De Rham's Theorem
December 2023
De Rham's theorem provides an important connection between topology and smooth manifolds---that for manifolds, the de Rham cohomology is exactly the (real) singular cohomology, and the isomorphism is given by integration. Read moreCovering Spaces
August 2022
Covering spaces are a cool thing in topology. Finding covering spaces is like unwrapping a topological space. There are intimate connections between the fundamental group and covering spaces. Read moreNullstellensatz
July 2022
Hilbert's Nullstellensatz is an important theorem in providing a bridge between algebra and geometry. This article walks through topics in commutative algebra before arriving at a proof of the theorem. Read moreSylow Theorems
December 2021
The Sylow Theorems are fundamental to the study of finite groups. We build up to their proof by (1) studying group actions, (2) focusing on existence, and finally (3) studying conjugation of Sylow subgroups. Read moreProducts and Coproducts
December 2021
We get our feet wet in category theory by going through the universal properties of products and coproducts. We'll see some examples and prove uniqueness of products. Then, we prove uniqueness of coproducts by duality. Read more