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The Napkin project (v1.5)

Flowchart for Napkin
(click to enlarge)


Download the most recent draft of Napkin.

Project Status#

The recent v1.5 is a new update which revises many of the earlier chapters and adds 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 a listing of periodic snapshots. You can also view the source code on GitHub; the most recent version is automatically compiled from that source.

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


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. Bonus: Topological Abel-Ruffini Theorem
  35. Measure spaces
  36. Constructing the Borel and Lebesgue measure
  37. Lebesgue integration
  38. Swapping order with Lebesgue integrals
  39. Bonus: A hint of Pontryagin duality
  40. Random variables (TO DO)
  41. Large number laws (TO DO)
  42. Stopped martingales (TO DO)
  43. Multivariable calculus done correctly
  44. Differential forms
  45. Integrating differential forms
  46. A bit of manifolds
  47. Algebraic integers
  48. Unique factorization (finally!)
  49. Minkowski bound and class groups
  50. More properties of the discriminant
  51. Bonus: Let’s solve Pell’s equation!
  52. Things Galois
  53. Finite fields
  54. Ramification theory
  55. The Frobenius element
  56. Bonus: A Bit on Artin Reciprocity
  57. Some topological constructions
  58. Fundamental groups
  59. Covering projections
  60. Objects and morphisms
  61. Functors and natural transformations
  62. Limits in categories (TO DO)
  63. Abelian categories
  64. Singular homology
  65. The long exact sequence
  66. Excision and relative homology
  67. Bonus: Cellular homology
  68. Singular cohomology
  69. Application of cohomology
  70. Affine varieties
  71. Affine varieties as ringed spaces
  72. Projective varieties
  73. Bonus: B'ezout’s theorem
  74. Morphisms of varieties
  75. Sheaves and ringed spaces
  76. Localization
  77. Affine schemes: the Zariski topology
  78. Affine schemes: the sheaf
  79. Interlude: nineteen examples of affine schemes
  80. Morphisms of locally ringed spaces
  81. Interlude: Cauchy’s functional equation and Zorn’s lemma
  82. Zermelo-Fraenkel with choice
  83. Ordinals
  84. Cardinals
  85. Inner model theory
  86. Forcing
  87. Breaking the continuum hypothesis
Updated Fri 19 Jan 2024, 22:04:39 UTC by 2b78c4f2c42c