Number Theory

You never know when you're going to find your calling. For some, it's a lifelong search. For Zebediah Engberg, it took a few math classes at Hampshire.

"I didn't fall in love with math until I got here," says Engberg. "I thought I would do philosophy, and those were good classes. But the math classes surpassed my expectations so much that I knew I would have to focus on that."

Number theory, in particular, is what really catches Engberg's attention. Essentially a study of the properties of integers (negative and positive numbers and zero, excluding fractions), it's a field that has captivated some of history's greatest mathematical minds.

"There are a lot of open problems that just haven't been solved yet, that are fascinating and extremely important in the field," he says, noting that even when answers are discovered, constructive use is minimal. "There are no real-world applications. It's completely abstract, I think it's more like art or poetry."

As a rock climber, Engberg has earned a reputation internationally as a top competitor.  His passion for climbing is similar to what he feels for mathematical problems, where the end result isn't always as important as the challenge, the analysis of all the factors involved and the attempt to overcome what can sometimes seem like insurmountable odds. 

Engberg credits Hampshire mathematics professor Ken Hoffman (his Div III chair) and associate professor of mathematics David Kelley as two of his biggest influences on campus. For his Div III, he focused on a family of equations studied by the Greek mathematician Diophantus in the third century A.D. Questioning whether there are any solutions to the problems, the finite or infinite nature of any solutions, and whether all of the solutions can be found are some of the things he investigated.

"I first saw the problem two or three years ago in a number theory class. It seems so pure. And you don't need analytic techniques to work on it. You can start collecting data immediately, just add and multiply the numbers together. Nothing sophisticated is needed," says Engberg.

The apparent simplicity of the computations is deceptive. Engberg is pleased with the progress he made in the Div III project, but a definitive solution remains elusive. For the next five years he will be studying number theory (and possibly devoting some time to continuing his current work) at Dartmouth College in a Ph.D. program. It was a choice that he says was made possible by the freedom at Hampshire College.