Topoi and Internal Logic in Algebraic Geometry
A final project I wrote in my senior year as an undergrad for my algebraic geometry class (Math 256A at UC Berkeley). The first half of the paper is devoted to establishing the foundations of topoi and their internal logic (Kripke-Joyal semantics, although apparently I transposed the names throughout the paper). The latter half explores some immediate applications of the theory of topoi to prove a few simple results about schemes and their modules. One can take this much further (see this paper), this project was more an opportunity for me to learn more about topos theory and its applications outside Abstract Nonsense™.