# Writings

Here, you can find many of my expository writings. Most are unpolished and not necessarily intended to be read; use with caution. I will happily welcome feedback/errata; feel free to send me an email.

What are Spectra? This is an ongoing project, meant essentially to be a notebook of my own while learning stable homotopy theory. It is eccentric, informal, ideosyncratic, and sometimes just weird; nonetheless, I am posting it here in the hopes that someone may find my ramblings useful.

The Leray-Serre Spectral Sequence. This was my final paper for Math 231a: Algebraic Topology, taught by Peter Kronheimer. It is a particularly short exposition of the Serre spectral sequence, culminating in a computation of the cohomology ring $$H^*(\mathbb{C}\mathbb{P}^n;\mathbb{Z})$$. Most of the exposition is lifted from McCleary's book A User's Guide to Spectral Sequences; in particular, I do not claim any originality of proofs.

Toward a Synthetic Theory of (oo, 1)-Categories, Part I: Preliminaries. This paper is my attempt to understand the Riehl-Verity definition of an "$$\infty$$-cosmos" as a category of $$(\infty, 1)$$-categories. I had once intended for it to be the first in a series of papers explaining the theory, but quickly became overwhelmed with unfamiliar concepts and notation. Nonetheless, this article-written as a midterm paper for Eric Peterson's Advanced Algebraic Topology course-serves to introduce the theory of monoidal categories, enriched categories, and model categories necessary to end with a definition of an $$\infty$$-cosmos.

Notes on the Adams Spectral Sequence. Very incomplete. These notes mark a final paper for Eric's class that I started and one day hope to finish; at the moment, they are full of inaccuracies and gaping holes. Perhaps these notes will flourish into notes on Doug Ravenel's Complex Cobordism and Stable Homotopy Groups of Spheres; perhaps they will remain unfinished.