Jason McCullough, who joined Iowa State last fall as an assistant professor, just shattered the case for the 32-year standing Eisenbud-Goto conjecture.
“I rank this as the best result in the field of commutative algebra in the last decade,” said Hal Schenck, professor and chair of the Department of Mathematics. “Hiring McCullough instantly raised the national profile of Iowa State Mathematics.”
The Eisenbud-Goto conjecture was put forward in 1984 by David Eisenbud and Shiro Goto. It theorizes that for certain algebraic varieties, or geometric spaces defined by polynomials, the regularity, a measure of computational complexity, would be less than the degree, a geometric measure of its twistedness in space.
Many mathematicians had worked on this conjecture in the past and tried to prove it true. Through this collective work the conjecture had been proven for many special cases, including all curves and smooth surfaces. McCullough, whose parents were both mathematicians and jokes he was bred for mathematics, knew of the Eisenbud-Goto conjecture. He was looking for interesting varieties to study when he came across an example with intriguing properties that hadn’t been studied further. He transformed the example into a form that would work to investigate the Eisenbud-Goto conjecture and used computer software to study it.
“I can become almost an experimental mathematician by harnessing the power of the computer,” he said. “I can say, ‘Give me two million examples of this’ and then I can sift through them and say, ‘All right, this idea I had clearly isn’t going to work, but these other ideas I’ve been having seem more plausible.’”
The new example gave him a new approach to the problem and a new path to investigate. To his own surprise, he found counterexamples to the long-standing conjecture, showing instances where the computational complexity is, in fact, higher than the degree.
“We did later prove that there is a bound on the complexity. It was just much, much bigger than the conjectured bound,” he said. “We can’t tell you what it is. We just know it exists. And figuring out the actual answer is something I’m working on still. I don’t have a good guess right now. All we could prove is that it’s not polynomial. So, it has to be pretty big.”
McCullough and his coauthor, Irena Peeva, professor of mathematics at Cornell University, presented their work at a commutative algebra conference in July 2016. The conference doubled as a birthday surprise for Craig Huneke, a professor at the University of Virginia and well-known commutative algebraist who had helped McCullough identify the Eisenbud-Goto conjecture as an important outstanding problem.
“We had a little bit of fun with this,” McCullough said. “We didn’t tell people what we were going to do in advance. Our abstracts were very vague about what we were going to say, and we had back-to-back talks, so we tried to maintain the suspense throughout the talks.”
“I rank this as the best result in the field of commutative algebra in the last decade. Hiring McCullough instantly raised the national profile of Iowa State Mathematics.”
The two received quite a reaction and much applause, even going viral on Facebook among mathematicians. After the talks Huneke addressed the audience.
“When I was seven years old, I wanted a bicycle. I really, really wanted a bicycle. And I got it for my birthday, and that was the best birthday present to me ever — until this present right now,” Huneke said.
The paper detailing the work was accepted in 2017 and appeared in the April 2018 volume of the Journal of the American Mathematical Society, considered by many to be the best journal in mathematics, accepting only five percent of submissions.
But research related to the Eisenbud-Goto conjecture will go on as the problem simply becomes more precisely defined. Though it was disproven generally, the counterexamples all had kinks, like a crease mark, causing a sharp angle in an otherwise smooth surface. McCullough suspects it is still true in smooth cases, many of which have already been proven true.
“I think of it as bifurcating the question,” McCullough said. “Does it still hold in this really nice case in which our examples don’t say anything and what is the better statement in the general setting?”
That better general statement, with more precise wording, remains elusive for the time being. But with the help of computer-run calculations to provide fresh examples and new perspectives, he hopes to someday find it.