SCHOOL OF NONLINEAR STUDIES AND COMPLEX SYSTEMS
The physical world is a manifestation of nonlinear interactions and solutions of ill-posed inverse problems. By this we mean that whatever we perceive is the result of interactions that are probably not linear in nature, even though linear models may be very good and reliable approximations for many such processes. A careful study of the state of the current development in the field leaves one with the feeling, however, that although the pervasive role of nonlinearity in the sciences and engineering is well established, not enough appears to have been achieved in the theorization of the necessary mathematical tools so that a rigorous approach to nonlinear problems in applications is possible. A classic example of this is of deterministic chaos: this phenomenon has been recognized in almost every field of our existence1 even though no rigorous and universally accepted definition of chaos exists in the literature today; mention may also be made of ill- posed inverse problems that lead naturally to set-valued analysis that have been studied by mathematicians for its own sake without an integration with inverse problems. The best example of theorization leading to mathematics needed for a complete understanding of nature is contained in the developments in Hilbert space theory that was necessitated by quantum mechanics; in nonlinear physics however, much it would appear remains to be done. The International Federation of Nonlinear Analysts (IFNA) in USA reflects this specific need; in this country no concentrated efforts appear to exist toward an integration of (pure) mathematics with the physical sciences. The purpose of the School of Nonlinear Studies and Complex Systems will be to bridge this gap and our efforts at the School will be directed towards a complete mathematical understanding of nonlinear physical phenomena and complex dynamical systems. We aim at mathematical (as opposed just to numerical) rigour; nevertheless the critical role of numerical experimentation in the nonlinear sciences will be reflected in our endeavours.
The object of the School is to have a research centre of excellence that will focus on an integrated study of the physical and mathematical aspects of the nonlinear world that we live in. Our goal will be to encourage and stimulate mathematicians to interact with physicists, biologists chemists and economists at the School to generate a mathematical framework of nonlinear physics based on an analysis and interpretation of the physical world from the viewpoint of inverse and ill-posed problems suitably integrated with the tools of set-valued analysis. The focused objective of the School will be to formulate, interpret and understand the laws of nature by going beyond the apparatus of linear functional analysis: it is felt that investigating nature from the viewpoint of inverse ill-posed problems in multifunctional space will add significantly to our understanding of nonlinear phenomena. We wish to strive for a common platform from which the mathematician and others will be able to communicate with each other and to promote continuous independent development of the basic "nonlinear mathematics". We believe in the fact that "we should continue to value academic specialization but must also work for integration of knowledge and the reestablishment of truly liberal learning as the coherent intellectual core of academic institutions"[2]. For achieving this goal, it is essential that a competitive living and working condition be provided that will attract the best talent available. For this, we need to provide the sound basic infrastructural facilities at the School which is to be fully residential in character. Due to the importance of numerical experimentation, computing facilities to include PC's for all faculty and readily accessible laser and inkjet printers are to be provided. The following is an indication of the possible research profile of the School.
.
<<<<<<<<<<<<<<<<<<Because of its very specialised nature (to the best of our knowledge, no comparable institution exists in this country), it will be necessary for the School to formulate its own policy regarding recruitment and retainment of its faculty that is to be a part of the attractive living and working conditions that the School wishes to provide. In order to keep pace with international developments it is essential to have an effective mechanism for research collaboration with Indian and foreign scientists, and travel and contingency funds will have to specifically provided for supporting national and international visitors to and from the School. The thrust of the school will be to promote and stimulate research in the nonlinear sciences and have a healthy and meaningful interaction with leading researchers in the field worldwide with the aim of building up a center of excellence in nonlinear sciences.
[1] The following is a selection of papers presented at a symposium on the Impact of Chaos on Science and Society held at the University of Tokyo in 1991. Reference: The Impact of Chaos on Science and Society, C Grebogi and J A Yorke editors. United Nations University Press, Tokyo, 1997.