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 .

2 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. 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.

3 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 )

4 Volunteered to address the need for thisinterplay between the mathematical sciences and other sciences and engineering in a report to theDivision of Mathematical sciences . 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.

5 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. 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.

6 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. 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.

7 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 .

8 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.

9 And a section on computation would have conveyed its key role in allareas of Science . 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.

10 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.