The Middlebury Campus

Math Professor Speaks About Unproven Conjecture


Whether they were math majors or hadn’t taken a class since high school pre-calculus, every member of the audience in Twilight Auditorium was enraptured by Professor of Mathematics David Dorman’s lecture “Right Triangles, Elliptical Curves, and the Conjecture of Birch and Swinnerton-Dyer.” The lecture was held on Wednesday, January 18 as part of the Carol Rifelj lecture series. The series is named after the late Carol de Dobay Rifelj and aims to showcase faculty research. Dorman’s research involved the history of a conjecture in the field of number theory that defines the set of points that make up an elliptic curve.

The idea for the lecture, Dorman says, came from a dinner-party conversation with a non-mathematician who was curious about his research. Rather than trying to explain the complexities of number theory at the dinner table, Dorman took the conversation to the lecture hall. He prefaced his talk by warning that it can take many years of graduate school to understand the problems he’s trying to solve, let alone solve them.

The Conjecture of Birch and Swinnerton-Dyer is named after the mathematicians who developed it in the 1960s: Bryan Birch and Peter Swinnerton-Dyer. The conjecture is such a central problem in mathematics that the Clay Mathematics Institute listed it as one of seven Millennium Prize Problems. The Clay Institute has offered $1,000,000 to anyone who can prove the conjecture or any of the other six extraordinarily challenging prize problems.

Dorman’s lecture set out not to solve the Birch and Swinnerton-Dyer conjecture and collect its million-dollar bounty, but merely to explain it.

“It’s actually very difficult to give a mathematics talk to non-mathematicians, because you can’t assume anything,” Dorman said. In order to make the lecture accessible to mathematicians and non-mathematicians alike, Dorman decided to start his explanation of the Birch and Swinnerton-Dyer conjecture at the very beginning: the basics of right triangles, congruent numbers and the Pythagorean theorem. As the talk progressed, it touched on more complex topics such as elliptic curves, ranks of curves, and finally L-functions, all subjects that can take years of study to comprehend.

“You try and layer math talks that way, “ Dorman said. “Make sure everyone understands what’s going on at first, and then you say, ‘How many people am I willing to lose, if any, and when should I lose them?’ I spent weeks trying to figure out how to make that easy.”

To supplement the discrete mathematical aspect of the lecture, Dorman included a rich history of the conjecture that started in 1225 B.C. with the study of congruent numbers, or positive integers that are the areas of right triangles with three rational-number sides. Generating congruent numbers and discovering which numbers are congruent is an arduous process and one for which there exists no natural ordering. It wasn’t until the 1950’s that Birch and Swinnerton-Dyer discovered a connection between elliptic curves and finding congruent numbers, whence their conjecture was born.

This thousand-year process of identifying and solving problems is at the crux of studying mathematics. Mathematicians don’t spend all their time on the “big” problems; rather, they slowly chip away at them while entertaining smaller problems.

“You have a number of things that you’re working on at the same time,” Dorman said to describe how mathematicians approach their research. “It’s like a stove, so you’ve got something simmering on the back burner and it’s gonna be cooking for a long time. Those are the hard projects. And there are the ones in front that you’re working on, that hopefully are easier and that you can make progress on.”

When a theory is finally proven, it can open up the door to an entire new field of mathematics and an infinite number of new problems to solve. Discovering a new problem, like the points that make up elliptic curves, can suddenly become central to solving a potentially centuries-old problem like the congruent number problem. That is how conjectures such as Birch and Swinnterton-Dyer’s become so famous.

“A lot of people are working on this problem, so it became an important problem. It’s not like someone says, ‘OK, this one is important;’ people just realize that it is central to the topic. So if we solved this one, we’d be a lot closer to solving the other problem,” Dorman said.

Though the implications of proven theories may not always reach far beyond the world of mathematics, Dorman believes there can be great beauty in the process itself.

“It’s like poetry. Is the world really better because of this haiku? And the answer is, actually, yes.”

Dorman also fundamentally believes in the beauty of a solving a simpler problem: getting students to understand difficult concepts. When not examining the world’s most challenging math problems, Dorman teaches a number of algebraic geometry and number-theory-related math classes, including “The Magic of Numbers” and “Abstract Algebra.” Some of his most successful math days, he says, are when he is able to boil down complex subjects to make them more accessible.

Dorman’s lecture could certainly be considered a successful math day. The professor said it finally provided an answer to his dinner party guest’s seemingly innocuous question — and they’re still friends, too!

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