The Napkin project (v1.5)

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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.

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
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