Modul: MAT770 Oberseminar: Algebraische Geometrie

## A Galoisian approach for walks in the quarter plane

Dr. Charlotte Hardouin talk

Speaker invited by: Prof. Dr. Joseph Ayoub

**Date:** 04.03.19 **Time:** 13.15 - 14.45 **Room:** Y27H25

A walk in the quarter plane is a path between integral points with a prescribed set of steps that is confined in the quarter plane. In the recent years, the enumeration of such walks has attracted the attention of many authors in combinatorics and probability. The complexity of their enumeration is encoded in the algebraic nature of their associated generating series. The main question is: are these series algebraic, holonomic (solutions of linear differential equations) or differentially algebraic (solutions of algebraic differential equations)? In this talk, we will show how the algebraic nature of the generating series can be approached via the study of a discrete functional equation over a curve E, of genus zero or one and the Galois theory of difference equations. In the first case, the functional equation corresponds to a so called q-difference equation and the generating series is differentially transcendental. In the genus one case, the dynamic of the functional equation corresponds to the addition by a given point P of the elliptic curve E. If the point P is torsion then any generating series is holonomic. When P is non torsion, the nature of the series is captured by the orbit configuration with respect to the dynamic induced by P of a certain set of points of the curve. This work combines some collaborations with T. Dreyfus (Irma, Strasbourg), J. Roques (Institut Fourier, Grenoble) and M.F. Singer (NCSU, Raleigh).