Here is a collection of various mathematical notes.

Perfectoid rings. A colletion of basic facts about perfectoid rings. Included is a general tilting equivalence, a proof of almost purity for fields and miscellany on completions.

Witt vectors from deformation theory. Finding addition and multiplication of Witt Vectors from deformation theory. Included is a recollection on deformation theory.

Six-functor formalisms. Lecture notes from Marc Hoyois course at the 2023 YTM. Updated from the YTM website version.

Moduli and automorphisms. Quick note to explain the saying that “automorphisms are obstruction to representability of moduli”.

Adjoint functor theorems. Notes on presentable categories and the adjoint functor theorem.