The Mathematical Sciences in 2025 (2013) / Chapter Skim
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3 Connections Between the Mathematical Sciences and Other Fields
3 Connections Between the Mathematical Sciences and Other Fields
Pages 58-92
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From page 58... ...
When mathematical sciences research produces a new way to compress or analyze data, value financial products, process a signal from a medical device or military system, or solve the equations behind an engineer ing simulation, the benefit can be realized quickly. For that reason, even govern ent agencies or industrial sectors that seem disconnected from m 58
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From page 59... ...
At present, much of the work in these growth areas -- for example, bioinformatics, Web-based companies, financial engineering, data analytics, computational science, and engineering -- is handled primarily by people who would not necessarily be labeled "mathematical scientists." But the mathematical science content of such work, even if it is not research, is considerable, and therefore it is critical for the mathematical sciences community to play a role, through education, research, and collaboration. People with mathematical science backgrounds per se can bring different perspectives that complement those of computer scientists and others, and the combination of talents can be very powerful.
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From page 60... ...
and applied mathematics, plus statistics and operations research, and extends to highly mathematical areas of other fields such as theoreti cal computer science. The theoretical branches of many other fields -- for instance, biology, ecology, engineering, economics -- merge seamlessly with the mathematical sciences.1 The 1998 Odom report implicitly used a similar definition, as embodied in Figure 3-1, adapted from that report.
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From page 61... ...
Overall, the array of mathematical sciences share a commonality of experience and thought processes, and there is a long history of insights from one area becoming useful in another. Thus, the committee concurs with the following statement made in the 2010 International Review of Mathematical Sciences (Section 3.1)
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From page 62... ...
The mathematical sciences aim to understand the world by performing formal symbolic reasoning and computation on abstract structures. One aspect of the mathematical sciences involves unearthing and understanding deep relationships among these abstract structures.
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From page 63... ...
In light of that "unreasonable effectiveness," it is even more striking to see, in Figure 3-2, which is analogous to Figure 3-1, how far the mathematical sciences have spread since the Odom report was released in 1998. Reflecting the reality that underlies Figure 3-2, this report takes a very inclusive definition of "the mathematical sciences." The discipline encompasses the broad range of diverse activities related to the creation and analysis of mathematical and statistical representations of concepts, systems, and processes, whether or not the person carrying out the activity identifies as a M Fin ic s ar an m er Ma ke ce ono put ce al n g nu tin g ...
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From page 64... ...
But many other areas of science and engineering are deeply concerned with building and evaluating mathematical models, exploring them computationally, and analyzing enormous amounts of observed and computed data. These activities are all inherently mathe atical m in nature, and there is no clear line to separate research efforts into those that are part of the mathematical sciences and those that are part of computer science or the discipline for which the modeling and analysis are performed.3 The committee believes the health and vitality of the discipline are maximized if knowledge and people are able to flow easily throughout that large set of endeavors.
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From page 65... ...
While this exercise was necessarily subjective and far from exhaustive, it gave an indication that NSF's support for the mathematical sciences is de facto broader than what is supported by DMS. It also lent credence to the argument that the mathematical sciences research enterprise extends beyond the set of individuals who would traditionally be called mathematical scientists.
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From page 66... ...
(The diagram also shows how the teaching foci of mathematics and nonmathematics departments differ from their research foci.) IMPLICATIONS OF THE BROADENING OF THE MATHEMATICAL SCIENCES The tremendous growth in the ways in which the mathematical sciences are being used stretches the mathematical science enterprise -- its people, teaching, and research breadth.
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From page 67... ...
:693-720, Figure 8. The num Bitmapped bers correspond to the following Zentralblatt MATH classifications: 05 Combinatorics 60 Probability theory 11 Number theory 62 Statistics 14 Algebraic geometry 65 Numerical analysis 15 Linear, multilinear algebra 68 Computer science 20 Group theory 74 Mechanics of deformable solids 26 Real functions 76 Fluid mechanics 32 Several complex variables 80 Classical thermodynamics 34 Ordinary differential equations 81 Quantum theory 35 Partial differential equations 86 Geophysics 37 Dynamical systems 90 Operations research 42 Fourier analysis 91 Game theory, economics 46 Functional analysis 92 Biology 53 Differential geometry 93 Systems theory, control 57 Manifolds, cell complexes 94 Information and communications 58 Global analysis
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From page 68... ...
Many mathematical scientists and academic math departments have justifiably focused on core areas, and this is natural in the sense that no other community has a mandate to ensure that the core areas remain strong and robust. But it is essential that there be an easy flow of concepts, results, methods, and people across the entirety of the mathematical sciences.
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From page 69... ...
While such knowledge can often be found through targeted conversation, seeing the complete picture would be beneficial for individual researchers, and it might alter the way the mathematical sciences community sees itself. In a discussion with industry leaders recounted in Chapter 5, the committee was struck by the scale of the demand for workers with mathematical science skills at all degree levels, regardless of their field of training.
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From page 70... ...
This is already well recognized in the areas of search tech nology, financial mathematics, machine learning, and data analytics. No doubt new research challenges will continue to feed back to the mathematical sciences research community as new applications mature; • The demand for people with appropriate skills will be felt by math ematical science educators, who play a major role in teaching those skills to students in a variety of fields; and • The large number of career paths now based in the mathematical sciences calls for changes in the curricula for mathematics and s tatistics undergraduates and graduate students.
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From page 71... ...
In addition to the steps identified in Recommendations 3-1 and 3-2, annual collection of the followng i information would allow the community to better understand and improve itself: • A compilation of important new technology areas, patents, and business start-ups that have applied results from the mathematical sciences, and estimates of employment related to these developments; • A compilation of existing technology areas that require input and training from the mathematical sciences community, and estimates of employment related to these areas; • A compilation of new undergraduate and/or graduate programs with significant mathematical science content; • The ratio of the number of jobs with significant mathematical sci ence content to the number of new graduates (at different levels) in the mathematical sciences; • An analysis of current employment of people who received federal research or training support in the past, to determine whether they are now in U.S.
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From page 72... ...
Before discussing these two major drivers, it is critical to point out that a great deal of mathematical sciences research continues to be driven by the internal logic of the subject -- that is, initiated by individual researchers in response to their best understanding of promising directions. (The oftenused phrase "curiosity driven" understates the tremendous effectiveness of this approach over centuries.)
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From page 73... ...
For example, the process of searching data, whether in a database or on the Internet, requires both the products of computer science research and modeling and analysis tools from the mathematical sciences. The challenges of theoretical computer science itself are in fact quite mathematical, and the fields of scientific computing and machine learning sit squarely at the interface of the mathematical sciences and computer science (with insight from the domain of application, in many cases)
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From page 74... ...
It was a tour de force of computational simulation -- based on cutting-edge mathematical sciences and computer science -- that would not have been feasible until recently and that enables novel investigations into complex biological phenomena. As another example, over the past 30 years or so, ultrasound has progressed from providing still images to dynamically showing a beating heart and, more recently, to showing the evolution of a full baby in the womb.
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From page 75... ...
education, training and workforce development of the next generation of computational scientists. The mathematical sciences contribute in essential ways to all the items on this list except the fourth.
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From page 76... ...
The theory of differential equations, for example, is challenged to provide structures that enable us to analyze approximations to multiscale models; stronger m ethods of model validation are needed; algorithms need to be developed and characterized; theoretical questions in computer science need resolution; and so on. Though often simply called "software," the steps to represent reality on a computer pose a large number of challenges for the mathematical sciences.10 A prime example of how expanding computational and data resources have led to the "mathematization" of a field of science is the way that biology became much more quantitative and dependent on mathematical and statistical modeling following the emergence of genomics.
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From page 77... ...
Business, especially finance and marketing, is increasingly dependent on methods from the mathematical sciences. Some topics in the humanities have also benefited from mathematical science methods, primarily data mining, data analysis, and the emerging science of networks.
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From page 78... ...
However, the role of the mathematical sciences in this area is not a lways recognized. Indeed, the stated goals for the OSTP initiative, to advance state-of the-art core technologies needed to collect, store, pre serve, manage, analyze, and share huge quantities of data; harness these technologies to accelerate the pace of discovery in science and engineering; strengthen our national security; and transform teaching and learning; and to expand the workforce needed to develop and use Big Data technologies, seem to understate the amount of intellectual effort necessary to actually enable the move from data, to knowledge, to action.
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From page 79... ...
Strong mathematical scientists who work in this area combine best practices in data modeling, uncertainty management, and statistics, with insight about the application area and the computing implementation. These prediction problems arise everywhere: in finance and medicine, of course, but they are also crucial to the modern economy so much so that businesses like Netflix, Google, and Facebook rely on progress in this area.
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From page 80... ...
Ideas from statistics, theoretical computer science, and mathematics have provided a growing arsenal of methods for machine learning and statistical learning theory: principal component analysis, nearest neighbor techniques, support vector machines, Bayesian and sensor networks, regularized learning, reinforcement learning, sparse estimation, neural networks, kernel methods, tree-based methods, the bootstrap, boosting, association rules, hidden Markov models, and independent component analysis -- and the list keeps growing. This is a field where new ideas are introduced in rapid-fire succession, where the effectiveness of new methods often is markedly greater than existing ones, and where new classes of problems appear frequently.
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From page 81... ...
The mathematical sciences contribute in important ways to the development of new algorithms and methods of analysis, as do other areas as well. Simplifying the data so as to find their underlying structure is usually essential in large data sets.
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From page 82... ...
The need for privacy and security has given rise to the areas of privacypreserving data mining and encrypted computation, where one wishes to be able to analyze a data set without compromising privacy, and to be able to do computations on an encrypted data set while it remains encrypted. CONTRIBUTIONS OF MATHEMATICAL SCIENCES TO SCIENCE AND ENGINEERING The mathematical sciences have a long history of interaction with other fields of science and engineering.
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From page 83... ...
The biotech industry heavily uses the mathematical sciences in modeling the action of drugs, searching genomes for genes relevant to human disease or relevant to bioengineered organisms, and discovering new drugs and understanding how they might act. The imaging industry uses ideas from differential geometry and signal processing to procure minimally invasive medical and industrial images and, within medicine, adds methods from inverse problems to design targeted radi tion therapies and is moving to incorporate the new field of computa a tional anatomy to enable remote surgery.
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From page 84... ...
Another recent report on the mathematical sciences in industry came to the following conclusions: It is evident that, in view of the ever-increasing complexity of real life applications, the ability to effectively use mathematical modelling, simula tion, control and optimisation will be the foundation for the technological and economic development of Europe and the world.15 Only [the mathematical sciences] can help industry to optimise more and more complex systems with more and more constraints.16 However, that report also points out the following truism: [Engineering]
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From page 85... ...
into reach. This is of vital importance to industry." This is a very important development, and it opens up new challenges for the mathematical sciences, such as how to efficiently explore design options and how to characterize the uncertainties of this computational sampling of the space.
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From page 86... ...
One more example of the role of the mathematical sciences in industry comes from the NRC report Visionary Manufacturing Challenges for 2020,20 which identified R&D that would be necessary to advance national capabilities in manufacturing. A number of these capabilities rely on research in modeling and simulation, control theory, and informatics: • ltimately, simulations of manufacturing systems would be based on a U unified taxonomy for process characteristics that include human char acteristics in process models.
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From page 87... ...
, for example, employs roughly 1,000 mathematical scientists, although the number might be half that or twice that depending on how one defines such scientists.22 They include people with backgrounds in core and applied mathematics, probability, and statistics, but people with computer science backgrounds are not included in that count. NSA hires some 40-50 mathematicians per year, and it tries to keep that rate steady so that the mathematical sciences com 21 White House, Office of Science and Technology Policy, 2012.
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From page 88... ...
The mathematical sciences are also essential to logistics, simulations used for training and testing, war-gaming, image and signal analysis, control of satellites and aircraft, and test and evaluation of new equipment. Figure 3-5, reproduced from Fueling Innovation and Discovery: The Mathematical Sciences in the 21st Century,23 captures the broad range of ways in which the mathematical sciences contribute to national defense.
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From page 89... ...
Science and technology to enhance human- machine interfaces, increasing productivity and effectiveness across a broad range of missions.24 While the mathematical sciences are clearly of importance to the first and third of these priority areas, they also have key roles to play in support of all of the others. Advances in the mathematical sciences that allow simulation-based design, testing, and control of complex systems are essential for creating resilient systems.
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From page 90... ...
underpin tools for control and communications in tactical operations. 34 FIGURE 3-5 Mathematical sciences inside the battlefield.
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From page 91... ...
CONNECTION BETWEEN MATHEMATICAL SCIENCES AND OTHER FIELDS 91 Inside... The Battlefield Signal analysis and control theory are essential for drones.
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From page 92... ...
Computational neuroscience, which relies heavily on the mathematical sciences, is also a promising area for future developments in human-machine interfaces. The realm of threat detection in general requires a multiplicity of techniques from the mathematical sciences.
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