New Interdisciplinary Approaches Using Mathematics and Methods of Their Implementation in Global Education
(Mathematics in Education)
[Project proposed in 2012 for the section “Expanded Access to Education” of Alumni Exchange Innovation Fund of Commission ECA of US Dpt. of State, by: Dr. Ovidiu-Florin Păsărescu, Principal Researcher, “Simion Stoilow” Institute of Mathematics of the Romanian Academy (IMAR), Bucharest, Romania, Senior Fulbright alumnus, 2007-2008]
0. Ante Scriptum.
I want to be clear from the beginning for everybody who will spend some time reading this Proposal: I have all my consideration for all the workers of this world: writers, painters, scientists, engineers, musicians (I am a violin player; I studied in a school of music until the end of my Junior High School; only later I progressively moved to Mathematics), artists, politicians, historians, and so on (sorry to not include an exhaustive list!), for the people from villages and cities, for the people from underdeveloped and developed arias,… I consider that a society can work well only if everybody do his/her job where he/she works. The human being evolved not only through science, but also through arts, live experiences, sports,… work and personal life in general (because all these things forced him/her to think). My Project Proposal addresses to everybody, trying to connect somehow all the aspects of the spiritual life and of the material live, in the measure of possibilities. Through Mathematics. I think that Mathematics represents a prominent unifying method (like arts, sports,…) and these unifying aspects should be thought through the world insuring an expanded access to continuous education. And, of similar importance, one of the purposes of this Project is to make the Mathematics-related things more pleasant (through Game Theory and Applications, through the use of computer, PlayStation, Wii, Facebook, etc.) and easier to understand by the people.
I include the following text from the book “Taming the Infinite. The Story of Mathematics from the First Numbers to the Chaos Theory” (Quercus Publishing PLC(UK), 2007) by Ian Stewart, a world renowned mathematician and popularizer of Mathematics, winner of the Royal Society’s Michael Faraday Medal (UK)(the following text represents my retranslation in English of a Romanian translation from English done by Humanitas Publ. House,2011; so, some differences from the original text will appear, but the ideas will be, certainly, the same):
“The present society would not work without Mathematics. Practically, all parts of the our days reality, from television to the cells, from the high-capacity jet planes to the satellite navigation systems of the cars, from the schedules of the trains to the medical scanners, are based on mathematical ideas and methods. Sometimes the applied Mathematics is thousands of years old, sometimes it was discovered a week ago. Most of us do not realize that Mathematics are permanently present, acting behind and making possible the miracles of the modern technology.
This is regrettable, because we could believe that the technology works by magic, arriving in such a way to look every day for new miracles. On the other hand, it is fair normal: we want to use these miracles as easy as possible, and without using too much our head. If every passenger would be obligated to pass a trigonometry exam before taking the plane, not too many of us will ever fly. And if in such a way we could reduce, maybe, the carbon emissions, our world will become, in the same time, very small and provincial.”
I add that many other applications of Mathematics are listed in section 3.
In the period June 27-30, 2013 a Joint Meeting AMS (American Mathematical Society)--RMS (Romanian Mathematical Society), with the collaboration of the “Simion Stoilow” Institute of Mathematics of the Romanian Academy (IMAR), www.imar.ro, is scheduled in Romania at Alba Iulia: http://www.ams.org/meetings/international/internmtgs http://imar.ro/ams-ro2013/description.php . About 300 North American Professors and Researchers and many Romanian and other foreign (non-US) mathematicians are expected to participate.
The schedule of this big event cover, through its 21 sections, most of the arias of Pure and Applied Mathematics, in connection with other sciences (Engineering, Economics, Biology, Physics, Game Theory, Sociology, Meteorology, Informatics etc.) in an interdisciplinary way, but having Mathematics as a pivotal science. I was involved from the beginning in discussions about the possibility to organize such a meeting, starting in 2003, when I visited MSRI (Mathematical Science Research Institute) Berkeley where I had discussions with Prof. David Eisenbud, Director of MSRI and President of AMS at that time, and continuing in 2007-2008 during my Fulbright fellowship at Colorado State University (and with the occasion of other visits which I had in USA), when I contacted many American colleagues on this subject.
I want that this project proposal be a continuation of the relations already developed between Romania and USA in Mathematics, in connection with other sciences. In June, with the occasion of the Joint AMS-RMS Meeting, one can discuss with many participants to further organize 3 more Conferences/Workshops during the next year (starting with the Autumn of 2013 and ending in August 2014), on arias covering most of the topics of the AMS-RMS Meeting, but seen from a different point of view also, consisting in the distribution of the scientific results through the societies using e-learning, production of the new books and courses etc. , insuring in such a way an expanded access to continuous education, discussing, in parallel with the scientific parts, on the methods to be used in order to better achieve this goal: by teaching courses at all levels, by e-learning, by writing books both for students and more senior people(continuous ed-ucation), including for popularization, etc. For these purposes, during the following years the co-operative Programs of the European Commission (and not only) can be also used, but I will detail these aspects in Section 4. The topics of the Conferences/Workshops will be explained in the forthcoming section. Now I only add the fact that, although based on an American-Romanian collaboration, the Conferences/Workshops should be open to any other specialist from the world, US Dpt. of State alumni included, all the new ideas being encouraged. The AMS-RMS Meeting represents a very good opportunity to propose such a Project, relating the scientific discoveries with their implementation in societies through continuous and expanded education. This is the reason for which I make this proposal for this call, believing that now it is one of the best moments to run it, including from the point of view of its feasibility.
3. Activities and Time-Lines.
The Project will run in the period August 26, 2013 – August 25, 2014. It will consist, mainly, in 4 steps, three of them being directly related with the organization of 3 Conferences/Workshops, open to specialists from all over the world, namely:
i) Conference/Workshop on “Didactics in Mathematics, e-learning, Problem Solving, Interdisciplinary approaches, and Exchange of Experience Between USA and Romania, Insuring an expanded access to global education” – Autumn, 2013.
This Conference will be organized at the beginning of the Project, in order to produce the main lines of the forthcoming activities. Several Professors from USA will be invited, and also Professors from other countries. Certainly, a powerful Romanian representation will be ensured, consisting in both Professors, Teachers, and official persons from the RMS and Romanian Ministry of Education and Research (like most of the Mathematics inspectors of the Romanian counties, for example).The workshop will focus on what and how to teach, at all levels, using the experience of the participants. We will also focus on trying to better understand how Mathematics can help the e-learning and vice-versa. This will also be a preparatory Workshop towards inter-disciplinary approaches. Methods of spreading knowledge around the world will also be considered.
ii) Conference/Workshop on “New trends and Methods in Pure Mathematics: Algebra, Geometry, Analysis, Topology, Mathematical Logic” - July 2014.
In order to apply Mathematics in the real life, we must to be, as much as possible, well informed on the recent progress already made. The inspiring sources of progress in Mathematics comes both from the interactions which the mathematicians have with specialists from other sciences (which is a topic of the next Conference/Workshop) – extrinsic influence, and based on its own, intrinsic logic. In order to better benefit from the interactions between Mathematics and other sciences, it is very well to understand the progresses made in pure Mathematics, in order to produce new methods of mathematical modeling, applicable in several other sciences. This Conference is dedicated to review and produce new directions of research in pure Mathematics, in its intrinsic logic, and to see how the results can be spread through the society using the methods of continuous and enlarged education, with a focus on Commutative Algebra, Algebraic, Complex and Differential Geometries, Topology, C*-Algebras, Real and Complex Analysis etc.
iii) Conference/Workshop o “Old and New Methods in Mathematical Modeling, with an interdisciplinary overview on Mathematical Physics, Mathematical Biology, Mathematical Economics, Mathematical Statistics, Sociology, Informatics, and Game Theory (Connections with Economics, Meteorology, Environmental protection, Conflict Resolution etc.); Methods of Distribution of the Ideas Through Societies, Insuring an Expanded Access to Continuous Education” – August 2014.
These two last Conferences will be organized almost one after another; in such a way, several participants will be able to attend both of them. These last two Workshops will represent the most important part of the Project, the last one relating the extrinsic development of Mathematics, under the influence of other sciences, with the extrinsic development of those sciences under the influence of Mathematics. As exemplification, I include at the end of the Proposal a very short list of scientific papers resulted from the collaboration between economists and mathematicians in Mathematical Economics (certainly, all the arias from the title of the Conference should be considered, but I do not want to write a too long Proposal). And, last, but not least many discussions on how to spread the information using expanded access to education at all levels (from junior and high school to doctoral and post-doctoral education) are planned. This is very important, because, unfortunately, there are many cases when, due to the lack of communication, possible very fruitful scientific collaborations do not take place. New subjects for courses and methods of teaching are expected to be found.I add that even humanistic sciences may take some profit from this Project; for instance, in Romania there are important contributions to Mathematical Poetics (by Academician Solomon Marcus) and Mathematical Music Theory (by Senior Researcher Dan Tudor Vuza, IMAR). Also,a better understanding of the human brain may be considered, with applications in quantum informatics and conversely, using methods of Topoi Theory and C*-Agebras (I am afraid that here I became quite technical!).
All the three Conference/Workshops will take place In Romania, organized jointly with some host Universities, to be established later.
The Core Step will run between the 1st and the 2nd Conferences/Workshops. It will mainly consist in permanent contacts with several specialists participating both to the AMS-RMS Meeting from June 2013 and to the 1st Conference/Workshop: producing original new results both in pure Mathematics and finding Mathematical Modeling Methods, with interdisciplinary applications. During this period, most of the Professors have teaching obligations, so the co-operations will consist in e-work and short bilateral visits Romania – SUA/other countries. Nevertheless, it will be a period when the e-learning methods will be used intensively, finding methods for their improvement. This period will produce the scientific and didactical materials to be presented during the last two Conferences/Workshops.
4. Innovative Ideas, Feasibility, and Relevance of the Project.
A) INNOVATIVE IDEAS:
i) to put together mathematicians and experts from several other sciences (students and seniors) aiming to use Mathematics in their research activities;
ii) pointing out essential methods in Mathematical Modeling and spreading them around the world using e-learning, courses, books and International Networks of Universities, insuring an expanded access to continuous education;
iii) producing contributions to a better understanding of the role and importance of Mathematics at a global level (see the second paragraph of the text of Prof. Ian Stewart from Section 1); namely, understanding that Mathematics is not (only) “Art for Art”, it is not “Art for Sale”, but it is mainly an “Art for Welfare”, both from technological, economical, and social point of view.
As the title of the Ian Stewart book (referred in section 1) says, Mathematics started with numbers. The number “4” represents four objects, doesn’t matter which one. Then the elementary operations (addition, multiplication,…) applies to everything, from money to atoms, from friends to planets. In a similar way (but, certainly much more sophisticated) one can use mathematical modeling methods in order to discover that arias apparently not connected in any way (like Quantum Physics and Mathematical Economics, or Voting Theory and Biological Evolution) can be modeled in a similar way, learning that they have, in fact, many similar (“isomorphic”) parts.
Based on this we can make analogies (quantum thinking), to produce conclusions in very difficult to observe arias studying more understandable things (analogies which already produced in the past very surprising - at the beginning - conclusions, like the Cosmologic Einstein’ s Relativity Theory, or the Micro-cosmologic Quantum Physics, for instance). Such kind of new discoveries should be one of the outputs of the Project, aiming to provide, in time, a new philosophic-al approach on many facts from our world (from the virtual so-called “bitcoins” to “continuous money”, from environmental protection to entrepreneurship and youth employment), generated by new technologies and new ways to see our world (or Universe, or Multiverse if we consider ourselves inside the context of Celestial Mechanics). An example of such analogy can be found in Reference  from the end of this Proposal (between Mathematical Economics and Thermodynamics), or see the analogies between the Mechanics of continuous media and the Continuum Economies of Aumann (see  from References). Moreover, one could understand in parallel how to pass from continuum to discrete economies and from continuum media to quantum media, using the methods of Non-standard Analysis (already used in economy in the papers , , , , ,  from References). Even more new possible modeling methods are presented in . We arrive in such a way to what I consider to be the most important innovative idea, namely
iv) finding isomorphic subdomains from completely different sciences, using the methods of Mathematical Modeling (seen in some kind of opposition with the usual notion of Applied Mathematics to a concrete modeled topic); understanding of the essence of such kind of things;
v) produce, teach, and spread around the world courses of Mathematical Mo-deling with applications to multiple arias from different sciences, expanding in such a way the actual level of knowledge and methods of education (in- cluding, possibly, by considering in parallel humanistic and live sciences).
The Institution where I work is very much involved in world scientific research in the arias covered by this project, being selected as a Center of Excellence of the European Commission, a list of pilot research Institutions in Europe. I also have many scientific contacts around the world (I had talks in 14 countries from 6 continents, presenting my scientific results, including at Harvard University, MIT, University of California at Berkeley, Tokyo University, which are from world’s top 10, and I thought courses and I conducted Scientific Seminars at Universities from Romania, France, USA and Japan). I think that this call is the right moment to insure a maximum efficiency concerning the feasibility of this Project, being immediately after the AMS-RMS Joint Meeting, and continuing it, including by finding new ways to popularize science through permanent and expanded education (e-learning is an example).
After the end of the project, I intend to publish a proceedings book, collecting the materials of all the three Conferences/Workshops. Moreover, we intend to further stimulate both the inter-disciplinary research through Mathematics, and the spread of the new accumulated knowledge around the world, using during the followings years the already existing International Programs. Some examples are the Atlantis EU-US Program and the Erasmus Mundus European Program (and others, with Canada, Japan, Korea, Australia, New Zealand):
http://eacea.ec.europa.eu/bilateral_cooperation/eu_us/programme/about_eu_usa_en.php http://eacea.ec.europa.eu/erasmus_mundus/ http://eacea.ec.europa.eu/bilateral_cooperation/eu_canada/index_en.php http://eacea.ec.europa.eu/bilateral_cooperation/eu_ici_ecp/index_en.php
I am currently involved in proposing such programs at Atlantis (waiting for a new call) and Erasmus Mundus (cooperation EU-Latin America, including Puerto Rico).
We will stimulate, due to the subjects of the three Conferences/Workshops, the production of courses insuring a kind of simultaneous learning in several sciences through Mathematics, enlarging in such a way the material possible to be thought and learned without increase the difficulty and the number of hours of teaching (avoiding the repetitions which are not so evident from the beginning), at all levels. I think that this is very important in a world dominated by an exponential increase of the quantity of information, and, even more important, learning how to connect its pieces in an easier and more efficient way. This should be possible because our brain is essentially a kind of quantum computer, organized in a mathematical way. The Mathematics themselves will become easier to understand when connected simultaneously to more sciences. Even our brain could be better understood in such a way, helping both Quantum (Bio)Informatics and Education process The distribution of the resulting ideas will be realized in the future through e-learning, networks (as before), etc.
5. Impact of the Project.
In most of the developed countries, the Universities are classified (more or less official) as: Research Universities (like Harvard, MIT, Stanford, Princeton, Berkeley, Tokyo, Sydney etc.), Research and Teaching Universities, and Teaching Universities. All these are parts of the Educational Systems (a similar unofficial classification works for High Schools also, depending on the results obtained at various competitions). Moreover, it is generally recognized that the Universities having the highest impact in global education are those from the first category. The subject of the Project should be seen in a similar direction, combining Research and Education (otherwise, what and how to teach in order to produce an as powerful as possible impact through the education?). I had all the time in mind, when writing this project, that these two aspects of the human live should be closely related in order to increase the impact of the Project.
So, the Project will impact by:
i) producing new Theorems in pure Mathematics;
ii) producing new Mathematical Modeling methods;
iii) finding new arias where the already existing modeling methods applies;
iv) finding unexpected analogies in science through mathematical modeling; as well as by
v) developing the already existing methods of education (e-learning and the classical ones);
vi) proposing new methods to distribute the knowledge around the world as part of the extended and continuous education, like teaching using mathematical analogies coming from Mathematical Modeling;
vii) finding groups of sub-disciplines which can be taught together from Mathematical Modeling point of view, being isomorphic (see Section 4.A).
The Project will also stimulate
viii) the writing of books authored by some of the participants to the Conferences (including Proceeding with the materials of the three Conferences);
ix) built networks of Universities/Research Institutions on the international programs (like those from Section 4)in order to implement the previous expected results thro-ugh students and faculty members.
6. Involved partners.
- American Mathematical Society;
- Romanian Mathematical Society;
- “Simion Stoilow” Institute of Mathematics of the Romanian Academy;
- Romanian Fulbright Alumni Association.
7. Project Team and the Role of the Participants.
9. Some References (from Mathematical Economics).
 AUMANN, R., Markets with a Continuum of Traders, Econometrica, 34(1964), no. 1-2, pp. 39-50.
 AUMANN, R., Existence of Competitive Equilibria in Markets with a Continuum of Traders, Econometrica, 34(1966), pp. 1-17.
 AUMANN, R., A Survey of Cooperative Games without Side Payments, in Essays in Math. Ec. In Honour of Oskar Morgenstern, Princeton Univ. Press(1967), pp. 3-27.
 AUMANN, R., HART, S., Handbook of Game Theory (with Economical Applications), vol. 1-3, Elsevier Science Publ.
 AUMANN, R., MASCHLE, R., The Bargaining Set of Cooperative Games, Adv. in Game Theory, Ann. of Math. Studies 52(1964), pp. 443-476.
 BROWN, D., LOEB, P., The values of Nonstandard Exchange Economies, Israel J.of Math., 25 (1976), pp. 71-86.
 BROWN, D., ROBINSON, A., A Limit Theorem on the Cores of Large Standard Exchange Economies, Proc. Nat. Acad. Sci. USA, 69(1972), no.5, pp. 1258-1260.
 BROWN, D., ROBINSON, A., Nonstandard Exchange Economies, Econometrica, 43 (1975), no. 1, pp. 41-55.
 CHICHILNISHY, G.,ZHOU, Y., Smooth Infinite Economies, J.of Math. Ec., 29(1998), pp. 27- 42.
 CUTLAND, N., KOPP, E., A Nonstandard Approach to Option Pricing, Math. Fin., 1 (1994), no.4, pp. 1-38.
 HILDEBRAND, W., SONNENSCHEIN, H., Handbook of Mathematical Economics, vol. 1-4, Elsevier Science Publ.
 KHAN, A., SUN., Y., Nonatomic Games on Loeb Spaces, Proc. Nat. Acad. Sci. USA, 93 (1996), pp. 15518-15521.
 OPRIŞ,D., FERRARA, M.,UDRIŞTE, C., Economic Geometric Dynamics, Monographs and Text books 6(2004), Geometry Balkan Press.
 PĂSĂRESCU, A., Nonstandard Analytic Methods in the Study of Analytic Spaces, APPS Monograph Ser. 2(2003), Geometry Balkan Press.
 PĂSĂRESCU, A., PĂSĂRESCU, O., Study on Formal Convergent Series Using Non-standard Analysis, Bull. Sci. Univ. Pitesti 5(2000), pp. 85-103.
 PRASAD, K., Constructive and Classical Models for Results in Economics and Game Theory, Metroeconomica, 55(2004), no. 2&3, pp. 141-154.