Chen E. An Infinitely Large Napkin 2025

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Chen E. An Infinitely Large Napkin 2025

Textbook in PDF format From the Author Preface: "The origin of the name “Napkin” comes from the following quote of mine. 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 Grp, 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 was initially my attempt at those forty hours, but has grown considerably since then. About this book The Infinitely Large Napkin is a light but mostly self-contained introduction to a large amount of higher math. I should say at once that this book is not intended as a replacement for dedicated books or courses; the amount of depth is not comparable. On the flip side, the benefit of this “light” approach is that it becomes accessible to a larger audience, since the goal is merely to give the reader a feeling for any particular topic rather than to emulate a full semester of lectures. I initially wrote this book with talented high-school students in mind, particularly those with math-olympiad type backgrounds. Some remnants of that cultural bias can still be felt throughout the book, particularly in assorted challenge problems which are taken from mathematical competitions. However, in general I think this would be a good reference for anyone with some amount of mathematical maturity and curiosity. Examples include but certainly not limited to: math undergraduate majors, physics/CS majors, math PhD students who want to hear a little bit about fields other than their own, advanced high schoolers who like math but not math contests, and unusually intelligent kittens fluent in English. Source code The project is hosted on GitHub at https://github.com/vEnhance/napkin. Pull requests are quite welcome! I am also happy to receive suggestions and corrections by email. Philosophy behind the Napkin approach As far as I can tell, higher math for high-school students comes in two flavors: Someone tells you about the hairy ball theorem in the form “you can’t comb the hair on a spherical cat” then doesn’t tell you anything about why it should be true, what it means to actually “comb the hair”, or any of the underlying theory, leaving you with just some vague notion in your head. You take a class and prove every result in full detail, and at some point you stop caring about what the professor is saying. Presumably you already know how unsatisfying the first approach is. So the second approach seems to be the default, but I really think there should be some sort of middle ground here. Unlike university, it is not the purpose of this book to train you to solve exercises or write proofs,1 or prepare you for research in the field. Instead I just want to show you some interesting math. The things that are presented should be memorable and worth caring about. For that reason, proofs that would be included for completeness in any ordinary textbook are often omitted here, unless there is some idea in the proof which I think is worth seeing. In particular, I place a strong emphasis over explaining why a theorem should be true rather than writing down its proof. This is a recurrent theme of this book: Natural explanations supersede proofs. My hope is that after reading any particular chapter in Napkin, one might get the following out of it: Knowing what the precise definitions are of the main characters, Being acquainted with the few really major examples, Knowing the precise statements of famous theorems, and having a sense of why they should be true. Understanding “why” something is true can have many forms. This is sometimes accomplished with a complete rigorous proof; in other cases, it is given by the idea of the proof; in still other cases, it is just a few key examples with extensive cheerleading. Obviously this is nowhere near enough if you want to e.g. do research in a field; but if you are just a curious outsider, I hope that it’s more satisfying than the elevator pitch or Wikipedia articles. Even if you do want to learn a topic with serious depth, I hope that it can be a good zoomed-out overview before you really dive in, because in many senses the choice of material is “what I wish someone had told me before I started”. Preface Advice for the reader Starting Out Basic Abstract Algebra Basic Topology Linear Algebra More on Groups Representation Theory Quantum Algorithms Calculus 101 Complex Analysis Measure Theory Probability (TO DO) Differential Geometry Riemann Surfaces Algebraic NT I: Rings of Integers Algebraic NT II: Galois and Ramification Theory Algebraic Topology I: Homotopy Category Theory Algebraic Topology II: Homology Algebraic Geometry II: Affine Schemes Set Theory I: ZFC, Ordinals, and Cardinals Set Theory II: Model Theory and Forcing Appendix