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Download#

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.

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