I am a second year mathematics graduate student at Columbia University, advised by Johan de Jong. Geometry fascinates me, with algebraic methods my preferred means of exploration.
This is a draft of an exposition of Kollár's proof of the projectivity of the moduli space of stable curves. This was started at the Stacks Project Workshop during August of 2017 together with Yordanka Kovacheva, Carl Lian, Monica Marinescu, and Takumi Murayama, with guidance from Davesh Maulik. The source code for this project is available here.
This is an expository note on the proof due to Gross–Hacking–Keel–Kontsevich (see this) of the positiviy of the Laurent Phenomenon conjecture for cluster algebras. This was written for a course taught by Lauren Williams at Columbia in the fall of 2016.