Tutte Colloquium - Steve Melczer

Abstract:

Although sequences satisfying linear recurrence relations have been studied for centuries, and appear as some of the first examples of combinatorial sequences encountered in an introductory combinatorics class, there are natural examples of simply stated problems related to their basic behaviour whose decidability is unknown. In this talk we survey some open computability and complexity questions related to the positivity of linearly recurrent sequences, before examining a new approach to proving positivity using rigorous numerical methods for functions satisfying linear differential equations. As a consequence, we give the first proof (to our knowledge) of solution uniqueness in genus one for an influential model of Canham which predicts the shape of biomembranes. Joint work with Marc Mezzarobba.