Some Expository Notes I Wrote (Very Rough):
A Short Introduction to Gröbner Bases – I like to think of Gröbner basis as the natural consequence of extending division to polynomials in many variables. They form a foundational component of Computational Algebraic Geometry, or how we teach computers to help us understand polynomials and systems of polynomials. I try to give a general overview of their theory and uses here.
Degree Theory: An Introduction and Applications – A famous theorem is that one cannot comb the hairs on a coconut to be completely flat (see here). However, we could comb the hairs on a doughnut to be flat. Why is this? What exactly makes this process impossible on a sphere, but possible on a doughnut? The answer to this question tells us something very deep about the relationship between topology and differential equations. I really enjoy this topic, although I admit the notes can be quite dense (especially if you don’t know your manifolds from many folds) and the style I chose is quite dense. To me, the most magical thing about this subject is how constrained you are. If you pick a few reasonable assumptions, you are led directly to a deep and unique construction.