# Tag Archives: Math

## Introduction to Ideal Class Groups

Algebraic number theory is a really interesting subject, but unlike some other subjects, it’s not 100% clear what objects people study. This post provides an introduction to the class group of a finite dimensional field extension of $\mathbb{Q}$, an object often … Continue reading

Posted in Math, Number Theory, Uncategorized | Tagged , | 1 Comment

Assume $\{F_x\}_{x \in \Gamma}$ is a collection of subsets (of a not-so important set!) such that every two are comparable, i.e for any $x$ and $y$, either $F_x \subset F_y \ \$ or $\ \ F_x \supset F_y \ … Continue reading Posted in Analysis, Math | | 2 Comments ## Gauge Theory and Low-Dimensional Topology (Part II: Smooth Four-Manifolds) In the last post, I attempted to give an overview of the state of affairs in four-manifold topology leading up to the introduction of gauge theory. In particular, we discussed the correspondence between (topological) four-manifolds and their intersection forms afforded by … Continue reading Posted in Math, Topology | Tagged , | 2 Comments ## A Pretty Lemma About Prime Ideals and Products of Ideals I was trying to prove a theorem in algebraic geometry which basically held if and only if this lemma held. Here’s the lemma: Lemma: Given any ring$A$, a prime ideal$ \mathfrak{p} \subset A\$, and a finite collection of ideals … Continue reading

Posted in Algebra, Algebraic Geometry, Math | Tagged , , | 3 Comments

## Gauge Theory and Low-Dimensional Topology (Part I: Historical Context)

Hi! This month, I thought I would start a brief series of articles describing the uses of gauge theory in mathematics. Rather than discuss current research directions in gauge theory (of which there are many), I hope to give an … Continue reading

Posted in Math, Topology | Tagged , | 1 Comment