If you ask Karl Mahlburg about his mathematical breakthrough, he will, typically, smile a very shy smile, duck his head, and say something self-effacing.

But Mahlburg, a 25-year-old UW-Madison graduate student, has solved what may be the last part of a historic mathematical problem that has challenged the brightest minds in the field of number theory for 75 years. It is a feat that has drawn praise from the elite in the math world.

So daunting was this part of the problem that Mahlburg’s professor and mentor, Ken Ono, Manasse Professor of Letters and Science—who made his own startling breakthrough on another piece of the same problem—advised against him even trying.

But Mahlburg forged ahead. For one year, working mostly in his modest cubby hole of an office on the fourth floor of Van Vleck Hall, he labored away, scrawling equation after equation on pages of lined notebook paper. And in the last few weeks, he has succeeded.

“It is beautiful,” the renowned Princeton physicist Freeman Dyson said of Mahlburg’s work Thursday. “Beautiful and totally unexpected.”

Clever, and quite stubborn
Dyson is perhaps one of the ultimate and most learned sources on this particular accomplishment. Better known among lovers of science writing for books such as “Infinite in All Directions,” and among scientists for his seminal work in quantum theory and particle physics, Dyson also plays a crucial role in the history of the problem on which Mahlburg worked.

Though the math itself quickly becomes a thorny tangle to the lay person, the broad outlines of Mahlburg’s work can be understood and appreciated as well as a few of its implications, though it is early to discern many. Such work, Ono said, can have important applications in number-related fields such as cryptology.

As important, the story of Mahlburg’s effort affords a look into the brainy and infrequently revealed depths of scholarly life on the UW-Madison campus. And it shows, too, how math is done and has been done since the ancients, how theory and proof are slowly built upon through the decades, bricks becoming walls and walls eventually becoming a house.

As for this particular structure of mathematical thought, Mahlburg may have added the roof.

“He’s whip-smart,” Ono said of his student. “He’s very clever. And, frankly, he’s quite stubborn, stubborn in not letting go of a problem.”

Mahlburg’s problem has its roots in the life of one of history’s most unlikely mathematicians. Srinivasa Ramanujan was born in southern India in 1887. With no formal training, the short, pudgy boy who was slow to learn to speak, grew into one of history’s most famous and gifted mathematicians.

Much of Ramanujan’s work was in number theory, the study of the relationships and properties of numbers.

Partitions are a good and relatively uncomplicated place to start in understanding the complexities of what Mahlburg has done. A partition, simply, is the expression of a number as a sum of numbers less than or equal to the given number. Four, as an example, has five partitions. They are: 1+1+1+1, 2+1+1, 3+1, 2+2, and 4.

The interesting thing about partitions and one of the things that makes them intriguing to study is that they start out simple but quickly become complicated. While the number four has five partitions, for example, the number 10 has 42. And by the time you get to 100, you are dealing with a number that has 190,569,292 partitions.

Number theorists look for patterns and for predictability in those patterns when they look at long lists of numbers.

Eventually, studying lists of partitions, Ramanujan discerned a pattern that became a historic discovery. He found that, starting with four, the number of partitions for every fifth integer is a multiple of 5. For example, the number of partitions for 9 is 30 and for 14 is 135. Such a relationship is called a congruence.

Ramanujan found congruences for 5, 7, and 11. The congruences were an unexpected finding and mathematicians through the years have been obsessed with figuring out why they exist and why there were apparently only congruences for those three prime numbers. They wanted to know the reason behind the rhyme.

Years passed with little or no progress. Then, in the 1940s, a young Dyson took up the task. He made advances the math world had been waiting on for years. In 1944, he published a paper theorizing the existence of a mathematical tool he called a “crank” that would explain congruences and could be used to find more of them.

And there the so-called “crank conjecture” sat for 40 years. But, beginning in the 1980s, there came a series of discoveries related to the problem. The actual existence of the crank was proven by another famous mathematician, George Andrews of Penn State University and a student of his, Frank Garvan.

This is where the story turns toward Madison and the Van Vleck offices of Ono and Mahlburg.

Teacher warned student
After the work by Andrews and Garvan, it was widely believed there would be no new major discoveries regarding partition congruences. But after long hours spent studying a lost notebook of Ramanujan’s, Ono shocked the math world by proving in 2000 that Ramanujan’s three congruences were merely the easy ones to see.

In research Science News called “a remarkable tour de force,” Ono showed Ramanujan congruences are everywhere.

Once again, many in the math world believed the work on congruences would end with Ono’s accomplishments. But as Ono explained, mathematicians come in a couple of types: theoreticians who make the grand connections, and problem solvers who make the theories work on paper.

Mahlburg, Ono said, is a problem solver. Ono said that in his four years as a graduate student at UW-Madison, Mahlburg has already written three papers, more than enough to earn a Ph.D. So when the young man said he wanted to continue Ono’s work on congruences, Ono was nervous. He didn’t want his student to fail.

But Mahlburg could not be dissuaded.

“Not everyone,” Ono said, “has the ability to work out a problem that at the end of a year, might not work out at all.”

In summary, Mahlburg described exactly how the famous crank works. It took calculations that even Ono found exhausting, pages and pages of mind-bending work. Mahlburg, who plays classical piano to get the numbers out of his head when he needs a break, lived with his problem day in and day out. He did much of his best thinking, he said, just before going to sleep at night.

But he succeeded. Even as monumental a figure in the field as Andrews said Thursday that he is deeply impressed with Mahlburg’s work. Now, Andrews said, Mahlburg’s work, along with Ono’s breakthrough, allows for a more complete explanation of the intricacies of partitions and congruences than ever before.

“This is something way beyond what anybody has done before,” Andrews said. “This is a very bright guy who has done something very hard.”

Mudd Influence
During a busy time that included interviews and subsequent articles in  PhysOrg.com, New Scientist and WMTV-TV, Mahlburg ’01, who received HMC departmental honors in the humanities and social sciences as well as mathematics, responded by e-mail to the HMC Bulletin.

Training. “The greatest benefit I gained from HMC was due to the fact that the professors taught at a high level and constantly encouraged thinking about solving new problems. This training has helped me approach my research throughout graduate school, even in unfamiliar subjects.”

Professors. “I had a great deal of positive interaction with [mathematics professors] Francis Su and Art Benjamin. They are both very dynamic, exciting teachers with a passion for mathematics, and seem truly committed to working with talented students.”

Humanities. “I had played classical piano for many years, and was thankful to be able to continue studying at Scripps. I’ve continued to do the same through the music department at UW-Madison. I was also interested in literature, and public policy and world events in a vague sense, and although it was sometimes difficult to meet the various requirements, I am glad for all of the humanities courses that I took at Mudd.

Advice. “In terms of schoolwork and the like, there’s really no substitute for time and effort.”