Math 5810 Topics in Combinatorics: 

	"Analytic Combinatorics in one and several variables"

Analytic combinatorics involves estimation of combinatorial quantities
via asymptotic analysis of their generating functions.  The course
begins with the univariate case and the study of combinatorial 
enumeration, specifically but not exclusively via generating functions.
We look at a compendium of problems from computer science, genomics, 
graph theory, probability and elsewhere.  

Next we will address how asymptotic analysis is carried out via 
complex contour integration.  This requires only undergraduate
level techniques.    

The latter part of the course is on multivariate enumeration.  Again
we will begin with motivating problems, with connections to cluster
algebras, statistical physics, lattice recursions and enumeration of
recursive structures.  Extraction of coefficient asymptotics proceeds 
again by complex contour integration.  This time, the machinery required 
to carry out the asymptotic integration, in addition to classical
saddle point integration, includes Morse theory and computational algebra.
The most interesting cases involve topological analysis and the theory 
of hyperbolic polynomials.  Which of these techniques we focus on will
depend to some extent on the interests of those attending.

We will follow the second edition of "Analytic Combinatorics in 
Several Variables" by Pemantle, Wilson and Melczer (2024).