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Mathematics and Science1 Have a Long and Close Relationship That Is of Crucial and Growing Importance for Both

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Mathematics and Science1 Have a Long and Close Relationship That Is of Crucial and Growing Importance for Both Division of Mathematical Sciences Mathematics and Science Dr. Margaret Wright Prof. Alexandre Chorin National Science Foundation April 5, 1999 PREFACE Today's challenges faced by science and engineering are so complex that they can only be solved through the help and participation of mathematical scientists. All three approaches to science, observation and experiment, theory, and modeling are needed to understand the complex phenomena investigated today by scientists and engineers, and each approach requires the mathematical sciences. Currently observationalists are producing enormous data sets that can only be mined and patterns discerned by the use of deep statistical and visualization tools. Indeed, there is a need to fashion new tools and, at least initially, they will need to be fashioned specifically for the data involved. Such will require the scientists, engineers, and mathematical scientists to work closely together. Scientific theory is always expressed in mathematical language. Modeling is done via the mathematical formulation using computational algorithms with the observations providing initial data for the model and serving as a check on the accuracy of the model. Modeling is used to predict behavior and in doing so validate the theory or raise new questions as to the reasonableness of the theory and often suggests the need of sharper experiments and more focused observations. Thus, observation and experiment, theory, and modeling reinforce each other and together lead to our understanding of scientific phenomena. As with data mining, the other approaches are only successful if there is close collaboration between mathematical scientists and the other disciplinarians. Dr. Margaret Wright of Bell Labs and Professor Alexandre Chorin of the University of California-Berkeley (both past and present members of the Advisory Committee for the Directorate for Mathematical and Physical Sciences) volunteered to address the need for this interplay between the mathematical sciences and other sciences and engineering in a report to the Division of Mathematical Sciences. Their report identifies six themes where there is opportunity for interaction between the mathematical sciences and other sciences and engineering, and goes one to give examples where these themes are essential for the research. These examples represent only a few of the many possibilities. Further, the report addresses the need to rethink how we train future scientists, engineers, and mathematical scientists. The report illustrates that some mathematical scientists, through collaborative efforts in research, will discover new and challenging problems. In turn, these problems will open whole new areas of research of interest and challenge to all mathematical scientists. The fundamental mathematical and statistical development of these new areas will naturally cycle back and provide new and substantial tools for attacking scientific and engineering problems. The report is exciting reading. The Division of Mathematical Sciences is greatly indebted to Dr. Wright and Professor Chorin for their effort. Donald J. Lewis Director (1995-1999) Division of Mathematical Science National Science Foundation 1 Overview Mathematics and science1 have a long and close relationship that is of crucial and growing importance for both. Mathematics is an intrinsic component of science, part of its fabric, its universal language and indispensable source of intellectual tools. Reciprocally, science inspires and stimulates mathematics, posing new questions, engendering new ways of thinking, and ultimately conditioning the value system of mathematics. Fields such as physics and electrical engineering that have always been mathematical are becoming even more so. Sciences that have not been heavily mathematical in the past---for example, biology, physiology, and medicine---are moving from description and taxonomy to analysis and explanation; many of their problems involve systems that are only partially understood and are therefore inherently uncertain, demanding exploration with new mathematical tools. Outside the traditional spheres of science and engineering, mathematics is being called upon to analyze and solve a widening array of problems in communication, finance, manufacturing, and business. Progress in science, in all its branches, requires close involvement and strengthening of the mathematical enterprise; new science and new mathematics go hand in hand. The present document cannot be an exhaustive survey of the interactions between mathematics and science. Its purpose is to present examples of scientific advances made possible by a close interaction between science and mathematics, and draw conclusions whose validity should transcend the examples. We have labeled the examples by words that describe their scientific content; we could have chosen to use mathematical categories and reached the very same conclusions. A section labeled “partial differential equations” would have described their roles in combustion, cosmology, finance, hybrid system theory, Internet analysis, materials science, mixing, physiology, iterative control, and moving boundaries; a section on statistics would have described its contributions to the analysis of the massive data sets associated with cosmology, finance, functional MRI, and the Internet; and a section on computation would have conveyed its key role in all areas of science. This alternative would have highlighted the mathematical virtues of generality and abstraction; the approach we have taken emphasizes the ubiquity and centrality of mathematics from the point of view of science. 2 Themes As Section 3 illustrates, certain themes consistently emerge in the closest relationships between mathematics and science: · modeling · complexity and size · uncertainty · multiple scales · computation · large data sets. 1 For compactness, throughout this document “mathematics” should be interpreted as “the mathematical sciences”, and “science” as “science, engineering, technology, medicine, business, and other applications”. 2.1 Modeling Mathematical modeling, the process of describing scientific phenomena in a mathematical framework, brings the powerful machinery of mathematics---its ability to generalize, to extract what is common in diverse problems, and to build effective algorithms---to bear on characterization, analysis, and prediction in scientific problems. Mathematical models lead to “virtual experiments” whose real-world analogues would be expensive, dangerous, or even impossible; they obviate the need to actually crash an airplane, spread a deadly virus, or witness the origin of the universe. Mathematical models help to clarify relationships among a system's components as well as their relative significance. Through modeling, speculations about a system are given a form that allows them to be examined qualitatively and quantitatively from many angles; in particular, modeling allows the detection of discrepancies between theory and reality. 2.2 Complexity and Size Because reality is almost never simple, there is constant demand for more complex models. However, ever more complex models lead eventually---sometimes immediately- --to problems that are fundamentally different, not just larger and more complicated. It is impossible to characterize disordered systems with the very same tools that are perfectly adequate for well-behaved systems. Size can be regarded as a manifestation of complexity because substantially larger models seldom behave like expanded versions of smaller models; large chaotic systems cannot be described in the same terms as small- dimensional chaotic systems. 2.3 Uncertainty Although uncertainty is unavoidable, ignoring it can be justified when one is studying isolated, small-scale, well-understood physical processes. This is not so for large-scale systems with many components, such as the atmosphere and the oceans, chemical processes where there is no good way to determine reaction paths exactly, and of course in biological and medical applications, or in systems that rely on human participation. Uncertainty cannot be treated properly using ad hoc rules of thumb, but requires serious mathematical study. Issues that require further analysis include: the correct classification of the various ways in which uncertainty affects mathematical models; the sensitivities to uncertainty of both the models and the methods of analysis; the influence of uncertainty on computing methods; and the interactions between uncertainty in the models themselves and the added uncertainty arising from the limitations of computers. Uncertainty of outcome is not necessarily directly related to uncertainty in the system or in the model. Very noisy systems can give rise to reliable outcomes, and in such cases it is desirable to know how these outcomes arise and how to predict them. Another extreme can occur with strongly chaotic systems: even if a specific solution of a model can be found, the probability that it will actually be observed may be nil; thus it may be necessary to predict the average outcome of computations or experiments, or the most likely outcome, drawing on as yet untapped resources of statistics. 2.4 Multiple Scales The need to model or compute on multiple scales arises when occurrences on vastly disparate scales (in space, time, or both) contribute simultaneously to an observable outcome. In turbulent combustion, for example, the shape of the vessel is important and so are the very small fluctuations in temperature that control the
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