Mathematics and Science - NSF

1 National Science Foundation Division of Mathematical SciencesMathematics and ScienceDr. Margaret WrightProf. Alexandre ChorinApril 5, 1999 PREFACET oday's challenges faced by Science and engineering are so complex that they can only be solvedthrough the help and participation of mathematical scientists. All three approaches to Science ,observation and experiment, theory, and modeling are needed to understand the complexphenomena investigated today by scientists and engineers, and each approach requires themathematical sciences . Currently observationalists are producing enormous data sets that canonly be mined and patterns discerned by the use of deep statistical and visualization , there is a need to fashion new tools and, at least initially, they will need to be fashionedspecifically for the data involved.

2 Such will require the scientists, engineers, and mathematicalscientists to work closely theory is always expressed in mathematical language. Modeling is done via themathematical formulation using computational algorithms with the observations providing initialdata for the model and serving as a check on the accuracy of the model. Modeling is used topredict behavior and in doing so validate the theory or raise new questions as to thereasonableness of the theory and often suggests the need of sharper experiments and morefocused observations. Thus, observation and experiment, theory, and modeling reinforce eachother and together lead to our understanding of scientific phenomena. As with data mining, theother approaches are only successful if there is close collaboration between mathematicalscientists and the other Margaret Wright of Bell Labs and Professor Alexandre Chorin of the University ofCalifornia-Berkeley (both past and present members of the Advisory Committee for theDirectorate for Mathematical and Physical sciences ) volunteered to address the need for thisinterplay between the mathematical sciences and other sciences and engineering in a report to theDivision of Mathematical sciences .

3 Their report identifies six themes where there is opportunityfor interaction between the mathematical sciences and other sciences and engineering, and goesone to give examples where these themes are essential for the research. These examples representonly a few of the many possibilities. Further, the report addresses the need to rethink how wetrain future scientists, engineers, and mathematical 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 areasof research of interest and challenge to all mathematical scientists. The fundamentalmathematical and statistical development of these new areas will naturally cycle back and providenew and substantial tools for attacking scientific and engineering report is exciting reading.

4 The Division of Mathematical sciences is greatly indebted to and Professor Chorin for their J. LewisDirector (1995-1999)Division of Mathematical ScienceNational Science Foundation1 OverviewMathematics and science1 have a long and close relationship that is of crucial andgrowing importance for both. Mathematics is an intrinsic component of Science , part ofits fabric, its universal language and indispensable source of intellectual , Science inspires and stimulates Mathematics , posing new questions,engendering new ways of thinking, and ultimately conditioning the value system ofmathematics. Fields such as physics and electrical engineering that have always been mathematicalare becoming even more so.

5 sciences that have not been heavily mathematical in thepast---for example, biology, physiology, and medicine---are moving from description andtaxonomy to analysis and explanation; many of their problems involve systems that areonly partially understood and are therefore inherently uncertain, demanding explorationwith 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 incommunication, finance, manufacturing, and business. Progress in Science , in all itsbranches, 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 betweenmathematics and Science .

6 Its purpose is to present examples of scientific advances madepossible by a close interaction between Science and Mathematics , and draw conclusionswhose validity should transcend the examples. We have labeled the examples by wordsthat describe their scientific content; we could have chosen to use mathematicalcategories and reached the very same conclusions. A section labeled partial differentialequations would have described their roles in combustion, cosmology, finance, hybridsystem theory, Internet analysis, materials Science , mixing, physiology, iterative control,and moving boundaries; a section on statistics would have described its contributions tothe 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 allareas of Science .

7 This alternative would have highlighted the mathematical virtues ofgenerality and abstraction; the approach we have taken emphasizes the ubiquity andcentrality of Mathematics from the point of view of ThemesAs Section 3 illustrates, certain themes consistently emerge in the closest relationships betweenmathematics 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 mathematicalsciences , and Science as Science , engineering, technology, medicine, business, and other applications . ModelingMathematical 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 indiverse problems, and to build effective algorithms---to bear on characterization, analysis, and prediction inscientific problems.

8 Mathematical models lead to virtual experiments whose real-world analogues wouldbe expensive, dangerous, or even impossible; they obviate the need to actually crash an airplane, spread adeadly virus, or witness the origin of the universe. Mathematical models help to clarify relationshipsamong a system's components as well as their relative significance. Through modeling, speculations abouta system are given a form that allows them to be examined qualitatively and quantitatively from manyangles; in particular, modeling allows the detection of discrepancies between theory and Complexity and SizeBecause reality is almost never simple, there is constant demand for more complexmodels. However, ever more complex models lead eventually---sometimes immediately---to problems that are fundamentally different, not just larger and more complicated.

9 It isimpossible to characterize disordered systems with the very same tools that are perfectlyadequate for well-behaved systems. Size can be regarded as a manifestation ofcomplexity because substantially larger models seldom behave like expanded versions ofsmaller models; large chaotic systems cannot be described in the same terms as small-dimensional chaotic UncertaintyAlthough uncertainty is unavoidable, ignoring it can be justified when one is studyingisolated, small-scale, well-understood physical processes. This is not so for large-scalesystems with many components, such as the atmosphere and the oceans, chemicalprocesses where there is no good way to determine reaction paths exactly, and of coursein biological and medical applications, or in systems that rely on human cannot be treated properly using ad hoc rules of thumb, but requires seriousmathematical study.

10 Issues that require further analysis include: the correct classificationof the various ways in which uncertainty affects mathematical models; the sensitivities touncertainty of both the models and the methods of analysis; the influence of uncertaintyon computing methods; and the interactions between uncertainty in the modelsthemselves and the added uncertainty arising from the limitations of of outcome is not necessarily directly related to uncertainty in the system orin the model. Very noisy systems can give rise to reliable outcomes, and in such cases itis desirable to know how these outcomes arise and how to predict them. Another extremecan occur with strongly chaotic systems: even if a specific solution of a model can befound, the probability that it will actually be observed may be nil; thus it may benecessary to predict the average outcome of computations or experiments, or the mostlikely outcome, drawing on as yet untapped resources of Multiple ScalesThe need to model or compute on multiple scales arises when occurrences on vastly disparate scales (inspace, time, or both) contribute simultaneously to an observable outcome.