# The Napkin Project

**For most recent draft, click here to download.**

## Project Status

See link above for the most recent draft, and here for an archive of all drafts. I would very highly appreciate any corrections, suggestions, or comments.

Most of what remains to do is fill in the rest of the exercise problems, and slowly let all the typos get sent in to me.

(There is also a live Dropbox link which updates in real-time as I make changes. Feel free to email me to acquire this link.)

## 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.

#### I. Basic Algebra and Topology

- Chapter 1. What is a Group?
- Chapter 2. What is a Space?
- Chapter 3. Homomorphisms and Quotient Groups
- Chapter 4. Topological Notions
- Chapter 5. Compactness

#### II. Linear Algebra and Multivariable Calculus

- Chapter 6. What is a Vector Space?
- Chapter 7. Trace and Determinant
- Chapter 8. Spectral Theory
- Chapter 9. Inner Product Spaces

#### III. Groups, Rings, and More

- Chapter 10. Group Actions Overkill AIME Problems
- Chapter 11. Find All Groups
- Chapter 12. Rings and Ideals
- Chapter 13. The PID Structure Theorem

#### IV. Complex Analysis

- Chapter 14. Holomorphic Functions
- Chapter 15. Meromorphic Functions
- Chapter 16. Holomorphic Square Roots and Logarithms

#### V. Quantum Algorithms

- Chapter 17. Quantum states and measurements
- Chapter 18. Quantum circuits
- Chapter 19. Shor's algorithm

#### VI. Algebraic Topology I: Homotopy

- Chapter 20. Some Topological Constructions
- Chapter 21. Fundamental Groups
- Chapter 22. Covering Projections

#### VII. Category Theory

- Chapter 23. Objects and Morphisms
- Chapter 24. Functors and Natural Transformations
- Chapter 25. Abelian Categories

#### VIII. Differential Geometry

- Chapter 26. Multivariable Calculus Done Correctly
- Chapter 27. Differential Forms
- Chapter 28. Integrating Differential Forms
- Chapter 29. A Bit of Manifolds

#### IX. Algebraic Topology II: Homology

- Chapter 30. Singular Homology
- Chapter 31. The Long Exact Sequence
- Chapter 32. More on Computing Homology Groups
- Chapter 33. Bonus: Cellular Homology
- Chapter 34. Singular Cohomology
- Chapter 35. Applications of Cohomology

#### X. Algebraic NT I: Rings of Integers

- Chapter 36. Algebraic Integers
- Chapter 37. Unique Factorization (Finally!)
- Chapter 38. Minkowski Bound and Class Groups
- Chapter 39. More Properties of the Discriminant
- Chapter 40. Bonus: Let's Solve Pell's Equation!

#### XI. Algebraic NT II: Galois and Ramification Theory

- Chapter 41. Things Galois
- Chapter 42. Finite Fields
- Chapter 43. Ramification Theory
- Chapter 44. The Frobenius Endomorphism
- Chapter 45. A Bit on Artin Reciprocity

#### XII. Representation Theory

- Chapter 46. Representations of Algebras
- Chapter 47. Semisimple Algebras
- Chapter 48. Characters
- Chapter 49. Some Applications

#### XIII. Algebraic Geometry I: Varieties

- Chapter 50. Affine Varieties
- Chapter 51. Affine Varieties as Ringed Spaces
- Chapter 52. Projective Varieties
- Chapter 53. Bonus: Bezout's Theorem

#### XIV. Algebraic Geometry II: Schemes

- Chapter 54. Morphisms of varieties
- Chapter 55. Sheaves and Ringed Spaces
- Chapter 56. Schemes

#### XV. Set Theory I: ZFC, Ordinals, and Cardinals

- Chapter 57. Bonus: Zorn's Lemma and Cauchy's Functional Equation
- Chapter 58. Zermelo-Frankel with Choice
- Chapter 59. Ordinals
- Chapter 60. Cardinals

#### XVI. Set Theory II: Model Theory and Forcing

- Chapter 61. Inner Model Theory
- Chapter 62. Forcing
- Chapter 63. Breaking the Continuum Hypothesis