# Some wonderful conjectures (but very few theorems) at the boundary between analysis, combinatorics and probability

-
Alan Sokal , NYU
Fine Hall 224

PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.    Many problems in combinatorics, statistical mechanics, number theory and analysis give rise to power series (whether formal or convergent) of the form f(x,y) = sum_{n \ge 0} a_n(y) x^n, where a_n(y) are formal power series or analytic functions satisfying a_n(0) \neq 0 for n = 0,1 and a_n(0) = 0 for n \ge 2. Furthermore, an important role is played in some of these problems by the roots x_k(y) of f(x,y) --- especially the leading root'' x_0(y), i.e. the root that is of order y^0 when y tends to 0. Among the interesting series f(x,y) of this type are the partial theta function'' Theta_0(x,y) = sum_{n\ge 0} x^n y^{n(n-1)/2} which arises in the theory of q-series, and the deformed exponential function'' F(x,y) = \sum_{n\ge 0} (x^n/n!) y^{n(n-1)/2} which arises in the enumeration of connected graphs. These two functions can also be embedded in natural hypergeometric and q-hypergeometric families. In this talk I will describe recent (and mostly unpublished) work concerning these problems --- work that lies on the boundary between analysis, combinatorics and probability. In addition to explaining my (very few) theorems, I will also describe some amazing conjectures that I have verified numerically to high order but have not yet succeeded in proving. My hope is that one of you will succeed where I have not! Further information is available at http://www.maths.qmul.ac.uk/~pjc/csgnotes/sokal/