Kaczynski’s Dissertation Would Leave Your Head Spinning

By now, scholars, journalists and FBI agents have scoured every page of the Unabomber’s 35,000-word manifesto for insights into his motivation, methods and message.

But few have examined an earlier work of Theodore Kaczynski, the man being held in connection with the Unabomber case. The handful of mathematicians who are familiar with his 1967 doctoral dissertation, a 75-page consideration of something known as boundary functions, consider it impressive but somewhat difficult to explain to mere mortals.

“There are some topics that are not suitable for discussion,” said George Piranian, a University of Michigan mathematics professor who taught Kaczynski in the 1960s and declined to discuss the details of his student’s work. “I simply do not see a way of writing something that is readable for the intelligent layman.”

It’s not that Kaczynski’s work is long--word for word, it’s good bit shorter than the Unabomber’s manifesto. And the dissertation contains only five simple illustrations.

But it also contains dozens of equations, and assumes that the reader understands what a Riemann sphere, a Baire class and a Borel-measurable function are.

“The words that Kaczynski uses there are pretty easily understood by analysts,” said Piranian. “To the analyst, it is beautiful.”

But to most people, it’s an incomprehensible mess. And Kaczynski’s dissertation is like the Unabomber’s manifesto in another respect as well--it’s written in the first person plural.

As in, “In the first of the two chapters of this dissertation, we give a more direct proof of this result than McMillan’s, and we prove . . . .”

And, “If E is any subset of X, we will say . . . .”

Writing in the first person plural is common in mathematics, said Kenneth Ribet of UC Berkeley. So in hindsight, the Unabomber’s manifesto may provide an obscure hint at its alleged author’s former profession.

Kaczynski’s math career ran from fall 1962, when he entered the University of Michigan as a graduate student in mathematical analysis, to June 30, 1969, when he resigned as an assistant professor at Berkeley.

The simplest way to understand a small piece of Kaczynski’s mathematical career is to picture a situation similar to one he described in a paper titled, “On a Boundary Property of Continuous Functions.” It appeared in a 1966 issue of the Michigan Math Journal and was based on part of his dissertation work.

The paper considered a circular region. Think of it as a massive Frisbee, one that might be deformed into a landscape of mountains and valleys by twisting it, pushing through it to make bulges--anything, as long as it remains circular.

For some landscapes, there will be places on the disk’s edge that Kaczynski called points of curvilinear convergence. At those points, a person walking though the Frisbee landscape could find a path that gets flatter as he nears the edge.

In a completely flat landscape, that set would include any path. So every point on the circle would be a point of curvilinear convergence.

But in other landscapes there could be no such points. For example, the circle could contain a landscape of concentric hills, all the same height, separated by level valleys. That kind of rugged landscape might not offer any flattening approaches to the Frisbee’s edge.

“You’d always be going up the hill and across the valley and back up again,” said Pennsylvania State University mathematics professor Donald Rung.

What Kaczynski did, greatly simplified, was determine the general rules for the properties of sets of points of curvilinear convergence. Some of those rules were not the sort of thing even a mathematician would expect.

“The surprising thing is you can say anything at all,” said Peter Duren, a mathematics professor who was on the committee that considered Kaczynski’s dissertation at Michigan. “He was really an unusual student.”

Others are less impressed. Rung, who reviewed the 1966 paper before it was published but has never met Kaczynski, said that the work “proved some nice theorems,” but the paper by itself didn’t constitute a brilliant career.

“I thought that they were good beginnings,” Rung said of the efforts he saw.

The field that Kaczynski worked in doesn’t really exist today, mathematicians said. Most of its theories were proven in the 1960s, when Kaczynski worked in it.

“He probably would have gone on to some other area if he were to stay in mathematics,” Rung said. “As you can imagine, there are not a thousand theorems to be proved about this stuff.”