linear Algebra for MEDNET'U. This course was designed, written and illustrated together with Dr. N.K. Spanoudakis of the University of Crete. I am most grateful to him for his work, co-operation, advice and expertise for this course.

SUPERVISION OF DOCTORATES

N.K. Spanoudakis obtained his doctorate in 1993 under my supervision. I have been in the committee (but not the main supervisor) of the following other doctorates: E. Katsoulis (Athens University), M. Papadakis (Athens University), E. Deligianni (University of Crete), who worked in areas of Functional Analysis. I have also been in the committees for doctorates in the History of Mathematics or in Pedagogical Studies, of N. Andreadakis (University of Crete), F. Kalavasis (University of Athens), N. Kastanis (University of Thessalonica), M. Terdimou (University of Ioannina), J. Thomaides (University of Thessalonica) and A. Tokmakidis (University of Thessalonica).

PAPERS IN RESEARCH JOURNALS

My published papers in mathematics are

1) “Complete Atomic Boolean lattices”, Journal of the London Math. Soc., 15 (1977) 387-390. MR 57 #2891.

2) “Semi-simple completely distributive lattices are Boolean algebras”, Proceedings of the Amer. Math. Soc., 68 (1978) 217-219. MR 56 #3030.

3) “Rank one elements” (with J. A. Erdos and S. Giotopoulos), Mathematika, 24 (1977) 178-181. MR 57 #7176.

4) “Non-trivially pseudocomplemented lattices are complemented”, Proceedings of the Amer. Math. Soc., 77 (1979) 155-156. MR 80j:03087.

5) “Abelian algebras and reflexive lattices” (with W. E. Longstaff), Bulletin of the London Mathematical Society, 12 (1980) 165-168. MR 82b:47057.

6) “Approximants, commutants and double commutants in normed algebras”, Journal of the London Math. Society, (2) 25 (1982) 499-512. MR 84f:47053.

7) “Completely distributive lattices”, Fundamenta Mathematica, 119 (1983) 227-240. MR 85g:06008.

8) “Strong density of finite rank operators in subalgebras of B(X)”, Proceedings of the Centre for Mathematical Analysis, 20 (1988) 83-99. MR 90e:47035.

9) “On the rank of operators in reflexive algebras”, Linear Algebra and its Applications, 142 (1990) 211-235. MR 91k:47104.

10) “Counterexamples concerning bitriangular operators” (with W.E. Longstaff), Proceedings of the Amer. Math. Soc., 112 (1991) 783-787. MR 91j:47017.

11) “Unit Ball density and the operator equation AX=YB” (with W.E. Longstaff), Journal of Operator Theory, 25 (1991) 383-397. MR 94c:47026.

12) “Atomic Boolean subspace lattices and applications to the theory of bases” (with S. Argyros and W.E. Longstaff), Memoirs of the Amer. Math. Soc., 445 (1991) 1-96. MR 92m:46022.

13) “Finite rank operators leaving double triangles invariant” (with W.E. Longstaff), Journal of the London Math. Soc., (2) 45 (1992) 153-168. MR 93e:47056.

14) “Spectral conditions and reducibility of operator semigroups” (with H. Radjavi and W.E. Longstaff), Indiana Univ. Math. Journal, 41 (1992) 449-464. MR 94a:47069.

15) “The decomposability of operators relative to two subspaces” (with A. Katavolos and W.E. Longstaff), Studia Mathematica, 105 (1993) 25-36. MR 94h:47082.

16) “On some algebras diagonalized by M-bases of l₂” (with A. Katavolos and M. Papadakis), Integral Equations and Operator Theory, 17 (1993) 68-94. MR 95c:47048.

17) “On the reflexive algebra with two invariant subspaces” (with A. Katavolos and M. Anoussis), Journal of Operator Theory, 30 (1993) 267-299. MR 95i:47082.

18) “Spatiality of isomorphisms between certain reflexive algebras”(with W.E. Longstaff), Proceedings of the Amer. Math. Soc., 122 (1994) 1065-1073. MR 95b:47053.

19) “Some counterexamples concerning strong M-bases of Banach spaces” (with W.E. Longstaff), Journal of Approximation Theory, 79(1994)243-259. MR 96k:46014.

20α) “Spectral synthesis and reflexive operators” (with J. Erdos and N.K. Spanoudakis), Comptes Rendus Mathématiques La Société Royale du Canada, 18 (1995) 193-196. MR 96h:47008 (this is a preliminary version of the next paper).

20β) “Block strong M-bases and spectral synthesis” (with J. Erdos and N.K. Spanoudakis), Journal of the London Math. Soc., 57 (1998) 183-195. MR 99e:46012.

21) “Spectral decompositions of isometries on c_p” (with D. Karagiannakis and E. Mageiropoulos), Journal of Mathematical Analysis and its Applications, 215 (1997) 190-211. MR 99i:47067.

22) “Non-reflexive pentagon subspace lattices” (with W.E. Longstaff), Studia Mathematica 125 (1997) 187-199. MR 98f:47006.

23) “Pentagon subspace lattices on Banach spaces” (with A . Katavolos and W.E. Longstaff), Journal of Operator Theory, 46 (2001) 355-380. MR 2003a:47137.

24) “Small transitive families of subspaces in finite dimensions” (with W.E. Longstaff), Linear Algebra and its Applications, 357 (2002) 229-245.

I also have a) the paper “Automatic continuity and implementation in normed algebras”, which I circulated as a report at the University of Crete, b) four papers submitted. For two of them I am a single author and for the other two I worked with N.K. Spanoudakis.

My work is in the Theory of Invariant Subspaces, in Spectral Theory, in Basis Theory, in Banach Algebras and in Lattice Theory. Below I include the reviews of my papers 1 to 23 as they appeared in Mathematical Reviews. I also add a brief description of my most recently published paper (which has not been reviewed yet) and of my report “Automatic continuity and implementation in normed algebras” (which is a research paper, often quoted in the literature).

PUBLICATIONS IN THE HISTORY OF MATHEMATICS

One of my research interests is in the History of Mathematics. My publications there are all in Greek, so I include a brief description of each of my papers.

1) One field of my interests is mathematics in Greece during Ottoman occupation, that is, in the period 1453-1821. During this time many of the mathematical works remained unpublished, because authorities did not allow a printing press in Greece, or where published abroad, mainly in the Latin West. In any case, their source was western and they all attempted to revive the interest in mathematics of the ancient ancestors. For example, as mentioned in the introduction of one such book, the aim was “to help the Muses return to their original home, Mount Parnassus, as they flew away in these challenging times the nation goes through”. For my researches I had to study manuscripts that are to-day scattered in public or in monastic libraries, throughout Greece. As a matter of fact, I have discovered in small libraries some unknown till then manuscripts as, for example, early 19^th century translations of Christian Wolf’s Arithmetic and Voucerer’s Physics.

Some of my research results are included in my article “Non-elementary mathematics during Ottoman times: the case of Nikephoros Theotokis” (National Research Council, 1990 ). In this article I study the context and influence (as seen for example in correspondence between scholars of that time) of the first calculus book in modern Greece, the celebrated three volume Stoiheia Mathimatikon of Theotokis, printed in Moscow (1798-99).

A second paper concerning mathematics in the same period is “An attempt to duplicate the cube during Ottoman times, and the text of ‘Antipelargisis’ ”. (Euclides, 11 (1994) 41-67). Here we examine the “method” of Balanos Vasilopoulos to duplicate the cube using ruler and compass. He published this in Venice (1756) in a Greek and Latin edition which survives in only one copy, to be found in Library of the Romanian Academy in Bucharest. The “method” of Vasilopoulos (which incidentally was proposed before the impossibility of the construction was proved) is rather complicated, and the error is hidden. Naturally a scientific war ensued, with the author defending his method against criticisms by the competent scholars Eugenios Voulgaris and Nikephoros Theotokis. The story is too complicated to even summarise here, but in the end the opinion of Euler, Maria Agnesi (who was fluent in Greek) and Riccati was sought. Agnesi did not answer, Riccati gave his opinion verbally to Vasilopoulos’ student Nikolaos Zerzoulis (the subsequent translator of Newton into Greek) and Euler answered in writing. The two letters to Euler and the master’s reply survive. From all this evidence I was able to recreate the fascinating story of the events around the proposed duplication.

In the book “Sciences during Ottoman times” (collective work, edited by J. Kara’s, 750 pages, to be published in 2003) I have written three articles, a) “Trigonometry in Ottoman times” , b) “Conic Sections in Ottoman times” (co-authored with N. Kastanis) and c) “Infinitesimal Calculus in Ottoman times” (also co-authored with N. Kastanis). The aim of the book is to record scientific knowledge in pre Revolution Greece, and its western influence. Every chapter is written by an expert in the topic, and there is a critical evaluation of knowledge in Mathematics (separately Arithmetic, Geometry, Trigonometry, Algebra, Conic Sections and Infinitesimal Calculus), Astronomy, Physics, Chemistry, Medicine and Geography.

With the same team I have worked to create a large data bank of all scientific publications in pre-Revolution (1821) modern Greece. For each book we recorded a) all it’s classification data (such as title, author, translator, publisher, printer etc), b) it’s number in the various catalogues of the time or modern (such as Bibliographie Hellénique of E. Legrand or Elliniki Vivliografia of D. Ginis and V. Mexas), c) all public or monastic libraries that posses a copy, d) all definitions within the text (so one could see, for example, when mathematical concepts unknown to the ancients were introduced to modern Greeks), e) each and every name mentioned (more than 1500) and their original version for the case of hellenized ones (for example: Xylander = Holtzman), f) all place names appearing.

In co-operation with I. Mountriza I am preparing a critical edition of the oldest mathematical text in modern Greece. It is the elementary work Eisagogi Mathematikis, dated 1695, by Anastasios Papavasilopoulos. The text survives in six manuscripts. For the edition I have compared word for word the manuscripts to prepare the apparatus criticus and I have written extensive annotations.

2) I have also worked in ancient Greek mathematics. In this direction I have published (in Greek) “The cattle problem of Archimedes”. This includes a) rich bibliography since antiquity b) the forgotten article of Lessing who discovered in 1773 the manuscript of the Platonic Charmides which includes the cattle problem γ) mathematical analysis of the problem d) literary analysis, e.g. for the meaning of words like “bowl-like” and “apple-like” numbers.

I have also published on “Archimedes’ palimpsest”. There I have the history of the manuscript, including some behind the scenes activities, before the auctioning of the manuscript in 1998.

EDITOR OF JOURNALS

I am a member of the editorial board of the Bulletin of the Greek Mathematical Society, a research journal, and for many years I have been at the editorial board of two journals on elementary mathematics, the Euclidis (Section C) and Mathimatiki Epitheorisis, published by the Greek Mathematical Society (these last two resemble, more or less, the Mathematics Magazine of the AAM).

I am also an editor of the electronic journal Forum Geometricorum (http://forumgeom.fau.edu/ ) which publishes original articles in modern Euclidean Geometry. Other editors include J.H. Conway, R. Guy, Paul Yiu etc. Finally, I am an editor for the Romanian journal on elementary mathematics, Lucrarile Seminarului de Creativitate Mathematica.

Part 2

(popularising)

I have spent much of my time in popularising mathematics at all levels. This is reflected by the books I have written, the translations, the editing, my work with Mathematical Olympiads, the training of gifted school students, the teaching of refresher courses to Secondary School teachers, a large amount of public lectures, a series of articles in popular topics etc.

BOOKS

I have written the following books (all in Greek, so I will need to say a few word about each):

1) Mathematics for fourth grades in High Schools (National School Book Publishing House, Ministry of Education, 1985). All schools books in Greece are published by the Government. For each course there is a unique official textbook, followed by all students, which is given to them for free. This set textbook is chosen after a national competition or by direct assignment, by an official committee of the Ministry of Education. Authors get a, once and out, nominal fee. For the above book I worked with three school teachers and I was the main author. The book was taught for six years to 16 year olds, nationally.

2) Elements of Pascal language (1990). It is for 15-16 year olds.

3) English-Greek Dictionary of Mathematical Terms (Athens 1992). I co-authored this dictionary with two other mathematicians and a philologist. For each mathematical term in English, we give its translation into Greek, various contexts in which it appears and its correct pronunciation (using the international phonetic alphabet). The book is still in circulation, and quite successful.

4) Mathematics for the “second chance” school. ((National School Book Publishing House, Ministry of Education, 2000). The book is the set mathematics textbook for people who did not complete their mandatory secondary education but decided to do so later in life.

CONTRIBUTIONS IN ENCYCLOPAEDIAS

1) I have contributed the article Hypatia in Encyclopedia of Greece and the Hellenic Tradition, Fitzroy Deaborn Publishers, London-Chicago, 2000.

2) I have contributed three articles, the Emil Borel, Leopold Kronecker and Ferdinand Lindemann, for the current issue of Encyclopedia Americana. This last can be accessed electronically at ( http://auth.grolier.com/cgi-bin/authV2 ).

NOTES FOR STUDENTS

I have written notes for 8 courses. Some is standard material, but I believe my notes on:

α) Functional and Numerical Analysis,

β) Analytic Number Theory,

γ) O.D.E’s and P.D.E’s,

δ) History of ancient mathematics

have some merits of originality.

TRANSLATIONS

I have translated into Greek the following:

1) The classic booklet of G.H. Hardy, A Mathematician’s Apology (Cambridge University Press), for which I also wrote extensive notes. The Greek edition was published by the University of Crete Press (the translation, but not the notes, was in co-operation with another colleague).

2) The book of R. Smullyan, The lady or the tiger? And other logic puzzles. (Alfred Knopf, N.Y.). Published in Greece by Katoptro (the translation was in co-operation with another colleague).

3) The classic booklet of P. Alexandroff, Topology (Dover). Published here by Trohalia (the translation was in co-operation with another colleague).

Also I have translated the following that have not appeared in print yet, but soon will:

4) R. G. Bartle, Elements of Integration (John Willey).

5) The edition by G. Toomer of Diocles, On Burning Mirrors (Springer-Verlag). This is an ancient Greek book on conic sections whose Greek original is lost, but it survives in an Arabic translation. Toomer rendered it into English and my translation is, of course, back into Greek.

EDITORIAL WORK

I have edited the Greek version of the following books, for some of which I also provided detailed annotations that supplemented the text:

1) Howard Eves, Great Moments in Mathematics (2 volumes, Mathematical Association of America).

2) Martin Gardner, Aha! Gotcha (W.H. Freeman & Co).

3) Philip Davis and Reuben Hersch, Mathematical Experience (Penguin).

4) E. Nagel and J.R. Newman, Gödel’s Proof (New York University Press).

5) Hermann Weyl, Symmetry (Princeton University Press).

6) Michael Spivak, Calculus (Addison Wesley).

7) D.E. Littlewood, The Skeleton Key of Mathematics (Hutchison University Library).

8) Yakov Perelman, Mathematics can be Fun (2 volumes, Mir).

9) George Polya, Mathematical Discovery (John Wiley & Sons).

10) Malba Tahan, The Man Who Counted (W.W. Norton & Co).

11) Eli Maor, Trigonometric Delights (Princeton University Press).

12) Waclaw Sierpinski, 250 Problems in Number Theory (MacMillan).

13) Serge Lang, The Beauty of Doing Mathematics (Springer-Verlag).

14) Paul Nahin, An Imaginary Tale: The story of √(-1) (Princeton University Press).

Also, I was the Editor in Chief of the Greek edition of the enormously popular QUANTUM magazine (American Mathematical Association) for about 8 years. The Greek edition was a translation of the American but with many additions and improvements, much the same way as the American edition was to the corresponding Russian KVANT.

MATHEMATICAL OLYMPIADS, CRUX MATHEMATICORUM

For two years I was the trainer of the Greek team that took part in the International Mathematical Olympiad (ΙΜΟ) and the corresponding Balkan (ΒΜΟ), and on four occasions I was the Leader of the team (1996 IMO in Bombay, 1996 BMO in Bacau Romania, 1997 IMO in Mar del Plata Argentina and 1997 BMO in Kalampaka Greece).

For many years I taught mathematic to gifted or to motivated school children. This was on a voluntary basis, with regularity, usually on Sundays.

My engagement with Mathematical Olympiads made me an avid reader of the monthly magazine Crux Mathematicorum (www.cms.math.ca/CRUX/), which is devoted to problem-solving. I have submitted solutions to more that 350 problems proposed in its problems section. In fact, for two years running I had submitted the largest number of solutions among other readers (the editors publish a statistic), and I had the largest number of featured solutions appearing in the solutions section.

POPULAR TALKS AND ARTICLES

I have given a great number of talks, nationally or internationally, on topics in the History of Mathematics and on Recreational Mathematics. The audience was the general public or University students or school students, depending on the occasion. I list here some of the more representative titles, which I have repeated many times.

a) “Non elementary mathematics in Greece during Ottoman times”.

b)“What is papyrology and how it helps us in the study of the History of Mathematics”.

c) “Ancient Greek Mathematics”.

d) “Diocles on Burning Mirrors”.

e) “The birth of mathematical thinking”.

f) “Archimedes’cattle problem”.

g) “Archimedes’ palimpsest”.

h) “Burning mirrors from Archimedes to Buffon”.

I have written many articles (in Greek) for the general mathematical public. The most representative ones are:

a)“Polynomial equations from the ancient Babylonians to Galois Theory” (together with S. Giotopoulos and S. Exarhakos, in Euclidis, Section C). This is a historical overview about polynomial equations from the pre-algebra stage to ruler and compass constructions.

b) “How much is sin1^ο?” (together with N. Tzanakis, in Euclidis, Section C). The article has two parts. The first describes the ingenious method of Ptolemy in the Almagest to determine (with modern notation) the numerical value of sin1^ο. The second part contains the exact value as given by the solution of a cubic and discusses the following not too widely known “paradox”: Although sin1^ο is, of course, a real number, there is no algebraic expression of it that does not contain the square root of –1.

c) “The mathematical work of Ptolemy” (Quantum, April 2000). Here the mathematical part of Ptolemy’s Almagest is discussed. Namely, Book I of this monumental thirteen book astronomical work contains all the machinery required further down. For instance “Ptolemy’s Theorem” is there, so are the difference and half angle formulas of trigonometric functions, that culminate with his famous “Table of Chords”.

d) “With ruler and compass” (a sequence of three articles, Quantum, May, July and September 2000, respectively). The problem of ruler and compass constructions in antiquity is discussed from a historic and mathematical perspective. In particular there is discussion of an animadversion in Pappus’ Collectio where he claims (intuitively but without a proof) that the classic three problems (trisection, duplication, squaring) cannot be achieved with these two instrument, but rather one has to use higher curves. Next, many examples are given of geometric construction problems solved by the ancients using ruler and compass. Of course, there is an abundance of well known such cases in the Elements, but we draw our examples from lesser known texts, such as On Division of Figures by Euclid, and On the Cutting-off of a ratio, On the Cutting-off of an area and On Tangencies by Apollonius.

e) “Mathematics in the 20^th century: a quick journey through” (Quantum, January 2001). This is a short review of some of the greatest accomplishments or famous open problems of mathematics in the past 100 years. It is a journey through Hilbert’s problems, to Cantor, to Gödel, to the Gelfond-Schneider theorem, to Fields medals, to Bieberbach’s conjecture, to the Riemann hypothesis, to the birth of Functional Analysis, to the classification of simple groups, to Fermat’s last Theorem, and many more.

f) “The ‘Spherica’ of Menelaus” (Quantum, May 2001). A description and the achievements of Menelaus (2^nd century) in his Spherica are presented. This is the book that introduces spherical trigonometry into mathematics. The Greek original is lost, but Arabic translations survive, from where it was transferred into Latin. The text has many interesting theorems such as the spherical triangle analogue of the so called Menelaus’ theorem for transversals on a plane triangle.

g) In Quantum magazine I had a permanent column entitled Scripta Manent. Its purpose was to discuss proverbial phrases drawn from the history of mathematics. For each phrase I traced it’s origin (for example, the most ancient text that gives us the information), the story of the phrase, the legends surrounding it, etc. Famous phrases that I discussed in length included “μηδείς αγεωμέτρητος εισίτω” (that is, “no one unversed in geometry should enter” as in the entrance of Plato’s Academy), “αυτός έφα” (that is, “ipse dixit” or “he said it” concerning Pythagoras), “όπερ έδει δείξαι” (i.e. “quod erat demonstrandum”), “μη μου τους κύκλους τάραττε” (“noli turbare circulos meos” or “do not disturb my circles” of Archimedean fame), “anni mirabiles” and “hypotheses non fingo” of Newtonian fame, “tanguam ex ungue leonem” exclaimed by Johann Bernoulli when he saw an anonymously written solution to one of his problems and immediately recognised Newton as the author, and many others.

h) In the journal of the Greek Mathematical Society I had for many years a column entitled “Euclid answers”. This was a queries column. Anybody could send in mathematical questions for clarification or for answers to his problems.

Finally, I have written shorter notes for the daily or Sunday press, such as “Ada Lovelace, Lord Byron’s daughter” etc.