Can a Mathematician Be All Things to All People?March 8, 1998
Students learn mathematics "in courses that have been in existence for 30 or 40 years without much change in the curriculum," says Fan Chung Graham; "there is a nontrivial gap between classroom mathematics and the math used in current technology."
"Mathematics is the foundation of science and technology," says Fan Chung Graham. "A student with solid mathematical training has an advantage in dealing with all sorts of tasks in this information age. Teaching is just one of the good things for mathematicians to do." Chung would also like to see, for example, "many more CEOs who are mathematicians."
Currently a professor of mathematics and computer science at the University of Pennsylvania, Chung is exceptionally broad-minded about the paths taken by her students, counting as success stories not only those who go on to do graduate work in combinatorics (her own field), but also a student who became an executive in a (family-owned) recycling company and another who accepted a position at Salomon Brothers.
Since her arrival at Penn three years ago, Chung has been working toward what she believes should be the goal of any successful mathematics department today: the generation of "dynamic students." She has pursued that goal in part by developing and teaching the predominantly undergraduate course whose graduates included the recycling executive and the Wall Street employee. Called Topics in Applied Mathematics, the course is being offered at Penn as part of the university's participation in the National Science Foundation's Mathematical Sciences and Their Applications Throughout the Curriculum undergraduate education initiative. The course, she tells SIAM News one afternoon over lunch, had its genesis in her 19 years at AT&T Bell Laboratories and Bell Communications Research.
Arriving at Bell Labs more than 20 years ago with a degree in mathematics, she recalls, she soon found her research enriched by problems arising in many areas of communication and computation. She collaborated with engineers (on switching networks, dynamic routing, optical codes, network design), with computer scientists (on algorithmic design and analysis, parallel architectures and computation), and with chemists and physicists (on metal clustering and on Buckminsterfullerene). Her research in combinatorics has also involved collaborations, with researchers in number theory, spectral geometry, and group representations, among other areas.
At Bellcore, when research vice president Alfred Aho requested an in-depth study of software reliability, Chung was selected to lead the project. She pulled together a group of experts and immersed herself for six months in both the global perspectives and the technical issues of protocols, networking, software engineering, and their mathematical foundations. She talked with hundreds of people in both engineering and research communities, and she read many software engineering papers-easier, she believes, given her background in mathematics and computer science, than it would have been for an engineer to acquire the necessary background in mathematics. She wrote the executive summary of the report, which later appeared in IEEE Communications. (Chung's prolific research, and especially the extent to which collaboration with other researchers has been important to her career, is discussed in depth in a new book on women in mathematics; see review .)
The best part about working at industrial labs, says Chung, who headed the mathematics division at Bellcore before being named a fellow toward the end of her ten years there, is that there are "no boundaries between disciplines. The mathematicians at the labs are dynamic-they can work on mathematical problems arising in all phases of communications." Some of the new mathematics graduates who were applying for jobs, in contrast, appeared to Chung to be "very narrow." Although in most cases "very intelligent, they are not able to apply their ideas to other related areas."
Why was this happening? Students learn mathematics "in courses that have been in existence for 30 or 40 years without much change in the curriculum," Chung points out; "there is a nontrivial gap between classroom mathematics and the math used in current technology." The world has changed, she says; "mathematics has its beauty and truth, but it also has power and impact, which are often revealed by its connecting with real-world problems. Sometimes this takes the form of connecting several different disciplines within mathematics."
Having been in positions at both Bell Labs and Bellcore to make sure that the mathematics budgets were justified, Chung has a large collection of real-world connections that couple theory with the practice of mathematics. Mathematics plays an essential role in permutation networks and routing, for example. Her mathematics group at Bellcore produced such specific products as a network optimization package for optical-fiber networks. Having also been responsible for building up a group in cryptography, she cites advances in time-stamping and cryptographic protocols as examples of that group's contributions.
Chung had her "dynamic" former colleagues in mind when she first began to develop her course at Penn, and in some cases she calls on them now to address her classes in person. In the one-semester course she has now taught twice, with a theme of probability and random processes, she invites four or five mathematicians from industry to visit the class. Speakers to date have discussed mathematical finance, applications of probability in communications, string-matching, and electronic digital cash. She prepares the students for each speaker and works with the speakers to make sure that they will address the students' interests at an appropriate level. Much to her gratification, the speakers have been impressed by the students' interest and by the nature of their questions, in some cases asking to return and sometimes even recruiting students on the spot. Looking ahead to next semester, when the theme of the course will be optimization, Chung envisions again a speaker in mathematical finance, another in game theory or combinatorial chemistry, and probably one in network optimization.
Enrollment in the course has increased, is in fact at the maximum she can handle, given the extent to which she interacts with the students individually, especially in relation to their projects. Project selection itself is a learning process, she explains; instead of giving a midterm exam, she meets with the students to help them figure out what they'd like to work on, which in many cases turns out to be a topic addressed by a visiting speaker; once the students' areas of interest have been determined, she gives them materials to read and refers them to other papers. The students later submit titles and outlines to her and finally make oral presentations-which must take the form of a story, with a beginning, a well-delineated research part, and an ending. After such preparations, it is quite easy to complete the written report. "Working on projects is an effective way for the students to have a taste of creative thinking and independent study. Such experience can be very useful for job interviews or graduate school applications."
Chung's own flexibility seems to have made her equally capable of succeeding in the corporate research laboratory environment of Bell Labs and Bellcore and in the very different world of academia (before joining the Penn faculty, she had spent leaves from the labs at the Institute for Advanced Study and Harvard). The soft-spoken, gentle manner that has helped endear her to her students, though, belies some forthright opinions that would be considered controversial and even subversive by many in academia. "Mathematical research means using existing knowledge to tackle the unknown. Having first-hand experience in using mathematics can greatly help teaching," she says. "The mathematics community needs to reach out"; otherwise, she believes, its current decline will continue.
It is not a gloomy prospect that she holds out, however. Today, she says, the rapidity with which technology is moving is pushing mathematicians to use all available techniques: "The time of the Renaissance mathematician is coming back." Her own work can be offered as evidence: In addition to the serious work on applications for which she is well known to the applied and computational mathematics community, people in group representations, finite fields, differential geometry, analysis, as well as computer science and communication systems . . . "all think I am one of them," she says; "I think mathematicians can be all things to all people."
Success in mathematics usually requires an individual to have a specialty in one topic, and to keep up with advances in that area; to that prescription, Chung would add, "Advances are often made by focusing on one special aspect. However, the research will have very little impact if it cannot be transported to more than one place." Depth and breadth are not conflicting goals for a mathematician, she believes; rather, they reinforce and complement each other. "Making connections is the key."