This talk explores the intricate interplay between the intrinsic geometry of complex projective varieties and the homological invariants of their defining ideals. We focus on the structure of syzygies—encoding higher order algebraic relations among defining equations—and the Castelnuovo–Mumford regularity, a key invariant governing the complexity of embedded varieties. Finally, we survey classic and recent developments regarding linear syzygies, the status of the Eisenbud–Goto conjecture, and the higher syzygies of secant varieties.

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.