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                * Princeton Discrete Math Seminar *
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Date: Thursday 17th October, 4:30 in Fine Hall 224.
 
Speaker: Alan Sokal (New York University and University College
London)

Title: 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_{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In this talk I will describe recent (and mostly unpublished)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/

                       -----------
Next week: Natan Rubin

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