I've spent a lot of my time learning about math, especially abstract math. Some of my topics of interest are:
Here are some things I've thought up in the wondrous world of mathematics.
A few years back (late 2021), I explored a simple extension to the Lotka-Volterra Equations to model population density distributions over a given space. I recently typset my old work into a PDF and polished it up a bit.
This is an exploration of a relation on subsets of a topological space that I feel provides a more intuitive understanding of what certain topological ideas mean. The relation attempts to codify the idea that one subset of a topological space is "attached" or "adjacent" to another, although its asymmetry is a bit strange. The original intent was to eventually produce a definition of topological spaces in terms of this relation, but I began to lose interest, especially after I realized that it would essentially just be the Kuratowski closure axioms.
Recently, I've been exploring what I call "Geodesic Spaces." Given a space where geodesics can be defined (currently only considering manifolds with a connection), the Geodesic Space is a topological space in which each point corresponds to a given geodesic, irrespective of reparamaterizations. The initial, motivating example, is the geodesic space of Euclidean 2-space (a plane), which I believe to be homeomorphic to the Möbius strip. I'm currently working on collecting my thoughts on this topic and typesetting them.
Here's a list of various mathematics-related resources and things that I like.