Stephen Melczer, Department of Combinatorics & Optimization, University of Waterloo
"A Geometric Approach to Analytic Combinatorics"
The field of analytic combinatorics adapts techniques from complex analysis, algebraic and differential geometry, topology, and computer algebra to create algorithmic methods for the study of combinatorial objects. Classical analytic combinatorics derives the asymptotic behaviour of a combinatorial sequence by manipulating univariate Cauchy integral representations. The newer area of analytic combinatorics in several variables (ACSV) attempts to generalize this approach to multivariate sequences and limit theorems. In this talk we describe the basics of analytic combinatorics and ACSV before discussing how (new modifications of) results from stratified Morse theory help characterize Cauchy integral deformations that lead to asymptotic expansions in the multivariate setting. We then show how combining Morse-theoretic decompositions with effective algorithms for numeric analytic continuation from computer algebra allow for the computation of certain homology coefficients which seem very difficult to compute using topology alone. Open topological problems in this area will be discussed, along with examples from a range of applications including queuing theory, the complexity of biological networks, algebraic statistics, and the analysis of algorithms.
QNC 2501