Mathematics Department SIXTH RAMANUJAN^{*} COLLOQUIUM by Professor Freeman Dyson ^{**} The Institute for Advanced Study on PLAYING AROUND WITH PARTITIONS
Abstract: I have been playing around in two ways, first with equations and second with numbers. Using equations, I discuss a well-known connection between pure mathematics and statistical physics. The grand canonical ensemble of statistical mechanics is applied to the counting of partitions. This picture borrowed from physics gives a simple way to find the leading term in the exact calculation of the partition function by Hardy and Ramanujan. Using numbers, I have been trying unsuccessfully to guess a simple formula for the spt-crank function defined in recent paper by Andrews, Garvan and Liang. Here spt stands for smallest-part, and the spt-crank was designed to help us understand the beautiful new congruence properties of partition smallest-parts discovered by Andrews. The Andrews congruences are like the famous Ramanujan congruences for the partition function modulo 5, 7, and 11, except that Andrews has 13 mysteriously replacing 11. I found a simple formula for the spt-crank which exactly fits more than half the values but fails miserably for the others. I challenge everyone in the audience to fix it so that it fits them all.
^{**} ABOUT THE SPEAKER: Freeman Dyson is Professor Emeritus at the Institute for Advanced Study in Princeton. He was born in England and worked as a civilian scientist for the Royal Air Force during the 2nd World War. He graduated from Cambridge University in 1945 with a Bachelor of Arts degree in mathematics. He went on to Cornell University as a graduate student in 1947 and worked with Hans Bethe and Richard Feynman. He is known for his work in quantum field theory, solid-state physics, astronomy, nuclear physics, and his thoughts on science and society. Dyson has also done work in a variety of topics in mathematics, such as topology, analysis, number theory and random matrices. He is winner of the Lorentz Medal, the Max Planck Medal, and the Templeton Prize, as well as a Fellow of the Royal Society and the American Physical Society, and a member of the US National Academy of Science. Ramanujan Colloquium * University of Florida * Mathematics * Contact Info |