If f is a continuous function defined on the open unit disk
in the complex plane, then a boundary function for f
is a function on the unit circle that is obtained by taking limits
of f along arcs.
One of the first theorems of Ted Kaczynski states that a boundary function of a homeomorphism is continuous on the complement of a countable set.
We've been bombarded lately with stories about Ted Kaczynski, the loner mathematician who is better known as the Unabomber suspect. I had my own brush with the story of the man in the orange prison garb.
It all started April 3. A caller said he was a reporter from the New York Times and asked whether I was the Peter Rosenthal who had received a Ph.D. in mathematics from the University of Michigan in 1967.
I said I was, and the reporter asked whether I remembered a Theodore Kaczynski. This Kaczynski fellow, too, had received a Ph.D. in math at the same university in the same year. And he had just been arrested in the Unabomber case.
I thought hard for a moment; the name wasn't familiar.
"His Ph.D. supervisor was Allen Shields," the caller continued.
"Allen Shields was a friend of mine," I replied. "He died a few years ago."
The reporter continued, "Kaczynski wrote a thesis called 'boundary functions' under the supervision of Allen Shields." I felt I should have remembered anyone who got his Ph.D. in 1967 under the supervision of Shields, but I didn't.
Over the next days, I got telephone calls from many U.S. reporters ( Washington Post, CBS News, Time -- interestingly, no Canadian media tried this local angle). I still had no memory of Kaczynski.
The reporters wanted any kind of information, but they were particularly interested in the possibility of Kaczynski's connections to left-wing political activity.
The University of Michigan was one of the centres of student politics in the 1960s. Students for a Democratic Society, the most important radical student organization in the United States, was founded in Michigan. The first teach-in against the war in Vietnam was held at the University of Michigan; the idea quickly spread across the United States.
The years that Kaczynski and I were at Michigan, 1962 to 1967, saw a great deal of student political activity concerning civil rights, the Cuban missile crisis, nuclear weapons, and, most of all, opposition to the Vietnam War.
I was actively involved in these student movements and knew each of the several other Michigan mathematics graduate students who were active. Kaczynski was not one of us.
There were also questions about the possible relations between Kaczynski's mathematics and the crimes he is suspected of committing. The tone of the questions was insinuating. Reporters were seeking even the vaguest hint that the math mind gone wrong was capable of the most dastardly -- and explosive -- deeds.
In fact, mathematicians in general would be less prone to physical violence than most people. Doing research in mathematics is an intellectual struggle to understand deep logical constructs. Mathematicians are immersed in these constructs, not in the real world of explosives and death.
It is extraordinarily satisfying to succeed in proving a new theorem. Moreover, there is a kind of beauty peculiar to mathematics, which mathematicians often describe as "elegance".
Whatever the outcome of the Unabomber case, it will be counter-productive for future sleuths to assume that mathematicians are prone to violence.
Nonetheless, after the first few calls from reporters, the sense of the absurd did start to take hold of me. I toyed with a more interesting response: "You said that his thesis was entitled 'Boundary Functions.' Ah, that explains it. Mathematicians have known for years that study of that particular topic tends to lead to violence.
"In fact, at the last annual meeting of the American Mathematical Society, in Orlando in January, there was a secret session at which that problem was discussed."
I pictured the reporter checking to find that there was indeed a mathematics meeting in Orlando in January, and then printing my allegations together with denials from officers of the American Mathematical Society that there was any secret session on the violent aspects of studying boundary functions.
Only one of my (and Kaczynski's) fellow graduate students has been quoted saying he remembers the man in question. Joel Shapiro recalls him as quiet, private and unassuming, and having unusual mathematical ability.
Shapiro is himself a fine mathematician, so his opinion of Kaczynski's talent carries weight. However, Kaczynski never really became established as a mathematician.
After getting his Ph.D., Kaczynski was an assistant professor of mathematics at the University of California at Berkeley. Mysteriously, however, he resigned in the middle of his second year and apparently stopped doing mathematics.
During his short career, Kaczynski published six mathematical papers. All but one concerns "boundary functions." There are some important theorems about boundary values of analytic functions but, unfortunately, Kaczynski's work was on a different topic, boundary values of continuous functions.
This topic was only of interest to a very small group of mathematicians and does not appear to have broader implications; thus, his work had little impact. Kaczynski might have quit mathematics because he was discouraged by the resultant lack of recognition.
He may or may not be the Unabomber; we should be cautious about accepting police "revelations" as facts until we hear the evidence.
I wish I could be Kaczynski's lawyer. It would be interesting to discuss both his legal situation and his mathematical papers. There are, however, two insurmountable barriers: I'm in the wrong jurisdiction and Kaczynski hates leftists. Oh well, I'll just have to wait until a Toronto mathematician gets charged with a spectacular crime.
This article appeared in the April 27, 1996 edition of the Toronto Star and is reprinted and corrected with the permission of Peter Rosenthal.
Last updated: April 29, 1996
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