Holden Swindell

Writing

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I have informally written on a number of topics in mathematics. You can find most of this writing below.

If you notice any errors or have any suggestions or ideas you want to share, you can contact me using the information here.

Research Project Participation

• "Approximating the Uniform Norm via Discrete Sampling" (REU 2023) (PDF)

Short Notes

Short notes on assorted topics, including proofs, observations, ideas for further theory, and resources for students — essentially mathematical versions of GitHub's gists.

• "A Proof of Induction" (PDF)
• "A Proof of the Fundamental Theorem of Calculus" (PDF)
• "Redundancy in the Axioms for a Topology" (PDF)
• "Using Connectedness to Show Constancy" (PDF)

Longer Works

• "What is the Volume of a Ball?" (in-progress)
  • The goal of this work is to rigorously state and prove the formulae for the volume of a hyperball and the surface area of a hypersphere. Along the way, we discuss measure theory from a primarily geometric perspective.
• "A General Theory of Generating Sets" (in-progress)
  • The concept of a generating set shows up frequently in mathematics in various guises: generating sets for groups, spanning sets for vector spaces, and subbases for topologies. We develop a general theory of generating a mathematical object with the help of category theory and universal properties.