Title: Algebraic de Rham Cohomology for Schemes
Abstract: Prismatic cohomology can be roughly regarded as a deformation of de Rham cohomology. Consequently, many structures on prismatic cohomology have counterparts for de Rham cohomology that are more classical and more explicit. Our goal of this talk is to understand Section 2.1 of Bhatt’s note, i.e. the notion of algebraic de Rham cohomology. We will first review the classical theory of de Rham cohomology for smooth manifolds and explain the issues arising when switching to complex manifolds and more general algebraic geometric objects like varieties. Then we will introduce the so-called algebraic de Rham cohomology for general schemes and explain why it is the correct cohomology we need. We will also go through the Hodge filtration structure as well as the conjugate filtration structure on algebraic de Rham cohomology, and how they interact with Poincare duality (if permitted). Examples will be given for a better understanding of the above notions.
Title: Derived Categories and t-structure
Abstract: In this talk, we will review derived categories and t-structure and show that the heart is abelian. If time permits, we will also introduce the stable infinity category, and see for any ring R, there exists a stable infinity category whose homotopy category is the classical derived category D(R).
Bhatt's note on primsatic F-gauge
Bhatt's yourtube video on geometrization
Alex Youcis' note on de Rham cohomology