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Before terrorist Theodore John Kaczynski (1942-) began sending mail-bombs to faculty members at various American universities, he had a promising career in mathematics. In particular, between 1964–69, he published a total of six single-authored research papers in renowned mathematical journals, including *The American Mathematical Monthly *and* Proceedings of the American Mathematical Society.*

The young Kaczynski did work in analysis, specifically geometric function theory in the narrow subfield of boundary values of continuous functions. The purpose of this article is to give an introduction to this work.

**Education (1958–67)**

Kaczynski grew up in Illinois, where he attended Sherman Elementary School and Evergreen Park Central Junior High school. At the age of 10 years old, his IQ was evaluated to be 167, and so he skipped the sixth grade (Chicago Tribune, 2017), an event later described as pivotal to his development (Chase, 2004):

“Previously he had socialized with his peers and was even a leader, but after skipping ahead he felt he did not fit in with the older children and was bullied.”

## Harvard University (1958–62)

Kaczynski entered Harvard University in 1958 at the age of 16 years old. A mathematical prodigy since he was a child, he was described by other undergraduates as* “shy”*,* “quiet”* and* “a loner”* who *“never talked to anyone“* (Song, 2012):

“He would just rush through the suite, go into his room, and slam the door […] When we would go into his room there would be piles of books and uneaten sandwiches that would make the place smell”

His personality notwithstanding, Kaczynski’s talent was however still recognized among his Harvard peers, one of which in 2012 stated:

“It’s just an opinion — but Ted was brilliant [...]. He could have become one of the greatest mathematicians in the country”

Kaczynski graduated Harvard with a B.A. in mathematics in 1962. When he graduated, his GPA was 3.12, scoring B’s in the History of Science, Humanities and Math, C in History and A’s in Anthropology and Scandinavian (Stampfl, 2006).

## University of Michigan (1962–67)

With an IQ of 167, Kaczynski had been expected to perform better at Harvard. After graduating, he applied to the University of California at Berkeley, The University of Chicago and the University of Michigan. Although accepted at all three, he ended up choosing Michigan because the university offered him an annual grant of $2,310 and a teaching post. The “darling of the math department”, he would graduate from the University of Michigan in 1964 with a M.Sc. in mathematics and markedly improved grades — 12 A’s and five B’s, which he himself later attributed to the standing of the university:

“[My] memories of the University of Michigan arenotpleasant […] The fact that I not only passed my courses (except one physics course) but got quite a few A’s shows how wretchedly low the standards were at Michigan”

Nonetheless, as the story goes, while there once a professor named George Piranian told his students — including Kaczynski — about an unsolved problem in boundary functions. Weeks later, Kaczynski came to his office with a 100-page correct, handwritten proof. Kaczynski graduated with a Ph.D. in mathematics in 1967. His dissertation, entitled simply *“Boundary Functions”* regarded the same topic as his proof of Piranian’s problem. His doctoral committee consisted of professors Allen L. Shields, Peter L. Duren, Donald J. Livingstone, Maxwell O. Reade, Chia-Shun Yin. Every professor approved it. His supervisor Shields later called his dissertation

“The best I have ever directed”

An additional testament to its quality was it being awarded the Sumner Myers Prize for the best mathematics thesis of the university, accompanying a prize of $100 and a plaque in the East Quad Residence Hall entrance listing his accomplishment. Of the complexity (or perhaps narrow implications) of his dissertation, one of the members of his dissertation committee, Maxwell Reade, said

“I would guess that maybe 10 or 12 men in the country understood or appreciated it”

Another, Peter Duren, stated

“He was really an unusual student”

**University of California, Berkeley (1967–69)**

In late 1967, at 25 years old Kaczynski was hired as the youngest-ever assistant professor of mathematics at the University of California at Berkeley. There, he taught undergraduate courses in geometry and calculus, although with mediocre success. His student evaluations suggest that he was not particularly well-liked because he taught *“straight from the textbook and refused to answer questions”*.

He resigned on June 30th, 1969 without explanation.

# Work (1964–69)

## Wedderburn’s Theorem

Kaczynski’s only published paper relating to topics other than boundary functions was his first journal paper, written before he started his Ph.D. It is entitled:

**Kaczynski, T.J. (1964).**“Another proof of Wedderburn’s theorem”.*The American Mathematical Monthly*71(6), pp. 652–653.

The paper concerned a 1905 result of Joseph H. M. Wedderburn that every finite skew field is commutative. His paper provided a group-theoretic proof of the theorem, which had previously been proved at least seven times.

## Boundary Functions

Kaczynski’s Ph.D. dissertation concerned boundary values of continuous functions and was entitled, simply

**Kaczynski, T.J. (1967).***Boundary Functions*. Ann Arbor: University of Michigan.

Let H denote the set of all points in the Euclidean plane having positive y-coordinate, and let X denote the x-axis. If p is a point of X, then by an arc at p we mean a simple arc γ, having one endpoint at p, such that γ = {p} ⊆ H. Let f be a function mapping H into the Riemann sphere.

**Boundary Functions**By a boundary function for f we mean a function φ defined on a set E ⊆ X such that for each p ∈ E there exists an arc γ at p for whichlim (s p, s ∈ γ) f(z) = φ(p)

Kaczynski’s dissertation begins by re-proving a theorem of J. E. McMillan which states that if f(H) is a a continuous function mapping H into the Riemann sphere, the the set of curvilinear convergence of F (the largest set on which a boundary function for f can be defined) is of a certain type. This proof also shows that if A is a set of the same type in X, then there exists a bounded continuous complex-valued function in H having A as its set of curvilinear convergence. The dissertation contains two additional new proofs related to boundary functions, and a list of problems for future research. Of the results, Professor Donald Rung later stated:

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.

Kaczynski would publish five journal papers related to the work from his dissertation between 1965–69:

**Kaczynski, T.J. (1965)**. “Boundary functions for functions defined in a disk”.*Journal of Mathematics and Mechanics.*14(4), pp. 589–612.**Kaczynski, T.J. (1966)**. “On a boundary property of continuous functions”.*Michigan Math. J.*13, pp. 313–320.**Kaczynski, T.J. (1969)**. “The set of curvilinear convergence of a continuous function defined in the interior of a cube”.*Proceedings of the American Mathematical Society*23(2), pp. 323–327.**Kaczynski, T.J. (1969)**. “Boundary functions and sets of curvilinear convergence for continuous functions”.*Transactions of the American Mathematical Society.*141, pp. 107–125.**Kaczynski, T.J. (1969)**. “Boundary functions for bounded harmonic functions”.*Transactions of the American Mathematical Society.*137, pp. 203–209.

## The Distributivity Problem

The only other trace of Kaczynski in a mathematical journal is two notes in the American Monthly in 1964 and 65:

**Kaczynski, T.J. (1964).**“Distributivity and (−1)x = −x (Advanced Problem 5210)”.*The American Mathematical Monthly.*71(6), pp. 689.**Kaczynski, T.J. (1965)**. “Distributivity and (−1)x = −x (Advanced Problem 5210, with Solution by Bilyeu, R.G.)”.*The American Mathematical Monthly*72(6), pp. 677–678.

In the first note, Kaczynski proposes the following problem, concerning group theory:

Let K be an algebraic system with two binary operations (one written additively, the other multiplicatively), satisfying:1. K is an abelian group under addition,2. K - {0} is a group under multiplication, and3. x(y+z) = xy + xz for all x,y,z ∈ K.Suppose that for some n, 0=1+1+1....+1 (n times). Prove that, for all x ∈ K, (-1)x = -x.

In the second note, the solution to the problem is — somewhat dismissively — provided by R. G. Bilyeu:

The last part of the hypothesis is unnecessary. If z denotes -1, then z+z+zz = z(1+1+z) = z, so zz = 1. Now z(x+zx) = zx+x = x+zx, so either x+zx = 0 or z = 1. In either case zx = -x.

# Conclusion

Theodore J. Kaczynski was a very promising young undergraduate, graduate and post-graduate student in the 1960s. His work — although pertaining to vary narrow topics — was undoubtedly, technically, first rate.

As is the case however, elegance or complexity do not themselves raise the importance of problems, achievements or for that matter, mathematicians. As expressed by his fellow graduate student Professor Peter Rosenthal in a 1996 Toronto Star article (after Kaczynski was charged):

[The] 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.

In another 1996 article, in the Los Angeles Times article, Professor Donald Rung similarly expressed:

“The field that Kaczynski worked in doesn’t really exist today […]. 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.”

## References

**Chase, A. (2004)**. A Mind for Murder: The Education of the Unabomber and the Origins of Modern Terrorism. W. W. Norton & Company.**Chicago Tribune (2017)**. “The Kaczynski brothers and neighbors”.*Chicago Tribune*.**Song, D. (2012).**Theodore J. Kaczynski.*The Harvard Crimson.***Stampfl, K. (2006)**. He came Ted Kaczinsky, he left The Unabomber*.The Michigan Daily.*March 16 2006.

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