1.5 Proofs of a Ridiculous Identity

Recently I posted the following Tweet, calling attention to the existence of a Ramanujan-like identity: https://twitter.com/likethebuilder/status/1382444319521509376 In fact, such an identity holds for any exponent which is 1 modulo 4. For example, for 13 and 17, we get (somewhat) simple fractions. https://twitter.com/likethebuilder/status/1382650197252145152 This series of identities begs a simple explanation. Here I present one-and-a-half proofs, …

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A Simple Derivation of the Electron Gyromagnetic Ratio

When I first began to learn quantum field theory, I was always left quite unsatisfied when I was told that the gyromagnetic ratio (the degree to which the electron spin couples to an external magnetic field) is (almost) $latex g=2&s=1$. The best justification that a young frustrated me could get from more advanced students or …

From Polygon to Ellipse: A Proof

Recently this tweet from Matt Henderson caught my eye. It demonstrates that, given a random collection of $latex N&s=1$ ordered points in the plane, if you repeatedly average neighboring points (neighboring with respect to the given order), then the limiting set of points lies on an ellipse (after appropriate rescaling). The limiting process from a …

Supersymmetric Localization: An Invitation

Disclaimer: This post is largely based on notes I took during David Skinner's "Advanced Quantum Field Theory" course in the spring of 2017. His lecture notes are available online here and are highly recommended as reading material. Introduction Supersymmetry has a different reputation depending on whom you ask. For some, it is a beautiful concept, …

Pour Me Some of the Good Stuff: A brief introduction to Moonshine

Mathematics is an absolutely enormous field, spanning thousands of years and millions of pages of "fundamental" reading for anyone wishing to master all of its major aspects. As such, it is convenient to divide math into several sub-fields (algebra, geometry, number theory, analysis, etc.). Most mathematicians, even the best ones, only work in one or …