In this talk, I will review my academic journey over the past 31 years. In particular, I will examine the domain decomposition method, which has played a central role in my work, as well as its applications in image processing and recent artificial neural networks. I will also discuss what constitutes good research, the aspects I find regrettable, and my future plans.

The classical Simpson correspondence describes complex linear representations of the fundamental group of a smooth complex projective variety in terms of linear algebraic objects, namely Higgs bundles. Inspired by this, Gerd Faltings initiated in 2005 a p-adic analogue, with the aim of understanding continuous p-adic representations of the geometric fundamental group of a smooth projective variety over a p-adic local field. The goal is to present recent joint work with M. Gros and T. Tsuji on the functoriality of the p-adic Simpson correspondence. We introduce a new method for twisting Higgs modules via Higgs-Tate algebras. This construction builds on our earlier joint work with M. Gros on the p-adic Simpson correspondence, which it recovers as a special case. The resulting framework provides twisted pullbacks and higher direct images of Higgs modules, making it possible to study the functoriality of the p-adic Simpson correspondence under arbitrary pullbacks and proper (log-)smooth direct images, including for morphisms that do not lift to the infinitesimal deformations used in the construction of the correspondence. In the colloquium lecture, we will give a brief introduction to the correspondence, starting from its p-adic origin, namely Sen theory. I will present the general ideas, while discussing the constructions and results only in the local setting. The more technical second lecture on Friday will present our global approach to the p-adic Simpson correspondence, twisted pullbacks, and higher direct images of Higgs modules, together with the corresponding functoriality results. I will also illustrate our approach with a new construction of the Sen endomorphism.