Universal Phenomena in Analytic Combinatorics

Our modern world is full of discrete mathematical structures, such as large data structures and computer networks in computer science or phylogenetic trees in biology. In many applications we are interested in their typical behavior as they become larger and larger. Experiments are no longer sufficient for this analysis, as the number of possible configurations in many problems quickly exceeds the number of atoms in the observable universe. For this reason, we need a sound mathematical understanding of these, but in many cases we cannot even determine their number. The aim of this project is to solve these challenges in recursive counting problems. Recursive means that large structures can be built up from smaller ones through relatively simple operations. This enables the underlying structure to be translated into equations and recursions of number sequences, which in turn can be solved using methods from combinatorics, complex analysis, and probability theory. The innovative aspect of our approach is the underlying interdisciplinarity on several levels: We first combine methods from several mathematical fields to study problems in various areas such as computer science, biology, mathematics, and physics, using several different combinatorial models such as lattice paths, graphs, and Young tableaux. We focus on three types of phenomena: 1) analytic phenomena, such as novel asymptotic terms (e.g., stretched exponentials a^(n^s) where n tends to infinity), 2) probabilistic phenomena, such as new probability distributions, and 3) combinatorial phenomena, such as new classes of combinatorial objects to better describe applications.