Mathematics

I'm trying to review the mathematics necessary to understand the models of climate change, and various other physics. Although my focus is "applied", my fascination lies more towards the "pure".

Pure mathematics is the art of the provable, but applied mathematics is the description of what happens. These heuristics illustrate the gulf between these realms. The domain of a theorem is bounded by extremes, even if unlikely. Heuristics are descriptions of what is probable, not the full range of what is possible.

— John P. Boyd, "The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series" §3

• Analysis (which I broadly define as subjects related to doing calculus in more general settings)
  • Real Numbers
  • Infinity and concerns with numerical analysis
  • Calculus (differential and integral)
  • Complex Analysis
  • Asymptotic Analysis for describing the behavior of functions and solutions to differential equations
  • Partial Differential Equations
• Algebra (which I broadly define as the study of…algebraic stuff…yes, "Algebra is what algebraists do", that's it.)
  • Rings and things
  • Differential Fields after a review of polynomials
  • Heegner Numbers, like (exp (* pi (sqrt 163))), which are near-integers
  • Category Theory (topics)
  • Group Theory (topics) studies symmetries
  • Lie Groups and Algebras
• Appendix: Foundations
  • Proof Steps in structured proofs ("formal proof sketches")
  • Set Theory
  • Writing Mathematics