Interdisciplinary mathematics research

SUMMARY: There is a sense in which mathematics is always interdisciplinary but there can be special benefit to approaching it collaboratively with researchers from different disciplines bringing disparate skills, knowledge, and methodologies to bear.

In our increasingly complex society, the problems that need to be solved often lie beyond the scope of a single academic discipline. Interdisciplinary research crosses these traditional boundaries. Frequently, this boundary crossing involves bringing together individuals with a variety of knowledge and skills into collaborative working groups. Interdisciplinary research can also refer to an individual who regularly works in a field that is inherently interdisciplinary, such as mathematical physics. Interdisciplinary research can be both highly productive and truly inspiring, creating connections that lead to new knowledge, both at the intersection of the participating disciplines and within the individual fields involved in the collaboration. Mathematics plays a role in collaborations with a wide variety of disciplines, and many early mathematicians were multidisciplinary researchers and explorers. In the twenty-first century, the sciences, social sciences, businesses, and even the liberal and fine arts are working ever more closely with mathematicians on interdisciplinary problems. Despite the apparent benefits and future promises of interdisciplinary research, those who are interested in pursuing such activities often face obstacles and disincentives.

Sometimes these are barriers of communication or culture, since different disciplines have their own vocabularies and ways of working. Other barriers are related to the tradition of organizing academic institutions into discipline-based departments, which sometimes carries over into support and professional structures like funding organizations, professional societies, and journals. At the same time, mathematicians with interdisciplinary skills and experience are highly sought by employers, resulting in a shift toward creating departments or programs that exist on these interdisciplinary boundaries. There are also interdisciplinary centers and workshops to educate new and current mathematicians in both the rewards and challenges of interdisciplinary research.

Funding and Support

Funding agencies for research are generally supportive of interdisciplinary research because they feel more confident that expert input will be available in all necessary fields and that the results will be usable. They also want to help researchers learn from each other. However, it is not easy to publish interdisciplinary research, as it may not seem sufficiently novel to each discipline. A bigger problem is the lack of academic employment opportunities. One solution is to create a new discipline, for example, mathematical biology, sports science, science policy, or computational science. In times of budgetary constraint, disciplines may be reluctant to share scarce resources in interdisciplinary activities. However, this can be a difficult endeavor, as a new discipline may not immediately be seen as legitimate until it has been established within the peer community—and perhaps in society at large—that its results are valid and important. Enthusiasm for interdisciplinary mathematics research is reflected in a wide range of interdisciplinary societies, Web sites, and emerging venues for interdisciplinary publication and presentation.

Benefits

Interdisciplinary mathematics research can reveal the connections between methods used in different disciplines hidden beneath different representations. The engineer’s assessment of smoothness by spatial correlation is basically the same as the economist’s assessment of temporal change by autocorrelation. Even within the mathematics community, nomenclature and approaches can differ. The term “normal” means one thing to a statistician and something entirely different to an algebraist, and the use of the term “dimension” within linear algebra somewhat differs from many other applications of the term. Proper communication is essential to clarify and accommodate these linguistic and conceptual differences. However, this is also true for single-discipline research. The extra effort put into ensuring good understanding and communication makes a successful outcome more likely.

All participants in an interdisciplinary group can benefit from the diverse perspectives of the various fields that are represented, but care must be taken to avoid incompatible levels of detail and complexity as well as confusion over discipline-specific use of language or jargon. For example, clarification of confidence intervals can save much misunderstanding in the public sector, and understanding how government works is very useful to mathematicians. In the field of algebraic geometry, algebraic problems may be translated to geometric problems that are more easily solved in that setting, or vice versa.

Lean Six Sigma, a business management strategy that draws heavily on modern quality-improvement techniques, statistical process control, and broader statistical methods, is a good example of interdisciplinary mathematics research. Company staff are trained in a range of statistical methods and have to apply their knowledge in work-based projects. Computational science emerged from the multidisciplinary overlap of computer science, mathematics, and scientific applications. At first it was seen only as the intersection of these disciplines. As it grows in scope, computational science is seen as an independent discipline with unique issues and content. Mathematics and biology have long been intertwined, but the increasing collaboration and interdependence will no doubt enrich not only the interdisciplinary field but also both of its parent disciplines.

Interdisciplinary researchers also influence mathematics by analyzing and forecasting disciplinary trends. For example, technology forecaster Alan Porter and science and technology policy researcher Ismael Rafols examined whether science was becoming more interdisciplinary. They analyzed work between 1975 and 2005 over six research domains using established metrics, a new “index of interdisciplinarity,” and a science mapping visualization method.

Their analysis showed large increases in the number of cited disciplines, references, and coauthors per article, but the citations tended to be in close disciplinary areas. This suggested that science has in fact become more interdisciplinary, but incrementally—first to closely related fields and only later to more disparate areas. This is consistent with the fact that close disciplines are more likely to share methods and vocabulary, as well as peer reviewers, conferences, and venues for publication.

An example of the convergence of mathematics and a different discipline altogether occurred first in professional baseball and then spread globally to other sports. This data analytic process is more popularly known as “Moneyball.” The term was popularized by author Micheal Lewis in his 2003 book of the same name. In the twenty-first century, the cost of computer memory storage plunged, and the power of computers increased. This trend enabled computers to analyze data sets that had previously been too large and cost prohibitive for such analyses. Lewis studied the manner the Oakland Athletics Major League Baseball team revolutionized data analytics to make personnel decisions. The Athletics, playing in the small market of Oakland, could not compete with wealthier teams such as the New York Yankees in acquiring top-level player talent. When the Athletics did produce an All-Star player, marquee teams would simply pay the player more to sign with them, and the Athletics would lose the time and money they had invested in developing the player. Under the leadership of Oakland general manager Billy Beane and figures such as Paul DePodesta, the Athletics turned to a data-driven approach to making personnel decisions. Instead of relying on baseball scouts to identify talent, as was the historical norm, Beane first used statistical analyses to identify the valued outcomes he needed his players to achieve. If, for example, the Athletics determined they needed 40 home runs, Bean would not sign an expensive All-Star hitter to achieve this total. Instead, he might sign three less well-known and inexpensive players to match the same output. Using Moneyball techniques, small-market Oakland was able to consistently field competitive teams against teams with much higher payrolls.   

Eventually, Beane became a victim of his own success. Other teams replicated Beane’s mathematical processes and Moneyball became a standard industry practice. Moneyball then spread to other sports. Two decades later in football, data analytics evolved to where football coaches were using analytics in real-time to make play-calling decisions in crucial situations. 

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