Blog Posts:

Most of these posts were originally written as Twitter threads, they have been adapted somewhat but any strangeness in pacing is likely because of the original format.

An Introduction to Manifolds

There tends to be a lot of confusion about what exactly a manifold is on a first, second, or even third encounter with them. This post attempts to give an intuitive understanding what we we mean when we say manifold by viewing them as a generalization of surfaces. At the end we take a quick look at the classification of surfaces and some analogous questions for manifolds.This is intended to be for a completely general audience, but be ready to stop at think about some concepts, and don't be discouraged if some ideas are challenging the first time.

A Closer Look at Projective Space

Once again, for a general audience, as this one is pretty geometric. We define the real projective plane as the set of lines through the origin, but what does this actually mean? We look at why one would care about something like this, and try to intuitively construct some projective spaces.

Introduction to Morse Theory

This post largely failed to do what I wanted it to. The intention was to make a general introduction to Morse theory but it realistically needs an intuitive idea of a good bit of topology.

Maths is largely the study of analysis, geometry, and algebra. Morse theory says analysis and geometry should be done together. Analysis is the study of functions, and geometry is the study of shapes. Shapes for us means smooth manifolds, and functions are smooth real valued. We want to study manifolds by looking at the possible functions on it, so what's the goal? We'd like a way to reconstruct a manifold given a function on it. Turns out this is actually very doable.

Introduction to Homotopy theory.

This post aims to explain how topologists tell spaces apart to a general audience. In topology shapes are like clay: we can bend and stretch them as much as we want, but not cut or tear them. A natural question is how do we tell spaces apart in topology if things are so flexible? We introduce the most common way to do this.

Smooth Vs Topological Manifolds

This post aims to explain how topologists tell spaces apart to a general audience. In topology shapes are like clay: we can bend and stretch them as much as we want, but not cut or tear them. A natural question is how do we tell spaces apart in topology if things are so flexible? We introduce the most common way to do this.

Book and Resource Recommendations

Some books, videos, and course notes I'd recommend. Updated occasionally.

Accessibility:

I try to keep my notes and articles from being too visually cluttered or from relying on many distinct colours. I am working to make anything I produce as accessible as possible, as such please let me know if there is anything I can do to make these resources more accessible.