
Even among mathematicians, J. Michael Boardman might be considered eccentric. Internationally recognized as a top expert in a field of mathematics called homotopy theory, Boardman recently was honored with a conference in his name. Largely because of his prowess in the field, 50 wellknown homotopy devotees from around the world showed up to speak during the fourday session.
So, his Johns Hopkins colleagues were amused to find Boardman staffing the information booth. After all, he was the man of honor, a revered theoretician who is practically a household name in that branch of mathematics; yet there he was, helping out like some common Joe. "If somebody needed to ask where there was a restaurant, or where to park, he was the guy manning the local arrangements booth, at a conference that had attracted 50 speakers in his honor," said fellow Hopkins mathematician Stephen Wilson, who also specializes in homotopy theory. The session was held in January in conjunction with a larger gathering, the annual Joint Mathematics Meetings of the American Mathematical Society, in downtown Baltimore, which drew about 4,000 mathematicians. The date fell near his 60th birthday, generally the age at which mathematicians are honored for their work. During a banquet for Boardman, he stood up in front of about 80 people and, in his distinguished English accent, deadpanned that his birthday was actually the next month, so the entire thing was some sort of a mistake. "He has something of a unique personality," Wilson said. "He's incredibly unassuming, but yet he is someone who has worked on and really set up the absolute foundations of several complete areas of the field. Anybody who works in the field ultimately uses stuff that depends on his work." But there is something else extraordinary about Boardman, a practicing Quaker who is wont to put in extra time on cleanup duty after religious events. "He's also an incredibly nice guy," Wilson said. JeanPierre Meyer, another Johns Hopkins mathematician who works in the same field, put it this way: "He's really the only person I know about whom I can say that I've never heard him say nasty things about anybody." Boardman, meanwhile, acknowledged that the conference was a high point in his career. "Definitely, one of a kind," said the mathematician, who came to Johns Hopkins in 1969 and has had a profound influence on the Department of Mathematics. His very presence here has attracted other experts in the field, forming one of the top halfdozen university departments in the nation specializing in homotopy, which is part of a larger field called topology. Topology concerns the use of mathematics to study shapes; with homotopy theory, algebra and other techniques are used to study the relationships between shapes and the spaces they occupy. In the process, polygons and circles can be stretched and deformed, and the boundaries of traditional geometry no longer apply. For example, a circle doesn't have to be round. "In other words, I don't care whether a circle is a square or a rectangle or a triangle because I can move one to the other," Wilson said. "The essence of the circle is still there." Homotopy theory, and topology in general, has applications in a variety of fields, such as engineering, economics and the study of DNA. "In mathematics, and in particular in homotopy theory, there is a whole family of very important topological statements that are really important for the study of shapes," Wilson said. "There was a basic structure, called an infinite loop space structure, that people were struggling to learn more about, and Mike Boardman was the guy who came up ultimately with the solutions to these problems." To accomplish this feat, Boardman developed a complex mathematical framework that Wilson refers to as a "huge machinery," which has since been extended and applied in many other areas of mathematics. He also devised an entire system, called stable homotopy theory, for studying shapes. "Everybody in the field has to use this all the time," Wilson said. "They are still revising this theory. But the point is, everyone needed this theory but couldn't set it up until Mike came and did it." Despite the gravity of his research, Boardman displays a subtle humor and nonchalant style that are evident in his lectures and research papers. "In his professional lectures he frequently presents very new and exciting mathematical material so that the lecture is one large string of puns," Wilson said. His trademark phrase is to "sort out" complex mathematical problems. "He'll say, 'Well, this hasn't been sorted out yet,' which means, basically, 'I don't understand it yet,'" Meyer said. Mathematicians spend much of their time "sorting out" complex problems, locked in fierce concentration with obscure concepts. "If you walk by Mike's office, or any of our offices, much of the time we'll just be sitting there, like this," Meyer said, staring intently as if trying to solve a problem. "He's not busily writing away because he doesn't know what to write. He's thinking and trying to figure out things." Then, once a problem is solved, it's time to move on. "That's something that we have to teach graduate students because they're not really used to it, the idea that it's a very frustrating discipline because most of the time you don't know what's going on, and as soon as you do, you're done and you have to do something else," Wilson said. Boardman, a native of Manchester, England, who earned a doctorate from Cambridge University in 1965, said he actually enjoys that aspect of mathematics. "Once it is clear that the end is in sight, then people start drifting out of the subject because there is not much left to do," Boardman said. "That's not going to happen in topology. It's never all done, nor is it going to be any time soon." And, while he does relish the elegance of mathematics, he believes utility has its merits, even in theoretical research. It is satisfying to know that his work has laid the groundwork for other mathematicians. "There is a lot of beauty in mathematics," he said. "Some people regard that as the only justification for doing it. But I think it has to be significant in some way, as well."
