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

Things Fourier

For some reason several classes at MIT this year involve Fourier analysis. I was always confused about this as a high schooler, because no one ever gave me the ``orthonormal basis'' explanation, so here goes. As a bonus, I also prove a form of Arrow's Impossibility Theorem using binary Fourier analysis, and then talk about… Continue reading Things Fourier

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