Professor solves math problem 65 years in making

IU mathematics professor Nets Hawk Katz, along with Larry Guth of the Institute for Advanced Study in Princeton, N.J., solved Paul Erdös’ distinct distances problem from 1946.

“The problem was about how you can arrange a set of points in the plane to have the fewest possible number of distinct distances,” Katz said.

“If you have any number of points on the plane, you can write down what all the distances are between them, and by arranging them in a special way, you can make that list as short as possible. The question is how short can that list be when the number of points gets very, very big.”

Katz said it took about three months from when they started working on the paper to when the problem was actually solved.

“It is based on earlier works of ours and other people,” he said. “So if you think of it somewhat more broadly, it was really something that happened over a number of years.”

Katz said in order to solve the problem, they had to prove something about when the set of points cannot exist.

“You are not finding a set of points,” he said. “You are kind of proving a negative. In order to do that, you have to sort of imagine all possibilities and that can be a little tricky. Our approach was to try to use the algebraic method, but we realized at a certain point that wasn’t efficient to solve the problem, and we ended up having kind of a combination of different approaches."

They are entitled to $125 for solving the problem, but Katz said that he’d rather have the original check signed by Erdös.

“There is a choice of a check that is a real check or a check that has Erdös’ signature, which of course can’t be cashed because he died 15 years ago,” he said. “It would be a really great thing to frame. My real hope is to receive that check, but I am slightly afraid that all copies of it got lost so I don’t think I will actually receive it.”

Katz said he hopes students will realize what a great math department IU has.

“It is not just me,” he said. “There are many productive and creative people in our math department, and if any students are interested in what they would have to do in order to solve a mathematical problem, I would strongly encourage them to talk to professors in the math department.”