In this post we'll make sense of a holomorphic square root and logarithm. Wrote this up because I was surprised how hard it was to find a decent complete explanation. Let $latex {f : U \rightarrow \mathbb C}&fg=000000$ be a holomorphic function. A holomorphic $latex {n}&fg=000000$th root of $latex {f}&fg=000000$ is a function $latex {g… Continue reading Holomorphic Logarithms and Roots

# Category: Topology

## Facts about Lie Groups and Algebras

In Spring 2016 I was taking 18.757 Representations of Lie Algebras. Since I knew next to nothing about either Lie groups or algebras, I was forced to quickly learn about their basic facts and properties. These are the notes that I wrote up accordingly. Proofs of most of these facts can be found in standard… Continue reading Facts about Lie Groups and Algebras

## Algebraic Topology Functors

This will be old news to anyone who does algebraic topology, but oddly enough I can't seem to find it all written in one place anywhere, and in particular I can't find the bit about $latex {\mathsf{hPairTop}}&fg=000000$ at all. In algebraic topology you (for example) associate every topological space $latex {X}&fg=000000$ with a group, like… Continue reading Algebraic Topology Functors

## Uniqueness of Solutions for DiffEq’s

Let $latex {V}&fg=000000$ be a normed finite-dimensional real vector space and let $latex {U \subseteq V}&fg=000000$ be an open set. A vector field on $latex {U}&fg=000000$ is a function $latex {\xi : U \rightarrow V}&fg=000000$. (In the words of Gaitsgory: ``you should imagine a vector field as a domain, and at every point there is… Continue reading Uniqueness of Solutions for DiffEq’s

## Constructing the Tangent and Cotangent Space

This one confused me for a long time, so I figured I should write this down before I forgot again. Let $latex {M}&fg=000000$ be an abstract smooth manifold. We want to define the notion of a tangent vector to $latex {M}&fg=000000$ at a point $latex {p \in M}&fg=000000$. With that, we can define the tangent… Continue reading Constructing the Tangent and Cotangent Space