Wei Xiangsheng: An attempt to prove Berger's conjecture

Abstract: In $p$-adic Hodge theory, various rings have $\phi$-module structure induced by a proper substitution map. Sometimes it is important to calculate the fixed function(or even "eigenfunction") of the substitution map, thus Laurent Berger poses a conjecture, which states that, under some mild conditions, the substitution map of fraction field of Robba ring has no nontrivial fixed function. In this talk, I will explain the conjecture in detail starting with the basic definitions. Furthermore I shall give some of my first observations and present some ongoing possible ideas/progress.