Two Math Whizzes Whose Work Counts

Like many other Harvard students who have an affinity for math beyond the AP Calculus test, Lauren W. Ancel '95 goes to Math 25 every week. After all, she too is here to learn things from her professors. However, if everything goes well, she may soon be the one doing the teaching.

Ancel recently invented her own mathematical theorem.

In layman's terms, Ancel describes the theorem as applicable to the area of pure math called "dynamical systems."

In technical jargon, Ancel explains her theorem as follows: "Given a surjective continuous transformation from a circle to itself, every aperiodic interval is one way."

Although the details of Ancel's theorem may be slightly unclear to most non-mathematicians, the significance of its discovery can be understood by all: Here is a person already making contributions to the scientific community--as a first-year college student.

Ancel began her path toward discovery at the beginning of her senior year in high school.

"I was at the Research Science Institute at UCSD [University of California, San Diego] in the summer of 1990, working in the field of cryptography," she says. "Completing this program sparked my interest in pure math, and motivated me to do some research."

After returning home, Ancel asked her father, a math professor at the University of Milwaukee, Wisconsin, to show her a few problems that she could look into, and decided that one in particular interested her the most.

"My father came across this particular problem in a graduate student's Ph.D thesis. It was a question that had gone unanswered, so I decided to take a shot at solving it," says Ancel.

Ancel worked on this problem for approximately two months before discovering a solution.

"I got the problem in September, read whatever I could find about it, and played with it in my mind for a while. Around late October I had one major insight, and then over the next few weeks everything else fell into place," she says.

As far as what happens now, Ancel explains that she and the professor who first came across the original problem--Michael W. Hero of Bradley University--are planning to submit it to a mathematical journal in the near future.

"Her work was very good. I'm currently working on a joint paper with her," says Hero. "Our theorem has some interesting corrolaries related to the topological dynamics of self-maps of the circle."

Although Ancel has not yet received any formal recognition from the scientific community regarding her discovery, she says it was probably the main reason why she was chosen as one of USA Today's top 20 scholars this year.

He Does Number Theory

Jim W. Cheung '95 is also a Math 25 student, and, like Ancel, is extensively involved in mathematical research. Although he has not yet published his own theorem, his research ideas were revolutionary enough to satisfy the "top-10" judges of this year's Westinghouse Science Competition.

Cheung says his research is applicable to number theory and the use of continued fractions.

"I used continued fractions over the ring of Eisenstein integers to approximate complex numbers. In my proofs, I was able to show that the continued fractions over this system do not necessarily converge," explains Cheung.

Like Ancel, Cheung's intense interest in mathematics was greatly spurred by summer math programs.

"I attended the Ohio State University Summer Math Program for three years during high school, in addition to doing some reading on my own," he says.

Cheung says that he got started on this particular topic in high school, while he was still at Bronx Science in New York.

"I started on this problem in the summer of '89 at the beginning of my junior year. I began the project by just experimenting and toying with the idea, and it just kind of grew from that," says Cheung.

When asked whether he plans to submit his findings to a scientific journal soon, Cheung seemed cautious.

"I probably won't submit my proposals right away. I still plan to do some more work with it before I feel that it is truly complete," he says.