Reintroduction

I haven’t taken any formal education in mathematics and with more experience, I’ve come to realise, it’s the best thing you can do for your brain. Math is beautiful.

Initially, want to complete at least undergraduate level courses.

Questions

• Through calculus describe, what a continuous function is?
  • a function $f$ is said to be continuous at a point $x_0$, if $\forall ϵ>0,\exists \delta >0$, such that if $|x-x_0|<\delta ⟹|f(x)-f(x_0)|<ϵ$.
• Fundamental theorem of Calculus: $F(b)-F(a)={\int }_a^b{F}^{\prime }(x)dx$ where $F(x)$ has a continuous derivative.
  • But sometimes we need a weaker condition to prove the theorem, so we assume a simpler condition, and give it a name to “make the proof work”.

  "If we don’t know how to prove the theorem we want to, we’ll often ask, “What extra condition could we assume that would make it possible to prove this theorem?” And then we assume that condition holds, and often give it a name like “tame” or “well-behaved.” The conditions aren’t special or elegant–but they work." - Where do axioms come from

• How did Riemann come up with the definition of shapes, and manifolds?¹

Improvement

• Write my own notation OR explain concepts formally on my own
• Solve more problems without looking at notes.
• Start making small hypothesis about next chapters/subtopics inside the chapters.
• Active learning than writing notes. Less and less notes from now on.

TODOs

• learn about matrix algebra, tensor algebra
• eigenvector, eigenspace
• Abstract algebra from dummit and foote
• Solving mathematical problems
• How to write proofs
• Then it’s either Calculus, or real analysis
• Cauchy sequences
• Mathematical logic
  • https://web2.qatar.cmu.edu/cs/15317/
• Algebraic geometry
  • abstract algebra
  • number theory
  • algebraic topology
• Geometric algebra: prerequisites
  • Geometry
  • Topology
  • algebraic topology
  • Index—The Stacks project
• Analysis
  • Real
  • Complex
    • Complex Analysis
    • What exactly is i? (And other weird questions about complex numbers)
  • Functional
  • Differential equations
  • Measure theory
  • Numerical analysis
• what are continuous functions?

Linear Algebra

Vector spaces

• start:
• definition and properties
• Subspaces
  • Algebraic relation with calculus is really interesting. Example: All continuous real valued functions are subspace of ${\mathbf{R}}^{\mathbf{R}}$. Similarly, all differentiable real-valued functions are also subspace of ${\mathbf{R}}^{\mathbf{R}}$.
  • What are the operations possible on subspaces? Can I add them?
  • How to find out all possible subspaces of a vector space?
  • I’ve seen math’s obsession with subsets of a current structure. How are these utilised in real-world applications. I mean, are there examples where a subspace of a vector space is needed to solve a problem? or can you model real world problems into mathematical ones for these subsets?
• complete: 31-05-25 Finite-dimensional vector spaces:
• started: 31-05-25
• Bases: vectors $v_i$ in a base of a vector space $V$ are linearly independent. Are all the elements derived using linear combination of the base unique?

Resources:

• Napkin Math - Evan Chen
• Mathematics for the adventurous self-learner
• How to Become a Pure Mathematician (or Statistician)
• Mathacademy
• Art of Problem solving
• Math notes
• OSSU math
• MIT syllabus
  • MIT catalog
  • MIT roadmap
• Caltech syllabus
  • Caltech catalog
• Oxford mathematics hub
• UCL maths modules
• Yale Mathematics major
• How to Learn Math and Physics - John Baez
• rossant/awesome-math
• Paul’s online math notes
• Darij Grinberg: Mathematical Problem Solving (Math 235), Fall 2020: Initial undergraduate course on problem solving with amazing putnam problem and notes

Footnotes

1. https://infinityplusonemath.wordpress.com/2017/02/18/asteroids-on-a-donut/ ↩