In January 2009, after scrutinizing 200 different careers, JobsRated.com ranked “mathematician” as the nation’s best gig. TCU Mathematics Department Chairman Dr. Robert S. Doran couldn’t agree more.

“To a mathematician, life really is largely concerned with the solving of problems,” said Doran, the John William and Helen Stubbs Potter Professor of Mathematics. “Mathematics has a personality of its own that most people don’t know about. Advanced mathematics is also something that society tends to respect. That’s why there are programs like Numbers on TV. People like the magical aspects of how mathematics does things and solves problems. So it’s fun to be involved with it.”

In this dream job, research is vital. Doran has been studying operator algebras and closely related parts of mathematics for some 40 years. His specialties involve the theory of symmetric Banach*-algebras (pronounced “symmetric Banach star algebras”), C*-algebras, harmonic analysis, approximate identities and representation theory.

“A special challenge for mathematicians is to somehow make our very abstract work intelligible to interested persons who do not have the background, training and terminology to make sense of what we are trying to say,” Doran said. “The things we do are often so esoteric that one simply can’t relate them to normal activities, so it’s difficult to talk about them to nonmath people — or even sometimes outside of one’s own math specialty.”

 Formula for romance

Though Doran admittedly “was not a good student” early on, that all began to change his junior year of high school when he wanted to invite the smartest girl in his class to prom. He asked, she accepted and his life was academically — and personally — changed. Bob and Shirley Doran have been married more than 50 years. She works by his side as an assistant in the math department, typing all his papers and books — “and with math typing, that takes a special talent,” he said.

After high school, Doran went into the military, serving in the U.S. Army Special Forces, 82nd Airborne Division as a Green Beret, Ranger and jungle survival instructor in Panama. He applied his military-learned discipline to academics in the summer of 1958, taking — and acing — three math classes at the University of North Carolina. 

When he transferred to the University of Iowa, however, a friend suggested calculus would be a different story and dared him to take it. Knowing math had to be safer than parachuting out of airplanes, Doran accepted the challenge and worked hard enough to catch the attention of his professor, who encouraged his efforts. “Basically I was transformed from a non-math person to one that was. I found my niche.”

 Solving the seemingly unsolvable

Arguably one of Doran’s biggest claims to fame came early in his career when in 1971 he solved a famous — and previously unsolved — mathematical problem questioning whether an algebra remains symmetric when an identity is put in. The abstract mathematical problem in the theory of symmetric *-algebras was left open in mathematical literature in 1949 by legendary mathematician Irving Kaplansky, former chairman of the University of Chicago math department. 

Doran “concentrated intensely” on the problem for almost four years before writing an elegant solution in its most general form by showing that it is true for arbitrary symmetric *-algebras. His proof took less than half a page — extremely brief by mathematical measures. Thanks to Doran’s and his PhD students’ continuing work in this area, TCU became known among mathematicians as the “symmetric *-algebra capital of the world.”

Since then, many more problems have been solved, but research methods remain largely unchanged. 

“The methods of ultimately solving problems in mathematics still requires just deep thinking,” Doran said. “Sometimes the loneliest job in the world is a mathematician trying to think his way through a problem. Nobody has a clue and you don’t either but you just keep working.” 

In a “yes or no” problem, half the time is spent trying to discover a proof and half trying to show it is false by finding a counter-example. 

 “And there’s one more little trick to it: There are actually mathematical questions that should have a yes or no answer, but they’re undecidable — there’s no way that you can determine the answer. And if you happen to be working on one of those, you’re going to waste the rest of your life,” Doran said, noting that no one knows going in which are the “unsolvable” problems. 

Though technology can be helpful in checking special cases, in the end mathematicians still must think their way through a proof. And in many cases, technology doesn’t help one bit. “If things involve infinity, a computer is a finite operating system,” he said.

Mathematicians often try to characterize complicated mathematical objects by showing that they look like familiar objects. “When objects are structurally the same, we say the objects are ‘isomorphic’ or ‘homeomorphic’ to each other,” Doran said

 A second strategy is breaking very complicated mathematical objects into much simpler pieces. “Then we reassemble the original object from the simpler pieces.”

A third technique is to map a complicated abstract mathematical system into a familiar system, hoping the process doesn’t destroy (smear) the properties of the original system. “In the end, we hope to get a reasonable ‘picture’ of what the unknown system looks like from the familiar one,” he said. “Very roughly speaking, this is what the subject of ‘representation theory’ is about.”

 Literary Legacies

Doran’s extensive research efforts have resulted in several important books: Approximate Identities and Factorization of Banach Modules (with Josef Wichmann, 1979, Springer-Verlag) remains today’s sole reference on approximate identities. Characterizations of C*-Algebras: The Gelfand-Naimark Theorems (with V.A. Belfi, 1986, Marcel-Dekker) has been equated to “a legendary classic.” And Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, Volumes 1 and 2 (with J.M.G. Fell, 1988, Academic Press) are still considered the principal references for researchers in this area. The two-volume Representations project totaled almost 1,700 pages and took Doran and Fell eight years to write.

 “This work was life changing,” Doran said. “The manuscript alone was over 3,000 pages to carry around. And in the 1980s, we didn’t have the word processing we have today. That was all done on an IBM Selectric, and every time you changed a result, there were 3,000 pages you had to go through by hand and change all the numbers.”

 Numbers of Awards

Thanks to his early efforts at solving world-class math problems, Doran was invited in 1981 to the Institute for Advanced Study in Princeton, N.J., where international scholars do extended research with the Institute’s permanent faculty members (which have included scholarly superstars like Albert Einstein and Robert Oppenheimer). Since the mid-1980s, Doran has served on the Institute’s board, and he was president from 1990-99.

In addition to his work at TCU, Doran served as a visiting scholar at the University of Oxford in England and Massachusetts Institute of Technology, as well as a visiting professor at the University of Texas at Austin. He has received almost every kind of teaching award that exists, including the exclusive National CASE-Carnegie Gold Medal in 1988. Both CASE-Carnegie and the Mathematical Association of America have named him “Texas Professor of the Year.” 

“It’s kind of a nice thing because if you don’t seek awards and they just happen naturally, it shows that students appreciate what you’re doing,” he said. “One of the reasons I’ve been enjoying my life is because I love students and mathematics and I love teaching, so it all goes together beautifully.”

 Administrative Formula

Extremely “research oriented” when he came to TCU in 1969, Doran is more administratively focused today — though it wasn’t until 1990 that he would accept the math chairmanship because he knew it would cut into his research time.

“I’m still a big promoter of mathematics. I’ve been producing and running all kinds of conferences, editing books of all sorts and reviewing mathematical papers,” he said, noting that he has written more than 325 reviews for the prestigious Mathematical Reviews.

In May, Doran and TCU colleague Greg Friedman ran a conference at TCU that was backed by a $33,080 National Science Foundation grant. “Topology, C*-Algebras, and String Duality” brought in about 65 of the world’s top mathematical string theorists. Doran and Friedman are now editing the conference proceedings.

Most recently, Doran and TCU colleague Loren Spice and University of Chicago Professor Paul Sally Jr. presented  “Harmonic Analysis and Representations of Reductive P-Adic Groups” at the American Mathematical Society’s national meeting in San Francisco.

 Going Beyond the Numbers

Though he’s renowned for his deep mathematical research, Doran has also spent years meticulously looking for ways to improve the learning environment. 

“There are techniques that you can use that anybody, if they’re willing to apply them, can help them become a better teacher,” he said.

Doran developed and published a “care package” of successful teaching techniques. Among his proven strategies:

• Doran quickly learns students by name, and even provides the meaning of students’ first names on the first day of class. 

• He always thanks students for coming to class and sends notes of appreciation when he notices acts of goodwill.

• Doran realizes that his course is only part of a student’s heavy academic load.

• Emphasizing mathematics for life, he brings in current math research journals to illustrate that math is ever changing — not a dead body of knowledge.

• Doran announces exams well in advance — no pop quizzes — and both tests and homework are quickly graded and returned.

• Puzzles, optical illusions and unusual examples keep lectures interesting.

• Doran brings a “thought for the day” to every class. He also keeps class lively with limericks, stories and even tales from his own experiences as a parachutist and jungle survival trainer. 

“My whole philosophy is that people are infinitely precious and special and so that’s to be reflected in our teaching — one must always be well prepared, respect students, and try to make class interesting, relevant, and exciting,” he said.

 Calculating tomorrow

As for the future, the 72-year-old Doran said he will turn over the department chairmanship at the end of his cycle in two years, but otherwise he has no plans of slowing down.

“I’ll try to stay just as enthusiastic as I presently am about learning new mathematics, continuing to write reviews, going to math seminars, teaching — I don’t have any intention whatsoever of retiring.”

Contact Doran at r.doran@tcu.edu.