Holomorphic Logarithms and Roots

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

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