Department of Physics
New York University
4pm Tuesday 15th March 2005
Room 5215, JCMB, King's Buildings
The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph $G$, or more generally on an arbitrary matroid $M$, and encodes much important combinatorial information about the graph (indeed, in the matroid case it encodes the full structure of the matroid). It contains as a special case the familiar two-variable Tutte polynomial --- and therefore also its one-variable specializations such as the chromatic polynomial, the flow polynomial and the reliability polynomial --- but is considerably more flexible.
I begin by giving an introduction to all these problems, stressing the advantages of working with the multivariate version. I then discuss some questions concerning the complex zeros of the multivariate Tutte polynomial, along with their physical interpretations in statistical mechanics (in connection with the Yang--Lee approach to phase transitions) and electrical circuit theory.
Along the way I mention numerous open problems. This survey is intended to be understandable to mathematicians with no prior knowledge of physics.