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Qnas with Karl Mahlburg QnAs QnAs with Karl Mahlburg with Karl Mahlburg n April 2006, PNAS awarded its first Paper of the Year prize to Karl Mahlburg, a mathematics graduate student studying with number theo- Irist Ken Ono at the University of Wiscon- sin, Madison. Mahlburg’s work (1) ‘‘adds a lustrous chapter,’’ as mathematician George E. Andrews said in a Commen- tary, to the study of a long-standing prob- lem involving partition theory and the crank function—all of which began with an observation by the famed Indian math- ematician 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 Karl Mahlburg at the spring PNAS Editorial Board Meeting, 2006. Photograph by Mark Finkenstaedt that, but because of my contact with George © 2006. All rights reserved. Andrews, who was quite interested in my work, he offered to submit it. He recently was elected to the Academy. Once that partitions of 14. You may notice that the that you pick, you could find a divisibility came up, it did seem like a wonderful idea. number of partitions are all multiples of 5. pattern in the partitions. It opened up new PNAS is, of course, a highly respected Ramanujan found these congruences, or horizons in the area. divisibility conditions, in the partitions. For journal. Ken Ono published a paper in PNAS: Where did you enter the story? any number that ends in 4 or 9, the number PNAS with Scott Ahlgren a few years ago on Mahlburg: The question that I was studying of partitions is divisible by 5. He also found partition congruences. Even though the was, ‘‘Given such a huge new collection of similar patterns for partition numbers that presence of math is light, what is there is of congruences, what was the relation of the were divisible by 7 and 11. Then Freeman the highest quality. crank to those?’’ And what I found was that Dyson, when he was a student at Cambridge the crank function played a similar role for PNAS: How did you become interested in in the 1940s, learned about the Ramanujan this infinite collection of congruences for partition theory as a subfield of number congruences, and he found it a little unsat- any prime [as it did with the Ramanujan theory? isfying that there wasn’t a combinatorial congruences]. Not entirely the same, but Mahlburg: Actually, the area of partitions in explanation. particular was probably what most drew my similar. The crank could be showed to be a attention to number theory. Ken Ono is very PNAS: Why are combinatorial explanations universal statistic and have an intimate re- well known for his work in partition theory, so important? lation with all of these partition congru- and when I was learning about the things Mahlburg: Combinatorial proofs are valu- ences. It was rather surprising that anything, that he had worked on, partition theory able and can be difficult to find because in let alone the crank, could be involved in the remained in the back of my mind as an area a certain sense they really allow you to get a expanded role of all the congruences. handle on things. So Freeman Dyson won- that I would love to be able to make a PNAS: Looking back, do you think graduate dered if there was a way to break up the contribution in. Over the course of my first school was a good time to tackle an ambi- partitions and see why the congruence for 5 few years of graduate school, as I was learn- tious problem? was there. He found a way, which he called ing a broad array of topics, I would come Mahlburg: I guess I’ve benefited from a the rank. Dyson’s rank also worked for the back and read some of this work on parti- certain sort of naivete´. I probably would not congruence for 7, but it didn’t work for 11. tions that Ken had done. And I would try have been so eager to work on this had I So he conjectured that there must be a way again to study it and to master it. known that it would take over a year, with all to get all three—some sort of ‘‘crank,’’ he of the frustrations and baby steps and set- PNAS: What’s the history behind your work called it. It took over 40 years before Frank backs along the way. Instead, not having with partition congruences and the crank? Garvan and George Andrews actually undertaken a project of this magnitude be- Mahlburg: It’s a very romantic story. Ra- found it, at a conference celebrating the fore, I just viewed it incrementally. At every manujan, a famous Indian mathematician, 100th anniversary of Ramanujan’s birth. step of the way, I was able to see the next was perhaps best known for his work in They were inspired by some of Ramanujan’s goal and would just work toward that. partitions. A partition is just the number of scribblings in one of his ‘‘lost’’ notebooks. ways you can take a number and write it as At the time that seemed like the end of Regina Nuzzo, Science Writer a sum of smaller numbers, regardless of the story. But then in 2000, Ken Ono pub- order. Take the number ‘‘4.’’ You could lished his work on partitions where he 1. Mahlburg K (2005) Proc Natl Acad Sci USA 102:15373–15376. write it as 4, or 3 ϩ 1, or 2 ϩ 2, or 2 ϩ 1 ϩ 1, showed that, in fact, there weren’t just those or 1 ϩ 1 ϩ 1 ϩ 1. So there are 5 partitions of three divisibility patterns for 5, 7, and 11— 4, and there are 30 partitions of 9, and 135 there were infinitely many. For any prime © 2006 by The National Academy of Sciences of the USA www.pnas.org͞cgi͞doi͞10.1073͞pnas.0604770103 PNAS ͉ September 12, 2006 ͉ vol. 103 ͉ no. 37 ͉ 13569 Downloaded by guest on September 25, 2021.
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