Trees, functional inversion, and the virial expansion


Trees are ubiquitous. Probabilists may think of branching processes and ask about extinction or survival. The recursive structure of trees leads to functional equations for generating functions, of interest in analytic combinatorics. Trees also help organize power series expansions in various areas of analysis and mathematical physics, from numerics (Butcher trees) to renormalization (Gallavotti-Niccolo trees). The talk presents yet another application, namely inverse function theorems for functionals in measure spaces for which Banach inversion is not possible. Combined with cluster expansions from equilibrium statistical mechanics, the theorem allows for a rigorous derivation, in a restricted parameter regime, of density functionals used in analytic models of materials. The talk is based on joint work with Tobias Kuna and Dimitrios Tsagkarogiannis (arXiv:1906.02322 [math-ph]) and considerably improves earlier results based on Lagrange-Good inversion (J., Tate, Tsagkarogiannis, Ueltschi, CMP 2014).

Valeria Hünniger

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Francesca Arici

Radboud University Nijmegen

Tatjana Eisner

Leipzig University

Barbara Gentz

University of Bielefeld

Angkana Rüland

Max Planck Institute for Mathematics in the Sciences

Rebecca Waldecker

Martin-Luther-University Halle-Wittenberg

Milena Wrobel

Carl von Ossietzky Universität Oldenburg