Tutte seminar - Alan Sokal

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

Many problems in combinatorics, statistical mechanics, number theory and analysis give rise to power series (whether formal or convergent) of the form $$ f(x,y) \;=\; \sum\limits_{n=0}^\infty a_n(y) \, x^n \;, $$ 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 $y \to 0$. Among the interesting series $f(x,y)$ of this type are the "partial theta function'' $$ \Theta_0(x,y) \;=\; \sum\limits_{n=0}^\infty x^n \, y^{n(n-1)/2} \;, $$ which arises in the theory of $q$-series, and the ``deformed exponential function'' $$ F(x,y) \;=\; \sum\limits_{n=0}^\infty {x^n \over n!} \, y^{n(n-1)/2} \;, $$ 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In this talk I will describe recent (and mostly unpublished) work concerning these problems --- work that lies on the boundary between analysis, combinatorics and probability.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!