Friday, November 22, 2013 — 3:30 PM to 4:30 PM EST

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

Speaker: Andrew Childs
Affiliation: University of Waterloo
Room: Mathematics and Computer Building (MC) 5158

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In this talk I will describe recent (and mostly unpublished) work concerning these problems --- work that lies on the boundary between analysis, combinatorics and probability.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!

Location 
MC - Mathematics & Computer Building
5158
200 University Avenue West

Waterloo, ON N2L 3G1
Canada

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