Mathematics in Engineering: Notes from a Foreign Correspondent

In a recent article in the proceedings of a symposium on the work of Norbert Wiener, Thomas Kailath (Electrical Engineering, Stanford) used the term "mathematical engineering," comparing it to "mathematical physics." He commented that while no one thinks twice about using the latter as a description of a field and of what some people do, the former doesn't seem to be in many people's vocabulary. At least not yet. I won't attempt a definition, but it is clear that many of the faculty in engineering, in all of the departments, regard this as their profession. This affects the curriculum, the research agendas, and the faculty hires.

Whatever it is, mathematical engineering is not identical with "applied mathematics." Engineering faculty, as much as they may try not to be, are still somehow more rooted in the real world than faculty in applied mathematics. A mathematical engineer might very likely be interested in bringing a product to market, or at least know people who are interested; that's where a lot of the action is, and where a lot of the funding for research comes from, after all. I haven't seen as much of that interest among applied mathematicians. Furthermore, most engineering students, undergraduate and graduate, are heading for careers where bringing products to market is exactly what they'll be doing, and the faculty know this.

Where's the math in bringing a product to market? Everywhere. As one of my colleagues put it, "Don't mathematicians wonder where all these start-up companies come from? Or what's in a cellular phone? Or a CD player? Math is in a cellular phone and a CD player. These days digital signal processing, for example, is all math plus computers." The math is mostly complex and functional analysis. But there's group theory in the coding and error correction in CDs and linear algebra in the control of the CD player mechanics.

It's the "plus computers" that may be the biggest change, in both design and implementation. For example, one of the defining trends in electronics is the design and production of essentially zero-cost, custom chips. These devices could not be designed without automated synthesis, which relies on optimization, graph algorithms, etc., and they could not be made without PDE solvers that verify the process. And this still doesn't take into account the electro-optics and the abstractions inherent in the serious software engineering effort.

It would be worth the effort to incorporate some of the ideas and examples of mathematical engineering in the part of the undergraduate mathematics curriculum that serves primarily nonmajors, and it wouldn't hurt the majors either. There is one proviso, however. As I've indicated above, computation is driving everything—from cutting edge research, to day-to-day practice, to beginning courses in any of the engineering disciplines. Putting computation into mathematics courses in a meaningful way is where the effort lies. It cannot be grafted onto the usual courses and the usual textbooks with the usual examples.The situation is especially acute in linear algebra, where the idea of, say, row reducing 3 x 3 matrices by hand is simply unbelievable to any working engineer (or engineering student). Mathematica and Matlab have become factory standards, and they are often readily available to students, as well as professionals.

What about the mathematics courses that engineering students now take? Any discussion of the nature and extent of "Mathematics Outside of Mathematics Departments" should be informed by the article of that title written by Sol Garfunkel and Gail Young (Notices of the AMS, April, 1990). They observed that the need for more advanced mathematics was increasing and so were the enrollments in upper division math courses—outside mathematics departments. Being the first time this sort of information was collected, the authors were reluctant to call what they found a definite trend. They're more certain now. They have just written a follow-up article with the unambiguous title "The Sky is Falling," (Notices of the AMS, February, 1998), that includes the following: "Whatever the reasons, one fact stands out—in the 10-year interval from 1985-1995, we have lost 30% of our enrollment in advanced mathematics. We believe that this decline is an extension of a trend we pointed out in 1990, namely that students are increasingly taking their advanced mathematical training in non-mathematics department. These trends taken together are costing jobs and we fear eventually whole departments."

I was interested in the situation at Stanford, which is probably similar to many universities with large engineering programs. At the Sixth Conference on the Teaching of Mathematics I gave a talk on what's available here. I went through the catalog and picked out fifteen or so mathematics courses that should be of interest to engineers. I showed only the course descriptions, not the titles, and played "Name that Department" with the audience. None of the courses were from the Mathematics Department, the most popular choice of the contestants. And the numbers? This fall I sat in on a course called "Linear Dynamical Systems," taught in Electrical Engineering, that had about 120 students. The fall quarter version of "The Fourier Transform and its Applications" (typically offered every quarter) had 180.

Finally, returning to the impact of computation, and computers in general, another interesting development to watch is some Departments of Computer Science reorganizing to become Schools of Computer Science. This has already taken place at Carnegie-Mellon University and is soon to occur at the University of Michigan. The growing diversity and size of the departments in this discipline support the argument that it makes more sense, for education, for research, and for strategic planning, to reorganize as a school housing several related departments than to try to maintain a single "big tent." Of course, schools are more autonomous and powerful units within the university than single departments; they form their own departments and set their own curriculum, by and large. A School of Computer Science might decide to house a Department of Computational Mathematics, for example, thus reversing the earlier history and spinning off their own math departments!

This last point is an especially important one. Whereas traditionally, mathematics has drawn from the physical sciences as a source of problems and phenomena, the massive computational power that is becoming routinely available could suggest a new genre of mathematical problems for mathematicians to study. It already has, but it is likely to increase. Consider that, according to Moore's Law, in the roughly ten years since Garfunkel and Young began the study for their first article, computer power has increased about one hundred-fold. ("Moore's Law" is due to Gordon Moore of Intel's estimation that computer power, as measured by the number of transistors on a chip, doubles every 18 months.) This trend is expected to continue for awhile; achieving essentially infinite computing power in a finite time—imagine! Where will the mathematicians who study these problems reside, and where will they be trained? (It will also be interesting to watch the development of the pure research group at Microsoft, affectionately known as "Bill Labs"; check out http://pathfinder.com/fortune/1997/971208/mic.html.)

There's a lot of action, and whatever happens, it will happen quickly. As one of my new colleagues puts it: "The bottleneck these days is cleverness—just about whatever we can dream up can be implemented. Mathematics has never been so important, never been so central to this as it is now. It's too bad that mathematics departments don't seem to realize this." The extent to which mathematics departments do realize this will go a long way toward determining the future of the profession.