Atlantic Graph Theory Seminar: Jo Ellis-Monaghan (University of Amsterdam)

2017 saw the centennial of William Tutte, one of the greatest mathematicians of modern times.  One of the testimonies to Tutte’s genius is that nearly everything he did proved to be a catalyst, triggering an explosion of further investigations and opening whole new vistas of mathematics.  The Tutte polynomial is one of many such examples in his legacy.   Here we will explore some of its salient properties and some of the many directions that propagated outward from the original Tutte polynomial.  These include several ways in which the Tutte polynomial may be defined and its universality, as well as some of its combinatorial and algebraic properties.  We will showcase information encoded in the Tutte polynomial as evaluations and specializations, as these inform nearly every aspect of combinatorics.   Furthermore, the scope of the Tutte polynomial is continually broadening through generalizations of either its domain or parameter space, and we will highlight some important examples, and touch on its interrelations with other combinatorial polynomials.  We will conclude with its particularly fruitful connections with biology and the Potts model of statistical mechanics, and offer some open questions.