3. Postdoctoral Program - MSRI

Year-End Report 2009–10
NSF Math Institutes Postdoctoral Fellowship
1 Summary
Karl Mahlburg
I have had a very productive year thanks to the support of my NSF Math
Institutes Postdoctoral Fellowship. I was able to attend and speak at a
number of conferences and workshops, as well as to make several extended
research visits with various collaborators. I also benefited greatly from being
hosted at Princeton University, where there has been a constant presence
of interesting visitors and speakers, thanks in part to the Special Year in
Analytic Number Theory at the Institute for Advanced Study.
At the beginning of 2009, Kathrin Bringmann and I began exploring
some surprising connections between integer partitions, hypergeometric qseries,
automorphic forms and Ramanujan’s mock theta functions, and percolation
models from statistical physics. Since that time, we have successfully
developed these initial observations into a full research program, with
applications in number theory, combinatorial probability, and statistical mechanics.
I had no prior background in many of these areas, and it has been
very satisfying to learn more about these subjects while discovering their
connections to number theory. Indeed, I have additionally established new
interests and collaborations within combinatorial probability, notably with
Alexander Holroyd and Lionel Levine on abelian sandpiles.
Looking forward to the second year of my Fellowship, I am excited to
pursue the continued development of my research program in combinatorial
probability and number theory, but I also hope to be more directly engaged
with my mentor Manjul Bhargava. I have spent much of my time this past
year simply learning new areas of mathematics, but now that I am more
comfortable with these subjects, I intend to devote more energy to other
aspects of number theory.
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