Newsletters interview with Panos Pardalos
Dr Panos Pardalos is a Distinguished Professor at the University of Florida. He is also the Director of the Center for Applied Optimization. Dr Pardalos obtained a PhD degree from the University of Minnesota in Computer and Information Sciences. He has held visiting appointments at Princeton University, the DIMACS Center, the Institute of Mathematics and Applications, the FIELDS In stitute, AT&T Labs Research, Trier University, the Linkoping Institute of Technology and Universities in Greece. He has received numerous awards including University of Florida Research Foundation Professor, International Educator Award, Doctoral Dissertation Advisor/Mentoring Award, Foreign Member of the Royal Academy of Doctors (Spain), Foreign Member of the Lithuanian Academy of Sciences, Foreign Member of the Ukrainian Academy of Sciences, Foreign Member of the Petrovskaya Academy of Sciences and Arts (Russia) and Honorary Member of the Mongolian Academy of Sciences. Dr Pardalos has received the degrees of Honorary Doctor from Lobachevski University (Russia) and the V.M. Glushkov Institute of Cybernetics (Ukraine). He is a fellow of AAAS, a fellow of INFORMS and in 2001 he was awarded the Greek National Award and Gold Medal for Operations Research. Dr Pardalos is a world leading expert in Global and Combinatorial Optimization. He is the Editor-in-Chief of the Journal of Global Optimization, the Journal of Optimization Letters and the Journal of Computational Management Science. In addition, he is the managing editor of a number of book series and a member of the editorial board of several international journals. He is the author of several books and edited volumes. He has written numerous articles and has developed several wellknown software packages. His research is supported by the National Science Foundation, the National Institute of Health and other government organizations.
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| Panos Pardalos |
Interdisciplinary work is reshaping mathematics re search. A great mathematician like A. Turing was also an influential computer scientist. Von Neumann was also a great physicist. Most recent examples include mathema ticians like Steve Smale, who did fundamental work in economics, dynamical systems, machine learning and cooperative systems.
In which field do you see the most influence of mathematics in the 21st century?
Mathematics has entered new dimensions with profound impact. These include biomedical sciences, drug design and social networks. You can see this trend in the announcements of several funding agencies. For example, government agencies have announced funding on several high scope problems with the first one to be research in “the mathematics of the brain”.
Please explain some of your recent work in biomedical science.
In the last few years, I have been working with a group of students, engineers and neuroscientists on brain dynamics. A problem we have studied extensively is the dynamics of the epileptic brain. Some fundamental questions we investigated include the prediction and control of epileptic seizures based on electro-encephalogram (EEG) data analysis. This work involves chaos theory, mathematics of networks, statistics and mathematical programming. For our work on epilepsy we received the “William Pierskalla Award” for research excellence in health care management science from the Institute for Operations Research and the Management Sciences (INFORMS). In addition, several patents have been issued with our new techniques in understanding brain dynamics.
We live in the information revolution. From the Internet to iPhones and personal computers, information flow has already made a great impact and change in our lives. What in your opinion is the contribution of mathematics in this revolution?
There is no doubt that we have moved from the industrial age to the information age. Understanding the dynamics of information systems can be accomplished with mathematical tools and data analysis. New journals and new network models have been established with many specific new scopes. The structure of the Web (the Internet) and its dynamics is the source of challenging mathematical questions. Random graph theory has been complimented with models of large-scale power-law distribution networks.
Can you give me an example where your work in this area has had an impact?
The proliferation of massive data sets brings with it a series of special computational challenges. The “data avalanche” arises in a wide range of scientific and commercial applications. With advances in computer andinformation technologies, many of these challenges are beginning to be addressed. A variety of massive data sets (e.g. the web graph and the call graph) can be modelled as very large multi-digraphs with a special set of edge attributes that represent special characteristics of the application at hand. Understanding the structure of the underlying digraph is essential for storage organization and information retrieval. Our group was the first to analyze the call graph and to prove that it is a self organized complex network (the degrees of the vertices follow the power law distribution). We extended this work for financial and social networks. Our research goal is to have a unifying theory and develop external memory algorithms for all these types of dynamic networks. In my recent joint work with DingZhu Du and Ron Graham, we introduced a new method which can analyze a large class of greedy approximations with non-submodular potential functions, including some longstanding heuristics for Steiner trees, connected dominating sets and power assignment in wireless networks. There exist many greedy approximations for various combinatorial optimization problems, such as set covering, Steiner tree and subset-interconnection designs. There are also many methods to analyze these in the literature. However, all of the previously known methods are suitable only for those greedy approximations with submodular potential functions. Our work will have a lasting impact in the theory of approximation algorithms for many network problems.
You mentioned before that understanding the dynamics of information systems can get serious help from mathematics. Can you give me some examples?
We try to understand the potential influence information has on the system and how that information flows through a system and is modified in time and space. Concepts that increase our knowledge of the relational aspects of information as opposed to the entropic content of information are an important area of research. Dynamics of Information plays an increasingly critical role in our society. Networks affect our lives every day. The influence of information on social, biological, genetic and military systems must be better understood to achieve large advances in their capability and understanding of these systems. Applications are widespread and include design of highly functioning businesses and computer networks, modelling the distributed sensory and control physiology of animals, quantum entanglement, genome modelling, multi-robotic systems and industrial and manufacturing safety. Classical Information Theory is built upon the notion of entropy, which states that for a message to contain information it must dispel uncertainty associated with the knowledge of some object or process. Hence, large uncertainty means more information; small uncertainty means less information. For a networked system, classical information theory describes information that is both joint and time varying. However, for networked systems, information theory can be of limited value. Entropy does not attend to the value or influence of information: in a network some information, though potentially large in its entropy, could have little value or influence on the rest of the network, while another, less entropic, piece of information may have a great deal of influence on the rest of the system. How information flows and is modified through a system is not dependent upon entropy but more likely on how potentially useful the information is. How the value of information is linked to the connectedness of the network (and vice versa) is critical to analyzing and designing high performing distributed systems, yet it is not well studied.
You have published several books on the mathematics of optimization and, in particular, global optimization. What is a global optimization problem and why are such problems considered hard?
Most existing methods in optimization focus on computing feasible points that satisfy optimality conditions. Under certain convexity assumptions these points are locally optimal. Finding globally optimal solutions is the key objective of global optimization. The distinction of global (optimal) versus local, with its various connotations, has found a home in almost all branches of the mathematical sciences. In many applications, checking the convexity of an objective function is a very difficult problem. From the complexity point of view, many problems in global optimization are very hard. Computational complexity can be used to analyze the intrinsic difficulty of many aspects of optimization problems and to decide which of these problems are likely to be tractable. In addition, the pursuit for developing efficient algorithms also leads to elegant general approaches for solving optimization problems and reveals surprising connections among problems and their solutions. Global optimization has been expanding in all directions at an astonishing rate over the last few decades. At the same time one of the most striking trends in optimization is the constantly increasing interdisciplinary nature of the field. I am working on all aspects of global optimization with several PhD students.
Which of your books has had the greatest influence?
My textbook, Introduction to Global Optimization (there is also a Chinese translation by Tsingua University Press), has been used as a graduate textbook in many universities around the world. This is one of the first textbooks in the field of global optimization in English. In addition, I co-edited two handbooks of global optimization. Furthermore, for several years I have been involved with the Encyclopedia of Optimization.
What was your motivation for working on a multivolume Encyclopedia of Optimization?
At the onset, I had no plans for editing an Encyclopedia. Such a mega-project evolved that way after an invitation from the publisher. Developing and working in such a project involves many challenging issues, such as designing a framework of the desired product, involving the best people to advise, identifying outstanding authors and referees, and dealing with the production team and the publisher. Since an encyclopedia is never in a final form, dealing with such a project is a lifelong, demanding activity. On the other hand, it is very satisfying to see that the Encyclopedia of Optimization is used by a wide audience of researchers.
As an editor-in-chief of the main journal in the field of global optimization, what do you see to be the new directions?
Global optimization has expanded to include several areas, such as generalized convexity, variational problems and problems with equilibrium constraints.
What are your thoughts on basic mathematics education at universities today?
In general, the situation is disappointing. You meet MBAs who cannot do basic algebra and graduates of engineering schools who do not know how to solve differential equations. There is a great need for good mathematical knowledge in engineering, medicine and social sciences. This is driven by the demand and funding of interdisciplinary research.
From our previous discussions you mention that you like philosophy and poetry. What are your interests in philosophy and poetry?
I have always been fascinated by the pre-Socratic philosophers. They touched all deep questions humans try to answer. All these philosophers were polymaths. The great ancient mathematicians like Pythagoras and Euclid not only studied mathematics but also the connections of mathematics with music, aesthetics and architecture. Many times in my life I have written poetry. Mathematics expresses the rigorous part of your character. In poetry we express things that we cannot formulate precisely; poetry, like music, is needed for understanding and communicating different parts of ourselves. I very much enjoy reading Elytis, Seferis and Kavafis.
Is there any magic behind the word mathematics?
′ The word μaθημaτικa (pronounced: mathematica, mean′ ing: mathematics) originates from the verb μaνθaνειν (to learn, to feel, to watch, to understand, to realize). ′ From the same verb originates the word μaθησις in ′ the Attic dialect, which has the form η μaθa in the Doric, ′ Aeolic and Macedonian dialects and the form ο μaθος ′ ′ in the Ionic dialect. It is the πρaξις του μανθaνειν that is the process, the action of learning, the learning, the knowledge, the education but also the teaching. ′ Hence, το μaθημa is what somebody is learning, is taught, the knowledge and the science. Therefore, ′ μaθητης is the person who learns something, the person being taught. ′ In plural, τα μαθηματa meant for the ancient Greeks the mathematical sciences, the mathematics, since it was ′ necessary to μανθaνειν in order to excel in that. Thus, ο μaθηματικo (the mathematician) came to mean the ́ς ′ person elaborating on the μaθημaτικa, the mathematical sciences, and consequently is what the mathematician works on. Surprisingly, the word η μου σα and therefore the ′ ′ words το μουσεiον (the museum),η μουσικη (the music) ′ and ο μουσικoς (the musician) all seem to have common ′ radix with the verb μανθaνειν. For example, Hesichius saved in his lexicon the word ′ η μεθηρη with the meaning “care” and it is believed that ′ the word η μουσα (the one who cares about and takes care of the music) originates from this rare word. It seems that the radix μεν-θ or μαν-θ, depending on whether the dialect is northwestern or southeastern, were used to construct words having the meaning “I turn my mind to something, I care about something, I take care of something, I attempt to achieve something”. Actually, if one accepts the assumption of a common Indo-European language base, this same radix seems to appear in other languages with a similar meaning. For instance, if I recall correctly, in Sanskrit, the word “medha” means “wisdom”.
Last but not least, what are your plans for the future?
I will continue my work on research, teaching and advising my graduate students. A couple of years ago I was the Doctoral Mentoring Award Winner at the University of Florida. The best reward for a teacher is to see his students succeed. It is important to be honest, friendly and available to students, to create the opportunity for students to develop short- and long-range educational goals, to understand themselves, to explore the world of research, to foster critical thinking and decision-making skills and to engage in academic planning. In these processes, the advisor serves as an expert in his field and as a provider of general and specific program information. An advisor should also create a positive research atmosphere, reward achievements and maintain an enthusiasm for learning. After all, this is the real meaning of mathematics and a mathematician.
Themistocles M.Rassias [rassias@math.ntua.gr] is a Professor of Mathematics at the National Technical University of Athens,Greece.He received his PhD at the University of California at Berkeley under the supervision of the Fields Medalist Steve Smale. An example of Dr Rassias’ contribution in the field of Mathematical Analysis is “Hyers-Ulam-Rassias stability” and “Cauchy-Rassias stability”, and in Geometry the “Aleksandrov-Rassias problem”. He is a well known author of several articles and books, mainly in the areas of Mathematical Analysis, Global Analysis, Geometry and Topology. He is a member of the Editorial Board of several international mathematical journals including the EMS Newsletter.
M.Rassias
Newsletters, март 2010 г., выпуск 75, страницы 35—37