Title: What is a crank function?
Speaker: Daniel Glasscock, The Ohio State University
Seminar Type: What is... ?
Abstract: The partition function p(n) is a central object in number theory and combinatorics. Searching for a combinatorial explanation of Ramanujan's three remarkable congruences for p(n), Freeman Dyson in 1944, still an undergraduate at Cambridge, defined the rank of a partition. Based on examples, he conjectured that the rank modulo 5 splits the partitions of 5n+4 into 5 equally sized classes, and similarly for the rank modulo 7 with partitions of 7n+5. Ten years later, Arthur Atkin and Peter Swinnerton-Dyer proved Dyson's conjectures, giving a satisfying combinatorial explanation to the first two of Ramanujan's congruences.
Mysteriously, the rank completely fails to explain the third of Ramanujan's congruences. Dyson knew this, leading him to postulate the existence of another combinatorial statistic, the crank of a partition, which would explain the last congruence. It was not until 1987 that George Andrews and Frank Garvan realized Dyson's crank and showed it to successfully explain not only the last congruence, but all three of Ramanujan's congruences at once!
Recent work of Ken Ono, Karl Mahlburg, Kathrin Bringmann, and others has revolutionized what we know about congruences for the partition, rank, and crank functions. In this talk, I hope to tell this fascinating story along with plenty of concrete examples.
