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Chipping away at the Fuchs’ problem: Sunil Chebolu makes mathematical strides

image of algebra

Warning: The following story contains math.

WAIT! Before all arithmophobes run screaming into the hallways, know this story is more about the puzzles and great mysteries that humans have contemplated since the time of the ancient Greeks.

Though he forgoes the traditional robes of Pythagoras and Aristotle, Illinois State University’s Associate Professor of Mathematics Sunil Chebolu explores mathematical concepts and problems in the same vein as his ancient colleagues. And he is making startling headway.

Chebolu, in collaboration with his colleague Keir Lockridge from Gettysburg College, recently solved an important case of a problem that has been plaguing mathematicians for more than 50 years. Known as the Fuchs’ problem, Chebolu and Lockridge are earning accolades for discovering a new way to tackle this complex math problem.

image of Sunil Chebolu's work on the Fuchs problem.

Sunil Chebolu’s work on the Fuchs’ problem.

[NOTE: For all you mathematicians and generally brilliant math people, pieces of Chebolu’s work look like the picture on the right.  For the rest of us mere mortals, read on for an explanation.]

Some mathematical problems have been around for centuries, even millennia. The Fuchs’ problem, unearthed by László Fuchs (a Hungarian-American mathematician) in the 1960s, had seen little or no movement since its inception.

In simple terms, the Fuchs problem asks how two objects in algebra are related. “There are two fundamental objects in algebra. One is called a group, and the other is called a ring,” said Chebolu. “Groups and rings are ubiquitous in algebra. They are very basic, like household items for every mathematician.”

[For those algebra novices: A group is a set with a single binary operation like the multiplication of invertible matrices. A ring is a set with two compatible binary operations like addition and multiplication of integers. This is the last set of brackets. Promise. ]

To create a visual for Fuchs’ problem, Chebolu invokes the image of two islands. “On the one hand you have an island of groups, which have only one binary operation. And on the other you have an island of rings, which have two binary operations,” he said, placing his hands to cradle the imaginary islands.

 Sunil Chebolu

Sunil Chebolu

“Now, you can take a ring and cook up a group simply by taking the set of invertible elements in a ring,” he said, then paused and leaned forward. “Think of this construction like a bridge between these two islands. So the problem posed by Fuchs is to find all the groups which are obtained from rings by crossing that bridge.”

To find the answer to this algebra problem, Chebolu and Lockridge involved another mathematical field–number theory. “When you have a problem in mathematics, you try to look at it in a lot of different ways, and try to reformulate it in a different setting or language. That is exactly what happened in my research,” he said.

The pair began to explore prime numbers—ones that do not have any proper divisors—and discovered a path to understanding how some numbers were trotting over that bridge. “When it comes to abelian groups that cannot be broken down (also called indecomposable), we can tell exactly which ones have crossed the bridge.”

Not only did their work unravel some of the secrets of the Fuchs’ problem, but it also led to a new way of looking at some well-known series of prime numbers. “Our work gave new characterizations of Mersenne primes and Fermat primes—two families of primes which have been around for centuries. And to me that is really, very satisfying,” he said.

When describing the experience of solving a mathematical problem, Chebolu explains it is similar to puzzle pieces suddenly falling into place. “You see one possible path, and then there can be a chain reaction with insights into another problem, and another problem. For me, the culmination of reactions is the solution to the Fuchs’ problem in one big case,” he said.

The work has been received and applauded at the highest levels of his field, appearing in the prestigious Journal of Algebra and the research was funded partly by Chebolu’s grant from the National Security Agency (NSA). The results even garnered praise from the man who posed the problem. “At 91 years old, László Fuchs at Tulane University is still turning out incredible work,” said Chebolu, who has long been an admirer of Fuchs’ work.

Chebolu was invited to give a talk on his discovery at Tulane University (Fuchs’ home base). He also presented the work leading up to this discovery at other places including Illinois State, the University of Illinois Urbana-Champaign, Northern Illinois, Central Michigan, and Western Ontario in Canada.

“In mathematics, our steady state is that we are stuck,” said Chebolu. “But occasionally, we do make progress, and for a short while we have a big mathematical high…and then we immediately return to our steady state of being stuck,” he said with a laugh.

While Chebolu and his colleague solved an important case of the Fuchs’ problem, there are still larger questions for them to tackle. “Honestly, I didn’t have any hope that we would make any progress on this problem when we began. But there is a good lesson for mathematicians—never give up,” said Chebolu. “That’s a good lesson for everyone, come to think of it.”

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