Each subevent has its own organizing committee.
Several experts in the field will be present.
The programme will include three schools and two workshops dedicated to specific areas considered presently to have great importance to the study of semigroups, algorithms, automata and languages. These areas were selected considering their huge recent development, motivation from other fields of mathematics and computer science, and their potential applications.
This initiative is mainly directed to young university graduates in these areas, including Portuguese as well as foreign researchers. The courses should be viewed as post-graduation courses.
It is expected, on one hand, to strengthen the already established research in semigroups and languages in Portugal, and, on the other hand, to develop links with related areas such as theoretical computer science, computational algebra, logic and geometry.
Using this interpretation, I would like to respond with the work of Gian-Carlo Rota which as Richard P. Stanley put it "lifted the subject of combinatorics from disrepute to eminent respectability." Rota started this change with the first (of ten) of his "Foundation's" papers in 1964 subtitled "Theory of Möbius functions." His further work (and his charisma, charm, and worldliness) greatly influenced the development of mathematics in the last half of the twentieth century.
As we begin the twenty-first century, combinatorics is on a firm footing and is not only respectable but is considered to be very important in many parts of mathematics. Combinatorial ideas and constructs are now essential and widespread from algebra (e.g. representation theory) to geometry and topology (e.g. Grassmannians and flag manifolds) to analysis (e.g. harmonic analysis). And, of course, as an area of mathematics in itself, combinatorics is flourishing and has some of the best mathematical minds.
Gian-Carlo Rota died in April of 1999. One can read about him and his work in the February 2000 issue of the Notices of the American Mathematical Society.
Richard Brualdi is Professor of Mathematics at the University of Wisconsin-Madison, having been department chair from 1993 to 1999. He has been Co-editor-in-chief of the journal "Linear Algebra and its Applications" since 1979. At present he is President of the International Linear Algebra Society.
I am born in Romania and my mathematical education was in that country, at the University of Jassy, where I graduated (equivalent to the MSc degree) in 1959, I obtained a doctorate in mathematics (equivalent to a PhD degree) in 1965, and I was a faculty member from 1959 until 1976.
Uzi Vaisman
The university of Jassy had a known school of differential geometry which from the 1920's to the 1950's specialized mainly in the study of the differential geometry of geometric objects (curves, surfaces, line congruences, etc.) in various spaces of Klein's Erlangen Programme, such as projective, affine, centro-affine, etc. More general spaces, on the line of E. Cartan's works, were also studied there. Then, in the late 1950's and in the 1960's a sharp change to modern differential geometry (differentiable manifolds, global problems) occurred.
Accordingly, I wrote a thesis on projective-symplectic differential geometry, where I developed a theory of curves, surfaces, etc. in an odd-dimensional, projective space, endowed with an invariant, bilinear, nondegenerate, skew-symmetric form in the homogeneous coordinates, and I also discussed manifolds with the tangent spaces equipped as above, and with a corresponding connection, à la Cartan. This thesis explains my later interest in the geometry of the symplectic and Poisson manifolds.
Then, I changed my research subject and studied various structures on differentiable manifolds: foliations (cohomology of differential forms, conformal foliations, etc.), locally conformal Kähler structures, which were not studied seriously before and turned out to be important in the study of the complex surfaces in particular, symplectic structures (geometric quantization, Maslov classes, symplectic connections, Lagrangian foliations) and Poisson structures (Poisson cohomology, Poisson-Nijenhuis structures, etc.).
In 1976, when my main research subject was locally conformal Kähler geometry, I emigrated from Romania to Israel. Here, I was nominated full professor at the Department of Mathematics of the University of Haifa, and I am still holding this position now. It was a crucial point in my career since it allowed me free contacts with the international mathematical community. I was able to travel a lot and visit important mathematical centers. Paris, Berkeley and Oxford were among these, as well as many universities in France, U.S.A., Canada, Spain, Italy, Belgium, Japan, etc. Modern mathematical research is best done within the framework of such contacts.
I am working in the field of mathematics, doing research and teaching undergraduate and graduate courses, for over forty years now. I published more than one hundred research articles and a few research monographs and textbooks. I can say that this is a demanding but intellectually rewarding activity, and I believe that, should I start my life again, I would most probably choose the same profession.
Besides yours I can mention a few names of Israeli mathematicians of outstanding reputation. For instance I have attended talks by Saharon Shelah and Aner Shalev at important international meetings. How is Israeli mathematics doing these days?
Israeli mathematics was and is characterized by the effort to achieve excellence, by incorporation in the main stream research, by openness, and by involvement in the activity of the best mathematical centers worldwide.
All along its history, the Israeli mathematical community was a mixture between an immigrational influx and a local component, educated in the Israeli universities. In particular, in the latest years, a number of first class Jewish mathematicians from the former Soviet Union have joined Israeli universities. On the other hand, usually, the fresh PhD graduates of the Israeli universities are taking jobs in foreign countries (the U.S.A., mainly) and, after having got an important research and teaching experience there, may later return to Israel. In these ways, Israeli mathematics is strongly connected with the places where the main stream of today mathematics is produced.
We had and we have very good schools of combinatorics, logic, algebra, functional analysis, partial differential equations, and others, in Israel. And, now, more fields are ourishing, such as algebraic geometry, low-dimensional topology, global analysis, symplectic geometry, applied mathematics, etc. Overall, in Israel we have a very active mathematical life, and a very strong research.
However, like all over the world, many mathematically gifted young students prefer computer science as a career, wanting a field with plenty of well paid jobs. In the same time, funding of theoretical research, and of pure mathematics in particular, is more difficult than funding of applied computer science research, seen as immediately profitable to industry and governments. These two processes have a negative influence on the mathematics departments in universities, and on mathematical research.
The Wolf prize for mathematics is really prestigious. Has it had any significant impact on Israeli mathematics?
The Wolf prize is an international prize. It is awarded by the Israeli parliament, the Knesset, following proposals from all over the world, in a process similar to that used for the Nobel prizes, because this was the way the donor wanted it to be. The benefit of Israeli mathematics is that the award of the Wolf prizes leads to visits of famous mathematicians, and strengthens our contacts with the first class world mathematics. On the other hand, the Wolf foundation also provides some funding for mathematical research, postdoctorate grants, etc.
You have written books on symplectic geometry. On the other hand Dusa McDuff and Dietmar Salamon wrote "An introduction to symplectic topology". Could you please make the distinction between symplectic geometry and symplectic topology precise to our nonexpert readers?
Since you want an explanation for a non expert, let me first tell that symplectic geometry and topology is a field of mathematics which studies symplectic manifolds and generalizations, these being objects which the non expert should think of as phase spaces of mechanical systems. There is a large overlap between symplectic geometry and symplectic topology. My feeling about the difference between the two is as follows. Symplectic topology is the study of the category of symplectic manifolds and their natural equivalences (symplectomorphisms). The main interest of symplectic topology is in the global invariants, which are able to distinguish diffeomorphic, symplectically nonequivalent manifolds. Such invariants were discovered in the 1980's and their study is based on hard analysis. In symplectic geometry, I include the general study of symplectic manifolds and their generalizations, under all the aspects which are of interest either in mathematics itself or in applications to mathematical physics. This means that I also include aspects of a differential geometric character, quantization theories and so on. Let me add the remark that, more generally, what differentiates differential geometry from differential topology is the presence of curvature phenomena.
You are now in Portugal lecturing an introductory course on Poisson and symplectic geometry. How does this type of activity fit in your mathematical work? Do you enjoy it? Do you regard it as a form of leading the way as far as the younger ones are concerned?
I enjoy very much being here in Coimbra and lecturing at the summer course, and I want to express my gratitude to the organizers for having invited me here. I am quite sure that this kind of meetings can be of a real help in the formation of the young generation of researchers, since it may open a new perspective to those who participate.
Izu Vaisman was born in Jassy, Romania. He studied Maths at the local university where he obtained his PhD in 1965. He stayed on as a member of the staff until 1976 when he left for Israel. He has been Full Professor at the University of Haifa since then.Professor Vaisman has published more than 100 research papers in Differential Geometry and has written 8 books at both textbook and research monograph levels.
In September 1999 Professor Vaisman was in Coimbra to lecture a 12 hour course on Poisson and Symplectic Geometry in the Summer School on Differential Geometry organized by CIM.
While I was a student, I had the opportunity of attending an extra-curricular seminar organised by some of the most respected mathematicians of the time, who had recently returned from studies abroad and were trying to do something about the academic backwardness that they found in Portugal. One of the subjects studied in this seminar was Group Theory, and the bibliography contained, amongst half a dozen books by foreign authors, one by a Portuguese: Elements of Group Theory (Elementos da Teoria dos Grupos) by A. Almeida Costa (1942, 153 pp.). Thus it was that I first heard of the scientist who, in the same year, had published Abelian Groups, Rings and Non-Commutative Ideals (Grupos abelianos, anéis e ideais não comutativos) (173 pp.) and, immediately afterwards, in 1943, Elements of Ring Theory (Elementos da Teoria dos Anéis) (282 pp.)
Almeida Costa
How had Almeida Costa become so enthusiastic about these matters?
In 1928/9, he assumed responsibility for the courses in Astronomy in the Faculty of Science in Oporto, and when, in this same year, the National Education Committee (Junta de Educação Nacional - henceforth J.E.N.) was set up, he immediately applied to them for a monthly subsidy in order to prepare his doctorate dissertation. His proposal was received favourably, since he clearly revealed exceptional intelligence and diligence, and his area of study (certain elements of surface theory by vector calculus methods) was recognised to be of topical interest. However, the scholarship was not awarded that year because "the candidate had not yet revealed his talent for research through written work".
The young Almeida Costa was not discouraged, and two years later, his first publications began to appear: Notes on Vector Calculus (Notas de Cálculo Vectorial) and On the Dynamics of Holonomic Systems (Sobre a dinâmica dos sistemas holonómos), which he presented with his application for a professorship.
In 1934, he once more requested a grant from the J.E.N., this time in order to travel abroad to pursue his studies into astronomy and geometry at the University of Paris, the latter under Élie Cartan (but with "the possibility of replacing, at his own responsibility, the study in geometry with physics, under the direction of Louis de Broglie").
However, the Committee was very demanding in its concession of the grant, particularly as regarded the subject matter proposed and national acceptance of the candidate's specialization. But Almeida Costa did not give up, and in this struggle, revealed his determination to fight for what he believed in, a quality that remained with him all his life. In successive applications, he revealed himself to be highly informed about what was happening in Europe, and in 1937 (three years after his first application), having personally explained to Prof. Max von Laue the aims of his scholarship (the study of theoretical physics, particularly quantum theory and theory of relativity), he finally received a grant to study at the Physikalischer Institut in Berlin, where he remained for 22 months.
There, he began to attend courses on Matrices and Group Theory, as well as Theoretical Electricity and Thermodynamics. When the Institute for High Culture (which had replaced the J.E.N.) observed that he should not diversify his attentions so much, he replied, "I cannot forget that a background in mathematics is absolutely indispensable for the serious study of theoretical physics. Pages and pages of physics books are given over to matrix theory, series expansions of complete functions, orthogonal or not ... group theory ... ".
The influence of Hilbert was obvious in the teaching of mathematics and theoretical physics at that time, and the last studies done by Almeida Costa in Berlin demonstrated his familiarity with methods of physical mathematics according to Courant and Hilbert, and quantum groups according to Hermann Weyl, another of Hilbert's disciples.
In conclusion: until September 1937, Almeida Costa had lived in the world of applied mathematics (essentially mechanics and astronomy) at the Faculty of Science in Oporto; the J.E.N. effectively rerouted him in the direction of theoretical physics, when he went to study in Berlin; and as a result of this, he dropped his interest in astronomy, and from the 1940s, moved into a completely different scientific world where he demonstrated zeal in transmitting to others what he had learned abroad.
In 1950, he became a Full Professor of Celestial Mechanics at the Faculty of Science in Oporto, and two years later, accepted the Chair in Algebra at the Faculty of Science in Lisbon, where he was able to lecture on subjects more in keeping with the choices he had made in the previous decade. This eventually made possible the modernization of the teaching of this subject.
It was thus in 1952 that I met him personally. Hardly had he arrived in Lisbon when he instigated a series of seminars on the topics that most interested him, seeking to transmit to the young academics his enthusiasm for "modern" algebra. During the academic year, there were regular expositions (at least once a week) on group theory, theory of rings and non-commutative ideals, and field theory. I became familiar with his massive work Hypercomplex Systems and Representations (Sistemas hipercomplexos e representações), and perhaps because of this, he himself suggested that I, still a teaching assistant, should be responsible for the course of Physical Mathematics, lecturing among other things group representations.
His enthusiasm for his subject was such that he even suggested that students who had passed his course with the minimum of 10 should resit the exam in the next session, since he could not be sure that they had attained sufficient mastery of the material! It was as if he felt, in his innermost self, the words of the famous algebrist Paul Halmos, "It saddens me that educated people don't even know that my subject exists!"
He was elected correspondent member of the Lisbon Academy of Sciences in 1959, and full member in 1972, and five years later, became its president. His activities in algebra did not cease upon his retirement in 1973, for the following year, the third volume of his Course in General Algebra (Cours d'Algèebre Générale) appeared, and he continued to present papers to the Academy until 1978, the year of his death.
Almeida Costa worked until the end, and was the greatest specialist in Semiring Theory, and the true founder of the Portuguese school of algebra.
F. R. Dias Agudo
Professor Jubilado da Faculdade de Ciências de Lisboa
Membro da Academia das Ciências de Lisboa
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