After completing the first year of graduate school (post on the industry $\to$ grad school transition forthcoming), I’ve finally got time to reboot the blog, with more focus. I’ll be posting on a $\sim$weekly basis about the interesting topics I come across in the course of research, class, and general academic meandering. Renewed areas of focus will include various mathematical, physical, or programming topics, such as:
Unravelling bundles (as in fibre bundles) Wormhole solutions in GR and energy conditions Managing environments in Python and developer workflows Fermi Figures - playful posts about solving various back-of-the-envelope questions Enthralling Theorem - spotlight of an interesting theorem / mathematical fact Found this week - weekly interesting discoveries, clearly and heavily inspired by the famous “This week’s finds” - an asymptote I only hope to approach over time.

I first encountered a diagram of algebraic structures at the end of Jeevanjee’s second chapter, “Vector Spaces”, which elegantly summarizes the high-level differences in structure between sets, vector spaces, and inner product spaces. 1
This diagram was immensely helpful to me, in that it helped show the relationships between various commonly used objects in mathematical physics. As I’ve encountered new structures, I’ve attempted to augment this map along two dimensions: a structure dimension that aims to measure the number of attributes an algebraic object has, and a specificity dimension which measures the amount of constraints placed on each attribute.

Motivation Python has increased in popularity to near ubiquity in the past five years. While the Python community (correctly) professes simplicity as a major accomplishment of the language, I still get a lot of questions about how to get a python environment setup properly. There are some lengthy guides out there on this - this post will aim to summarize and explain the relevant components to getting started.
Note: skip to bottom if you want quick install commands

Summary I picked up a copy of Nadir Jeevnajee’s An Introduction to Tensors and Group Theory for Physicists a few months ago with the intent of skimming through and spending most of my time in reference texts. To my pleasant surprise, I found this text to be self contained - requiring little to no references. The presentation is at once mathematically rigorous and physically intuitive, alluding to well-known examples from physics throughout.

A friend, who focuses primarily on experimental particle physics, recently asked me an interesting question about gravity. Specifically, he asked how the presence of electromagnetic fields impacts the gravitational field. Applying some modern-physics reasoning, he proposed that electromagnetic fields should exert gravitational influence because photons have momentum that can be viewed as mass in special relativity, and should interact gravitationally. I found this idea interesting, if a bit interpretive, and answered with the precise formulation of the impact of the electromagnetic field on the curvature tensor.

I’ve spent time focusing on the best chalkboards and chalks on my tools page, but -until recently- I’ve not spent much time thinking about the last part of the process – erasing. At the suggestion of several colleagues, I played around with several different methods of erasing chalk marks to find which is most effective. The results were decisive. This post outlines the results, and attempts to present a simple test as justification (though my own testing was more extensive).