The Napkin project (v1.5)


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For most recent draft, click here to download.

Project Status

The recent v1.5 is a new update which revises many of the earlier chapters as well as adding new content in real analysis, measure theory, and algebraic geometry. It is however even more visibly incomplete, with several chapters scheduled but not yet written. In addition, many chapters still lack problems or solutions.

See link above for the most recent draft, and here for an listing periodic snapshots. You can also view the source code on GitHub; the most recent version is automatically compiled from that source.

I would very highly appreciate any corrections, suggestions, or comments.

Description

The Napkin project is a personal exposition project of mine aimed at making higher math accessible to high school students. The philosophy is stated in the preamble:

I'll be eating a quick lunch with some friends of mine who are still in high school. They'll ask me what I've been up to the last few weeks, and I'll tell them that I've been learning category theory. They'll ask me what category theory is about. I tell them it's about abstracting things by looking at just the structure-preserving morphisms between them, rather than the objects themselves. I'll try to give them the standard example Gp, but then I'll realize that they don't know what a homomorphism is. So then I'll start trying to explain what a homomorphism is, but then I'll remember that they haven't learned what a group is. So then I'll start trying to explain what a group is, but by the time I finish writing the group axioms on my napkin, they've already forgotten why I was talking about groups in the first place. And then it's 1PM, people need to go places, and I can't help but think:

Man, if I had forty hours instead of forty minutes, I bet I could actually have explained this all.

This book is my attempt at those forty hours.

This project has evolved to more than just forty hours.

Current Table of Contents

  1. Groups
  2. Metric spaces
  3. Homomorphisms and quotient groups
  4. Rings and ideals
  5. Flavors of rings
  6. Properties of metric spaces
  7. Topological spaces
  8. Compactness
  9. Vector spaces
  10. Eigen-things
  11. Dual space and trace
  12. Determinant
  13. Inner product spaces
  14. Bonus: Fourier analysis
  15. Duals, adjoint, and transposes
  16. Group actions overkill AIME problems
  17. Find all groups
  18. The PID structure theorem
  19. Representations of algebras
  20. Semisimple algebras
  21. Characters
  22. Some applications
  23. Quantum states and measurements
  24. Quantum circuits
  25. Shor's algorithm
  26. Limits and series
  27. Bonus: A hint of p-adic numbers
  28. Differentiation
  29. Power series and Taylor series
  30. Riemann integrals
  31. Holomorphic functions
  32. Meromorphic functions
  33. Holomorphic square roots and logarithms
  34. Measure spaces
  35. Constructing the Borel and Lebesgue measure
  36. Lebesgue integration
  37. Swapping order with Lebesgue integrals
  38. Bonus: A hint of Pontryagin duality
  39. Random variables (TO DO)
  40. Large number laws (TO DO)
  41. Stopped martingales (TO DO)
  42. Multivariable calculus done correctly
  43. Differential forms
  44. Integrating differential forms
  45. A bit of manifolds
  46. Algebraic integers
  47. Unique factorization (finally!)
  48. Minkowski bound and class groups
  49. More properties of the discriminant
  50. Bonus: Let's solve Pell's equation!
  51. Things Galois
  52. Finite fields
  53. Ramification theory
  54. The Frobenius element
  55. Bonus: A Bit on Artin Reciprocity
  56. Some topological constructions
  57. Fundamental groups
  58. Covering projections
  59. Objects and morphisms
  60. Functors and natural transformations
  61. Limits in categories (TO DO)
  62. Abelian categories
  63. Singular homology
  64. The long exact sequence
  65. Excision and relative homology
  66. Bonus: Cellular homology
  67. Singular cohomology
  68. Application of cohomology
  69. Affine varieties
  70. Affine varieties as ringed spaces
  71. Projective varieties
  72. Bonus: B\'ezout's theorem
  73. Morphisms of varieties
  74. Sheaves and ringed spaces
  75. Localization
  76. Affine schemes: the Zariski topology
  77. Affine schemes: the sheaf
  78. Interlude: nineteen examples of affine schemes
  79. Morphisms of locally ringed spaces
  80. Interlude: Cauchy's functional equation and Zorn's lemma
  81. Zermelo-Fraenkel with choice
  82. Ordinals
  83. Cardinals
  84. Inner model theory
  85. Forcing
  86. Breaking the continuum hypothesis
Last modified: Sun Nov 24 23:36:26 2019.

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