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QnAs with Karl Mahlburg
In April 2006, PNAS awarded its first Paper of the Year prize to Karl Mahlburg, a mathematics graduate student studying with number theorist Ken Ono at the University of Wisconsin, Madison. Mahlburg's work (1) “adds a lustrous chapter,” as mathematician George E. Andrews said in a Commentary, to the study of a longstanding problem involving partition theory and the crank function—all of which began with an observation by the famed Indian mathematician Srinivasa Ramanujan in 1920. Mahlburg, who was funded by a National Science Foundation fellowship and will start the C. L. E. Moore Instructorship at Massachusetts Institute of Technology (Cambridge, MA) in the fall, took a break from writing his thesis to talk with PNAS.
PNAS:Why did you choose PNAS to publish your paper?
Mahlburg:PNAS certainly doesn't publish a lot of mathematics papers. I was aware of that, but because of my contact with George Andrews, who was quite interested in my work, he offered to submit it. He recently was elected to the Academy. Once that came up, it did seem like a wonderful idea.
PNAS is, of course, a highly respected journal. Ken Ono published a paper in PNAS with Scott Ahlgren a few years ago on partition congruences. Even though the presence of math is light, what is there is of the highest quality.
PNAS:How did you become interested in partition theory as a subfield of number theory?
Mahlburg:Actually, the area of partitions in particular was probably what most drew my attention to number theory. Ken Ono is very well known for his work in partition theory, and when I was learning about the things that he had worked on, partition theory remained in the back of my mind as an area that I would love to be able to make a contribution in. Over the course of my first few years of graduate school, as I was learning a broad array of topics, I would come back and read some of this work on partitions that Ken had done. And I would try again to study it and to master it.
PNAS:What's the history behind your work with partition congruences and the crank?
Mahlburg:It's a very romantic story. Ramanujan, a famous Indian mathematician, was perhaps best known for his work in partitions. A partition is just the number of ways you can take a number and write it as a sum of smaller numbers, regardless of order. Take the number “4.” You could write it as 4, or 3 + 1, or 2 + 2, or 2 + 1 + 1, or 1 + 1 + 1 + 1. So there are 5 partitions of 4, and there are 30 partitions of 9, and 135 partitions of 14. You may notice that the number of partitions are all multiples of 5.
Ramanujan found these congruences, or divisibility conditions, in the partitions. For any number that ends in 4 or 9, the number of partitions is divisible by 5. He also found similar patterns for partition numbers that were divisible by 7 and 11. Then Freeman Dyson, when he was a student at Cambridge in the 1940s, learned about the Ramanujan congruences, and he found it a little unsatisfying that there wasn't a combinatorial explanation.
PNAS:Why are combinatorial explanations so important?
Mahlburg:Combinatorial proofs are valuable and can be difficult to find because in a certain sense they really allow you to get a handle on things. So Freeman Dyson wondered if there was a way to break up the partitions and see why the congruence for 5 was there. He found a way, which he called the rank. Dyson's rank also worked for the congruence for 7, but it didn't work for 11. So he conjectured that there must be a way to get all three—some sort of “crank,” he called it. It took over 40 years before Frank Garvan and George Andrews actually found it, at a conference celebrating the 100th anniversary of Ramanujan's birth. They were inspired by some of Ramanujan's scribblings in one of his “lost” notebooks.
At the time that seemed like the end of the story. But then in 2000, Ken Ono published his work on partitions where he showed that, in fact, there weren't just those three divisibility patterns for 5, 7, and 11—there were infinitely many. For any prime that you pick, you could find a divisibility pattern in the partitions. It opened up new horizons in the area.
PNAS:Where did you enter the story?
Mahlburg:The question that I was studying was, “Given such a huge new collection of congruences, what was the relation of the crank to those?” And what I found was that the crank function played a similar role for this infinite collection of congruences for any prime [as it did with the Ramanujan congruences]. Not entirely the same, but similar. The crank could be showed to be a universal statistic and have an intimate relation with all of these partition congruences. It was rather surprising that anything, let alone the crank, could be involved in the expanded role of all the congruences.
PNAS:Looking back, do you think graduate school was a good time to tackle an ambitious problem?
Mahlburg:I guess I've benefited from a certain sort of naiveté. I probably would not have been so eager to work on this had I known that it would take over a year, with all of the frustrations and baby steps and setbacks along the way. Instead, not having undertaken a project of this magnitude before, I just viewed it incrementally. At every step of the way, I was able to see the next goal and would just work toward that.
Footnotes
 © 2006 by The National Academy of Sciences of the USA
 Photograph by Mark Finkenstaedt © 2006. All rights reserved.
References

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