Algebraic Combinatorics Seminar - Stephen Melczer

Title: From Combinatorics to Computer Algebra and Morse Theory - Making Sense of Multivariate Asymptotics

Abstract:

The  asymptotic study  of  multivariate  generating  functions comprises  the  domain  of  Analytic Combinatorics  in Several Variables (ACSV).​​​​​​​ Although the techniques of ACSV parallel a better known​​​​​​​ univariate theory, the pathologies which arise in the analysis of multivariate functions leads to many intriguing -- perhaps, in general, undecidable -- questions. This talk focuses on two issues: asymptotic transitions between different sequences encoded by one, typically rational, multivariate generating function, and the use of Morse theory to provide strong structure results for possible asymptotic behaviour. These results can be combined with computer algebra software to provide rigorous asymptotic proofs, and present the most promising attack on the "connection problem" for so-called P-recursive sequences. Applications discussed include quantum computing, queuing theory, and automatic proofs of transcendence.