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Some Glimpses of Mathematics in Industry

Paul Davis
Worester Polytechnic Institute

From the Notices, September, 1993, p. 800-802.

Most of us who practice our mathematics in academia know that particular village well; we have lived in it since birth. Once in a while, consulting gives a few of us a chance to catch a glimpse of industrial mathematics, as if we were peeking under a circus tent pitched at the edge of town.

The action in that tent must have some powerful attractions. Most of our undergraduates run away to join it, and some very capable graduates leave town to find satisfying careers there as well.

In fact a closer look suggests that business, industry, and government are not mere sideshows on the outskirts of town, but large and varied arenas in which the mathematical sciences have an important role. To get a better sense of how mathematics is practiced outside of academia, Peter E. Castro (supervisor, applied mathematics and statistics, Eastman Kodak Co.), I. E. Block (managing director, Society for Industrial and Applied Mathematics (SIAM)), and I, in various combinations, interviewed or visited about forty mathematicians working in business, industry, and government.

The observations gleaned from those conversations illuminate the remarkable differences between the culture and values of academia and those of business, industry, and government. For example, there is no assured niche for mathematicians in business, industry, or government as there is in a university, which never lacks a mathematics department.

The academic mathematician might be seen as an organism that has adapted to different environments, from teaching multitudes in a community college to nurturing a few graduate students in a research university. In contrast, the nonacademic mathematician exists in a variety of distinct forms: as a specialist in a large group of mathematicians or as a wide-ranging consultant, either singly or in a team. Sometimes mathematics is not explicitly recognized in titles or job descriptions.

Except perhaps for a common thread of problem solving, the skills valued in industry differ dramatically from those of classical academic scholarship. Teamwork, communication skills (speaking, writing, and listening), and learning new disciplines are valued in industry, but they are seldom critical to academic mathematics. In industry, breadth can be more important than depth, and a timely, incomplete answer to a complex but crucial question may be worth more than a lengthy, complete solution of a model problem. Computational skills and scientific interests outside of mathematics are commonly valued.

The observations that are described here represent a first step in the Mathematics in Industry project of SIAM. This article is based on the report, "Some Views of Mathematics in Industry," that is part of this project. The report is available two ways: (1) by anonymous ftp via; it is called siamrpt.dvi and is located in /pub/forum; and (2) from the SIAM Gopher server at This effort has two goals: helping mathematics faculty shape educational experiences that better prepare graduates for successful careers outside of academia and helping industry make better use of the productive potential of mathematics.

Formal training in mathematics is much less a prerequisite for employment in business, industry, or government than in academia. Literacy in some other field of science or engineering is often essential, however.

A consultant and head of a scientific computing group at a major pharmaceutical concern has a doctorate in chemistry. A member of the research group for one of the ``big three'' auto manufacturers has a Ph.D. in differential topology; others have more conventional backgrounds in engineering, scientific computing, or applied mathematics.

With a touch of hyperbole, one mathematician with both academic and industrial experience addresses industry's demand for breadth: ``You need literacy in some field of science or engineering to get credibility or you need computer expertise. Unless it already had an institutionalized mathematics effort, [my company] wouldn't hire Leonhard Euler without a chemistry course!''

Hiring Mathematicians

A harsh reality is the absence of an assured role for mathematics outside of academia. Few organizations in business, industry, or government have large departments that MUST be staffed with mathematicians. Demonstrating relevance is a key to survival outside of academia. A staff member in an automotive research laboratory says simply, "There is not a market niche for mathematicians." A semiconductor manufacturer "doesn't usually hire mathematicians specifically. It hires to fill technical weaknesses---for example, device physics and modeling. The new hire may be a mathematician, a physicist, or an electrical engineer. We know which universities produce the right people [at the doctoral level]."

Generalists can often function more productively in industry than narrower specialists, particularly when a small group is called upon to serve a diverse clientele. Critical judgment and problem solving ability are always essential.

When there is a conscious decision to hire a mathematician, the desired characteristics might include ``a broad background and interest in a variety of mathematical areas, computation, and science in general.'' The other criteria this mathematician in a pharmaceutical concern had when hiring included the ability "to take a problem out of the blue" and the promise of "day-by-day professional development motivated by intellectual curiosity."

Attributes of Successful Industrial Mathematicians

Much to the surprise of academic mathematicians, their industrial colleagues seldom list knowledge of specific subject matter when asked about the attributes of successful industrial applied mathematicians. Technical competence and (usually) knowledge of computation are taken for granted. The defining criteria are more cultural than purely intellectual.

A prospective industrial mathematician must "show curiosity and the ability to penetrate." "The key is an open mind and flexibility." Such individuals need a "taste for good methods and for good problems. We are all problem solvers in industry, whether we are mathematicians or marketers."

One group of applied mathematicians quickly agreed among themselves, "Communication skills are key." Another mathematician observed that, beyond technical skills, "You need visibility for success. You must show others how and why your ideas work."

"A mathematician needs communication skills to interact with chemists, physicists, and engineers of various stripes. Some cross-training helps you to get involved in problems at a much earlier stage. The cross-training that's important is not in a particular discipline. It is in the ability to approach a problem with an open mind, learning to translate from other disciplines into yours." The financial facts of life can make salesmanship essential. To find new funding in the face of budget cuts, one group at a government contractor now "must make friends and let them know our capabilities."

Listening is important, not just when consulting. One applied mathematician describes a group of colleagues in a pure research organization "who were insulated from the real requirements of [the petroleum industry]. They looked at real problems with disdain. They preferred model problems and they didn't know what the [real] business needs were. [People in the field] would ask, "What do I get for it? I can't use toy codes." This group's research charter lasted only as long as management was willing to protect it.

Problems and Solutions

Many problems are posed to industrial mathematicians by colleagues in other disciplines who may not yet understand the real problems they face. Problems need not be elegant, new, or well posed, just necessary to corporate welfare. Industrial problems are seldom selected by the natural evolution of classical scholarship. For example, two applied mathematicians (one originally trained as a chemist) at a pharmaceutical manufacturer include in their suite of problems developing a model of tumor heterogeneity. That problem was posed by the head of their laboratory.

Many mathematicians work either explicitly or implicitly in a consulting environment that can provide a natural flow of problems. Since their clients "may not yet know the real problem, small questions can grow into big problems."

An applied mathematician working with a major computer graphics manufacturer observes that the problems "don't even have to be interesting --- just necessary. If a group has hit the wall and their code release is next week, it's a good feeling when they make their deadline because you helped. You can go back to your other work with a sense of satisfaction." This same mathematician observes, "Every week I ask myself 'Is my job secure from what I'm doing? Am I relevant? Am I known by others?"

In most cases an acceptable solution is a new piece that fits nicely into a larger puzzle that a multidisciplinary team is working to solve. Relevance and quality of fit can determine the solution's value. Good solutions answer the question that really should have been asked, and they often are the consequence of deep involvement with problem formulation.

An experienced consultant within a photographic products manufacturer observes that an applied mathematician "must hear the question that's really being asked. You must lead clients to see the real problems, not just dump a quick answer to the first question they ask." Others warn, "Be prepared to ask questions. [Ask the client], 'What do you really want?" "You must speak to engineers in a variety of disciplines and understand what their problems really are." Communicating the solution to the user is important. External publication, with a few exceptions, is much less highly valued than in academia and may even be restricted by corporate policy.

In crafting a solution, mathematicians cannot be insulated from the competitive requirements of their businesses. Mathematicians "can't look at real world problems with disdain or prefer model problems or not know what their company's business needs are." "Someone has got to pay the bills."

The Working Environment

The working environment ranges from large groups similar in size to university mathematics departments to isolated individuals. In any case much of the work is joint, sometimes in rather large teams. The challenge of teamwork continues throughout a career, in contrast with the personally directed research path a tenured faculty member can choose to follow.

The dictates of teamwork "may mean you have to do what you don't want to do for a while." "You must have tolerance for a range of abilities and the wisdom to navigate the demands of teamwork and a diversity of personalities."

As one experienced woman makes clear, gender can be a factor in collegial relations. "Its tough for women if they are not aggressive. They must make sure people know what they did. They can't be afraid to say, "That's my idea." (But those recommendations appear to apply equally well to men.)

One mathematician's prescription for success describes the realities of a common working environment: "Learn how to work together in teams, have an openness of mind and people skills. Bring in customers and understand what they want, but understand that neither you nor they can know everything." And keep in mind that "there are few people in the world who can do pure mathematics in industry."

Broadly speaking, industrial mathematicians are supported in three ways. (The rare exception is the laboratory with a pure research charter only loosely related to corporate productivity.) They may be part of a staff whose mission is directly linked to the company's product, production cycle, or service. Examples would be a mathematician developing signal processing algorithms for a defense contractor or a statistician responsible for quality control in a manufacturing plant.

The other two modes of support hinge on consulting. Those who function as consultants may be funded either directly from the corporate operating budget, often called funding from overhead, or they may be supported by billing their time to sponsors inside or outside the organization. Regardless of the source of funding and of problems, most industrial mathematicians agree that "being in the middle of the action pays."

In any case mathematics is seldom the dominant technical discipline. At the corporate laboratories of a major, diversified chemical manufacturer, "Mathematics is always in the background. It is never in front with the physical problem. It is never in the limelight."

Differing Values

What the academic may scornfully dismiss as trivial, the industrial mathematician may need to exploit fully. Trivial problems can be important because they allow demonstrations of success and because their solution can build bridges to more important problems. An applied mathematician in the computer industry says, "You need a Mickey Mouse project where you can quantify progress."

Assisting in the solution of easy problems also provides opportunities to train new users, and hence additional advocates, of mathematics. For one internal consultant in a diversified chemical manufacturer, a chemist's request for help with the numerical solution of a system of seven ordinary differential equations was the beginning of a productive relationship that led to more challenging mathematics and significant contributions to profitable products. The easy response for the mathematician would have been "That's trivial. Use one of the packages in the computer center." But that answer would have pushed the chemist back across the disciplinary divide.

Breadth versus Depth

Although industrial employers do rely on narrow expertise, they often want breadth as well. In the pharmaceutical industry (and certainly elsewhere), "You do need years of experience to develop your craft," but practitioners also need breadth. "Industry wants breadth but relies heavily on narrow expertise as well." Of course, the latter alone is often the measure of mastery in an academic setting.

At a major corporate research laboratory, "The range of disciplines is so broad it doesn't matter what you know. Can you talk to others?" A mathematician who is an internal consultant to a petroleum company says, "I serve as a consultant. I can't specialize."

The size of the group with which the individual associates may determine the relative needs for depth and breadth. Larger groups of mathematicians can usually support a greater number of narrow specialists than smaller groups.

Corporate culture may favor certain disciplines (typically, an engineering discipline) over mathematics. That bias can make the introduction of mathematical approaches quite difficult. A kind of glass ceiling in the management structure may allow to pass those trained in one or two anointed disciplines into leadership roles, but hold back mathematicians. Questions about favoring other disciplines over mathematics might elicit an explanation like, "Nothing replaces the physical background."

There can be significant cultural barriers to introducing individuals trained primarily in mathematics. For example, engineers at a prominent defense contractor tell stories of lost competitive bids and design disasters that cry out for simple analyses and simulation. However, the corporate culture is not ready for mathematics. Facing the strains of the end of the Cold War, management has little interest in gambling on an unproven (and perhaps secretly threatening) discipline.

A cultural gap also separates academic and nonacademic mathematicians. Careers in industrial mathematics are often viewed as less acceptable than academic pursuits. An experienced independent consultant argues, "We may need to nurture attitude changes among ourselves that produce a comprehensive acceptance of a wide range of professional needs, not just those of the academic research mathematician." Another remarked, "My advisor and faculty treated me like I was lost when I decided to go into industry."

A Personal Perspective

My own reactions to this informal but in-depth examination of mathematics outside academia strengthened and refined educational ideas that had begun to develop when consulting first led me to poke my nose under the tent of industrial mathematics. Teaching communication skills---writing, speaking, and listening---is essential. Likewise, interdisciplinary problem solving must be a central component of mathematics education. It needs to be integrated into the daily classroom experiences of our students just as it is part of the daily professional life of the industrial applied mathematician. Perhaps we can gain more converts to the mathematical sciences by letting students solve problems in the "real world" than by delivering sermons to captives in a classroom.

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