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An Infinitely Large Napkinhttp://web.evanchen.cc/napkin.html

Evan Chen

Version: v1.5.20190220

http://web.evanchen.cc/napkin.html

When introduced to a new idea, always ask why you should care.

Do not expect an answer right away, but demand one eventually.

— Ravi Vakil [Va17]

If you like this book and want to support me,please consider buying me a coffee!

http://ko-fi.com/evanchen/

For Brian and Lisa, who finally got me to write it.

© 2019 Evan Chen. All rights reserved. Personal use only.This is (still!) an incomplete draft. Please send corrections, comments, pictures of kittens, etc.

to [email protected].

Last updated February 20, 2019.

http://ko-fi.com/evanchenhttp://ko-fi.com/evanchen/mailto:[email protected]

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 beenlearning category theory. They’ll ask me what category theory is about. I tell themit’s about abstracting things by looking at just the structure-preserving morphismsbetween them, rather than the objects themselves. I’ll try to give them the standardexample 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 rememberthat they haven’t learned what a group is. So then I’ll start trying to explain what agroup is, but by the time I finish writing the group axioms on my napkin, they’vealready forgotten why I was talking about groups in the first place. And then it’s1PM, 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 explainedthis all”.

This book was initially my attempt at those forty hours, but has grown considerablysince then.

About this book

The Infinitely Large Napkin is a light but mostly self-contained introduction to a largeamount of higher math.

I should say at once that this book is not intended as a replacement for dedicatedbooks or courses; the amount of depth is not comparable. On the flip side, the benefit ofthis “light” approach is that it becomes accessible to a larger audience, since the goal ismerely to give the reader a feeling for any particular topic rather than to emulate a fullsemester of lectures.

I initially wrote this book with talented high-school students in mind, particularlythose with math-olympiad type backgrounds. Some remnants of that cultural bias canstill be felt throughout the book, particularly in assorted challenge problems which aretaken from mathematical competitions. However, in general I think this would be agood reference for anyone with some amount of mathematical maturity and curiosity.Examples include but certainly not limited to: math undergraduate majors, physics/CSmajors, math PhD students who want to hear a little bit about fields other than theirown, high school students who like math but not math contests, and unusually intelligentkittens fluent in English.

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 thehair 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, leavingyou with just some vague notion in your head.

• You take a class and prove every result in full detail, and at some point you stopcaring about what the professor is saying.

v

vi Napkin, by Evan Chen (v1.5.20190220)

Presumably you already know how unsatisfying the first approach is. So the secondapproach seems to be the default, but I really think there should be some sort of middleground here.

Unlike university, it is not the purpose of this book to train you to solve exercises orwrite proofs1, or prepare you for research in the field. Instead I just want to show yousome interesting math. The things that are presented should be memorable and worthcaring about. For that reason, proofs that would be included for completeness in anyordinary textbook are often omitted here, unless there is some idea in the proof whichI think is worth seeing. In particular, I place a strong emphasis over explaining why atheorem should be true rather than writing down its proof. This is a recurrent theme ofthis book:

Natural explanations supersede proofs.

My hope is that after reading any particular chapter in Napkin, one might get thefollowing 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 whythey should be true.

Understanding “why” something is true can have many forms. This is sometimesaccomplished with a complete rigorous proof; in other cases, it is given by the idea of theproof; 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 ifyou are just a curious outsider, I hope that it’s more satisfying than the elevator pitch orWikipedia articles. Even if you do want to learn a topic with serious depth, I hope thatit can be a good zoomed-out overview before you really dive in, because in many sensesthe choice of material is “what I wish someone had told me before I started”.

More pedagogical comments and references

The preface would become too long if I talked about some of my pedagogical decisionschapter by chapter, so Appendix A contains those comments instead.

In particular, I often name specific references, and then end of that appendix has morereferences. So this is a good place to look if you want further reading.

Historical and personal notes

I began writing this book in December of 2014, after having finished my first semester ofundergraduate at Harvard. It became my main focus for about 18 months after that,as I became immersed in higher math. I essentially took only math classes, (gleefullyignoring all my other graduation requirements) and merged as much of it as I could (aswell as lots of other math I learned on my own time) into the Napkin.

Towards August of 2016, though, I finally lost steam. The first public drafts wentonline then, and I decided to step back. Having burnt out slightly, I then took a break

1Which is not to say problem-solving isn’t valuable; I myself am a math olympiad coach at heart. It’sjust not the point of this book.

Preface vii

from higher math, and spent the remaining two undergraduate years2 working extensivelyas a coach for the American math olympiad team, and trying to spend as much timewith my friends as I could before they graduated and went their own ways.

During those two years, readers sent me many kind words of gratitude, many reportsof errors, and many suggestions for additions. So in November of 2018, some weeks intomy first semester as a math PhD student, I decided I should finish what I had started.Some months later, here is what I have.

Source code

The project is hosted on GitHub at https://github.com/vEnhance/napkin.

Acknowledgementsadd moreacknowledg-ments

I am indebted to countless people for this work. Here is a partial (surely incomplete)list.

• Thanks to all my teachers and professors for teaching me much of the materialcovered in these notes, as well as the authors of all the references I have cited here.A special call-out to [Ga14], [Le14], [Sj05], [Ga03], [Ll15], [Et11], [Ko14], [Va17], [Pu02],[Go18], which were especially influential.

• Thanks also to dozens of friends and strangers who read through preview copies ofmy draft, and pointed out errors and gave other suggestions. Special mention toAndrej Vuković and Alexander Chua for together catching over a thousand errors.Thanks also to Brian Gu and Tom Tseng for many corrections.

• I’d also like to express my gratitude for many, many kind words I received duringthe development of this project. These generous comments led me to keep workingon this, and were largely responsible for my decision in November 2018 to beginupdating the Napkin again.

Finally, a huge thanks to the math olympiad community, from which the Napkin(and me) has its roots. All the enthusiasm, encouragement, and thank-you notes I havereceived over the years led me to begin writing this in the first place. I otherwise wouldnever have the arrogance to dream a project like this was at all possible. And of course Iwould be nowhere near where I am today were it not for the life-changing journey I tookin chasing my dreams to the IMO. Forever TWN2!

2Alternatively: “ . . . and spent the next two years forgetting everything I had painstakingly learned”.Which made me grateful for all the past notes in the Napkin!

https://github.com/vEnhance/napkin

Advice for the reader

§1 Prerequisites

As explained in the preface, the main prerequisite is some amount of mathematicalmaturity. This means I expect the reader to know how to read and write a proof, followlogical arguments, and so on.

I also assume the reader is familiar with basic terminology about sets and functions(e.g. “what is a bijection?”). If not, one should consult Appendix E.

§2 Deciding what to read

There is no need to read this book in linear order: it covers all sorts of areas in mathematics,and there are many paths you can take. In Chapter 0, I give a short overview for eachpart explaining what you might expect to see in that part.

For now, here is a brief chart showing how the chapters depend on each other; againsee Chapter 0 for details. Dependencies are indicated by arrows; dotted lines are optionaldependencies. I suggest that you simply pick a chapter you find interesting,and then find the shortest path. With that in mind, I hope the length of the entirePDF is not intimidating.

Ch 1,3-5

Abs AlgCh 2,6-8

TopologyCh 9-15,18

Lin Alg

Ch 16

Grp Act

Ch 17

Grp Classif

Ch 19-22

Rep Th

Ch 23-25

Quantum

Ch 26-30

Calc

Ch 31-33

Cmplx Ana

Ch 34-41

Measure/Pr

Ch 42-45

Diff Geo

Ch 46-50

Alg NT 1

Ch 51-55

Alg NT 2

Ch 56-58

Alg Top 1

Ch 59-62

Cat Th

Ch 63-68

Alg Top 2

Ch 69-73

Alg Geo 1

Ch 74-79

Alg Geo 2-3

Ch 80-86

Set Theory

ix

x Napkin, by Evan Chen (v1.5.20190220)

§3 Questions, exercises, and problems

In this book, there are three hierarchies:

• An inline question is intended to be offensively easy, mostly a chance to help youinternalize definitions. If you find yourself unable to answer one or two of them, itprobably means I explained it badly and you should complain to me. But if youcan’t answer many, you likely missed something important: read back.

• An inline exercise is more meaty than a question, but shouldn’t have any “tricky”steps. Often I leave proofs of theorems and propositions as exercises if they areinstructive and at least somewhat interesting.

• Each chapter features several trickier problems at the end. Some are reasonable,but others are legitimately difficult olympiad-style problems. Harder problems are

marked with up to three chili peppers ( ), like this paragraph.

In addition to difficulty annotations, the problems are also marked by how importantthey are to the big picture.

– Normal problems, which are hopefully fun but non-central.

– Daggered problems, which are (usually interesting) results that one shouldknow, but won’t be used directly later.

– Starred problems, which are results which will be used later on in the book.1

Several hints and solutions can be found in Appendices B and C.

Image from [Go08]

1This is to avoid the classic “we are done by PSet 4, Problem 8” that happens in college sometimes, asif I remembered what that was.

Advice for the reader xi

§4 Paper

At the risk of being blunt,

Read this book with pencil and paper.

Here’s why:

Image from [Or]

You are not God. You cannot keep everything in your head.2 If you’ve printed out ahard copy, then write in the margins. If you’re trying to save paper, grab a notebook orsomething along with the ride. Somehow, some way, make sure you can write. Thanks.

§5 On the importance of examples

I am pathologically obsessed with examples. In this book, I place all examples in largeboxes to draw emphasis to them, which leads to some pages of the book simply consistingof sequences of boxes one after another. I hope the reader doesn’t mind.

I also often highlight a “prototypical example” for some sections, and reserve the colorred for such a note. The philosophy is that any time the reader sees a definition or atheorem about such an object, they should test it against the prototypical example. Ifthe example is a good prototype, it should be immediately clear why this definition isintuitive, or why the theorem should be true, or why the theorem is interesting, et cetera.

Let me tell you a secret. Whenever I wrote a definition or a theorem in this book, Iwould have to recall the exact statement from my (quite poor) memory. So instead Ioften consider the prototypical example, and then only after that do I remember whatthe definition or the theorem is. Incidentally, this is also how I learned all the definitionsin the first place. I hope you’ll find it useful as well.

2 See also https://usamo.wordpress.com/2015/03/14/writing/ and the source above.

https://usamo.wordpress.com/2015/03/14/writing/

xii Napkin, by Evan Chen (v1.5.20190220)

§6 Conventions and notations

This part describes some of the less familiar notations and definitions and settles foronce and for all some annoying issues (“is zero a natural number?”). Most of these are“remarks for experts”: if something doesn’t make sense, you probably don’t have to worryabout it for now.

A full glossary of notation used can be found in the appendix.

§6.i Sets and equivalence relations

This is brief, intended as a reminder for experts. Consult Appendix E for full details.

An equivalence relation on a set X is a relation ∼ which is symmetric, reflexive, andtransitive. An equivalence relation partitions X into several equivalence classes. Wewill denote this by X/∼. An element of such an equivalence class is a representativeof that equivalence class.

I always use ∼= for an “isomorphism”-style relation (formally: a relation which is anisomorphism in a reasonable category). The only time ' is used in the Napkin is forhomotopic paths.

I generally use ⊆ and ( since these are non-ambiguous, unlike ⊂. I only use ⊂ onrare occasions in which equality obviously does not hold yet pointing it out would bedistracting. For example, I write Q ⊂ R since “Q ( R” is distracting.

I prefer S \ T to S − T .The power set of S (i.e., the set of subsets of S), is denoted either by 2S or P(S).

§6.ii Functions

This is brief, intended as a reminder for experts. Consult Appendix E for full details.

Let Xf−→ Y be a function:

• By fpre(T ) I mean the pre-image

fpre(T ) := {x ∈ X | f(x) ∈ T} .

This is in contrast to the f−1(T ) used in the rest of the world; I only use f−1 foran inverse function.

By abuse of notation, we may abbreviate fpre({y}) to fpre(y). We call fpre(y) afiber.

• By f img(S) I mean the image

f img(S) := {f(x) | x ∈ S} .

Almost everyone else in the world uses f(S) (though f [S] sees some use, and f“(S)is often used in logic) but this is abuse of notation, and I prefer f img(S) for emphasis.This image notation is not standard.

• If S ⊆ X, then the restriction of f to S is denoted f�S , i.e. it is the functionf�S : S → Y .

• Sometimes functions f : X → Y are injective or surjective; I may emphasize thissometimes by writing f : X ↪→ Y or f : X � Y , respectively.

Advice for the reader xiii

§6.iii Rings

All rings have a multiplicative identity 1 unless otherwise specified. We allow 0 = 1 ingeneral rings but not in integral domains.

All rings are commutative unless otherwise specified. There is an elaboratescheme for naming rings which are not commutative, used only in the chapter oncohomology rings:

Graded Not Graded1 not required graded pseudo-ring pseudo-ring

Anticommutative, 1 not required anticommutative pseudo-ring N/AHas 1 graded ring N/A

Anticommutative with 1 anticommutative ring N/ACommutative with 1 commutative graded ring ring

On the other hand, an algebra always has 1, but it need not be commutative.

§6.iv Natural numbers are positive

The set N is the set of positive integers, not including 0. In the set theory chapters, weuse ω = {0, 1, . . . } instead, for consistency with the rest of the book.

§6.v Choice

We accept the Axiom of Choice, and use it freely.

§7 Further reading

The appendix Appendix A contains a list of resources I like, and explanations of peda-gogical choices that I made for each chapter. I encourage you to check it out.

In particular, this is where you should go for further reading! There are some topicsthat should be covered in the Napkin, but are not, due to my own ignorance or laziness.The references provided in this appendix should hopefully help partially atone for myomissions.

Contents

Preface v

Advice for the reader ix

1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

2 Deciding what to read . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

3 Questions, exercises, and problems . . . . . . . . . . . . . . . . . . . . . x

4 Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

5 On the importance of examples . . . . . . . . . . . . . . . . . . . . . . . xi

6 Conventions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . xii

7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

I Starting Out 33

0 Sales pitches 35

0.1 The basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

0.2 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

0.3 Real and complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 37

0.4 Algebraic number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

0.5 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

0.6 Algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

0.7 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1 Groups 41

1.1 Definition and examples of groups . . . . . . . . . . . . . . . . . . . . . . 41

1.2 Properties of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.3 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

1.4 Orders of groups, and Lagrange’s theorem . . . . . . . . . . . . . . . . . 48

1.5 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.6 Groups of small orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.7 Unimportant long digression . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.8 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 51

2 Metric spaces 53

2.1 Definition and examples of metric spaces . . . . . . . . . . . . . . . . . . 53

2.2 Convergence in metric spaces . . . . . . . . . . . . . . . . . . . . . . . . 55

2.3 Continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.4 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.5 Extended example/definition: product metric . . . . . . . . . . . . . . . 58

2.6 Open sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.7 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61


15

16 Napkin, by Evan Chen (v1.5.20190220)

II Basic Abstract Algebra 65

3 Homomorphisms and quotient groups 673.1 Generators and group presentations . . . . . . . . . . . . . . . . . . . . . 67

3.2 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3 Cosets and modding out . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.4 (Optional) Proof of Lagrange’s theorem . . . . . . . . . . . . . . . . . . 73

3.5 Eliminating the homomorphism . . . . . . . . . . . . . . . . . . . . . . . 73

3.6 (Digression) The first isomorphism theorem . . . . . . . . . . . . . . . . 75


4 Rings and ideals 794.1 Some motivational metaphors about rings vs groups . . . . . . . . . . . 79

4.2 (Optional) Pedagogical notes on motivation . . . . . . . . . . . . . . . . 79

4.3 Definition and examples of rings . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.5 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.6 Generating ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.7 Principal ideal domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.8 Noetherian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87


5 Flavors of rings 915.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2 Integral domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3 Prime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.4 Maximal ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.5 Field of fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.6 Unique factorization domains (UFD’s) . . . . . . . . . . . . . . . . . . . 96


III Basic Topology 99

6 Properties of metric spaces 1016.1 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.3 Let the buyer beware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.4 Subspaces, and (inb4) a confusing linguistic point . . . . . . . . . . . . . 104


7 Topological spaces 1077.1 Forgetting the metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.2 Re-definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.3 Hausdorff spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.4 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.5 Connected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.6 Path-connected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.7 Homotopy and simply connected spaces . . . . . . . . . . . . . . . . . . 112

7.8 Bases of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114


Contents 17

8 Compactness 1178.1 Definition of sequential compactness . . . . . . . . . . . . . . . . . . . . 117

8.2 Criteria for compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8.3 Compactness using open covers . . . . . . . . . . . . . . . . . . . . . . . 119

8.4 Applications of compactness . . . . . . . . . . . . . . . . . . . . . . . . . 121

8.5 (Optional) Equivalence of formulations of compactness . . . . . . . . . . 123


IV Linear Algebra 127

9 Vector spaces 1319.1 The definitions of a ring and field . . . . . . . . . . . . . . . . . . . . . . 131

9.2 Modules and vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 131

9.3 Direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

9.4 Linear independence, spans, and basis . . . . . . . . . . . . . . . . . . . 135

9.5 Linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9.6 What is a matrix? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

9.7 Subspaces and picking convenient bases . . . . . . . . . . . . . . . . . . 140

9.8 A cute application: Lagrange interpolation . . . . . . . . . . . . . . . . . 142

9.9 (Digression) Arrays of numbers are evil . . . . . . . . . . . . . . . . . . . 143

9.10 A word on general modules . . . . . . . . . . . . . . . . . . . . . . . . . 144


10 Eigen-things 14710.1 Why you should care . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

10.2 Warning on assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

10.3 Eigenvectors and eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 148

10.4 The Jordan form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

10.5 Nilpotent maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

10.6 Reducing to the nilpotent case . . . . . . . . . . . . . . . . . . . . . . . . 152

10.7 (Optional) Proof of nilpotent Jordan . . . . . . . . . . . . . . . . . . . . 153

10.8 Algebraic and geometric multiplicity . . . . . . . . . . . . . . . . . . . . 154


11 Dual space and trace 15711.1 Tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

11.2 Dual space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

11.3 V ∨ ⊗W gives matrices from V to W . . . . . . . . . . . . . . . . . . . . 16111.4 The trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163


12 Determinant 16512.1 Wedge product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

12.2 The determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

12.3 Characteristic polynomials, and Cayley-Hamilton . . . . . . . . . . . . . 169


13 Inner product spaces 17313.1 The inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

13.2 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176


13.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17713.4 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17813.5 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 180

14 Bonus: Fourier analysis 18114.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18114.2 A reminder on Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . 18114.3 Common examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18214.4 Summary, and another teaser . . . . . . . . . . . . . . . . . . . . . . . . 18614.5 Parseval and friends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18614.6 Application: Basel problem . . . . . . . . . . . . . . . . . . . . . . . . . 18714.7 Application: Arrow’s Impossibility Theorem . . . . . . . . . . . . . . . . 18814.8 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 190

15 Duals, adjoint, and transposes 19115.1 Dual of a map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19115.2 Identifying with the dual space . . . . . . . . . . . . . . . . . . . . . . . 19215.3 The adjoint (conjugate transpose) . . . . . . . . . . . . . . . . . . . . . . 19315.4 Eigenvalues of normal maps . . . . . . . . . . . . . . . . . . . . . . . . . 19515.5 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 196

V More on Groups 199

16 Group actions overkill AIME problems 20116.1 Definition of a group action . . . . . . . . . . . . . . . . . . . . . . . . . 20116.2 Stabilizers and orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20216.3 Burnside’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20316.4 Conjugation of elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 20416.5 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 205

17 Find all groups 20717.1 Sylow theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20717.2 (Optional) Proving Sylow’s theorem . . . . . . . . . . . . . . . . . . . . 20817.3 (Optional) Simple groups and Jordan-Hölder . . . . . . . . . . . . . . . . 21017.4 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 211

18 The PID structure theorem 21318.1 Finitely generated abelian groups . . . . . . . . . . . . . . . . . . . . . . 21318.2 Some ring theory prerequisites . . . . . . . . . . . . . . . . . . . . . . . . 21418.3 The structure theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21518.4 Reduction to maps of free R-modules . . . . . . . . . . . . . . . . . . . . 21618.5 Smith normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21718.6 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 219

VI Representation Theory 221

19 Representations of algebras 22319.1 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22319.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22419.3 Direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

Contents 19

19.4 Irreducible and indecomposable representations . . . . . . . . . . . . . . 22719.5 Morphisms of representations . . . . . . . . . . . . . . . . . . . . . . . . 22819.6 The representations of Matd(k) . . . . . . . . . . . . . . . . . . . . . . . 23019.7 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 231

20 Semisimple algebras 23320.1 Schur’s lemma continued . . . . . . . . . . . . . . . . . . . . . . . . . . . 23320.2 Density theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23420.3 Semisimple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23620.4 Maschke’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23720.5 Example: the representations of C[S3] . . . . . . . . . . . . . . . . . . . 23720.6 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 238

21 Characters 24121.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24121.2 The dual space modulo the commutator . . . . . . . . . . . . . . . . . . 24221.3 Orthogonality of characters . . . . . . . . . . . . . . . . . . . . . . . . . 24321.4 Examples of character tables . . . . . . . . . . . . . . . . . . . . . . . . . 24521.5 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 247

22 Some applications 24922.1 Frobenius divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24922.2 Burnside’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25022.3 Frobenius determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

VII Quantum Algorithms 253

23 Quantum states and measurements 25523.1 Bra-ket notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25523.2 The state space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25623.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25623.4 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25923.5 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 262

24 Quantum circuits 26324.1 Classical logic gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26324.2 Reversible classical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 26424.3 Quantum logic gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26624.4 Deutsch-Jozsa algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 26824.5 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 269

25 Shor’s algorithm 27125.1 The classical (inverse) Fourier transform . . . . . . . . . . . . . . . . . . 27125.2 The quantum Fourier transform . . . . . . . . . . . . . . . . . . . . . . . 27225.3 Shor’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

VIII Calculus 101 277

26 Limits and series 27926.1 Completeness and inf/sup . . . . . . . . . . . . . . . . . . . . . . . . . . 279


26.2 Proofs of the two key completeness properties of R . . . . . . . . . . . . 28026.3 Monotonic sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

26.4 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

26.5 Series addition is not commutative: a horror story . . . . . . . . . . . . 286

26.6 Limits of functions at points . . . . . . . . . . . . . . . . . . . . . . . . . 287

26.7 Limits of functions at infinity . . . . . . . . . . . . . . . . . . . . . . . . 289


27 Bonus: A hint of p-adic numbers 29127.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

27.2 Algebraic perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

27.3 Analytic perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

27.4 Mahler coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299


28 Differentiation 30328.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

28.2 How to compute them . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

28.3 Local (and global) maximums . . . . . . . . . . . . . . . . . . . . . . . . 307

28.4 Rolle and friends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

28.5 Smooth functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312


29 Power series and Taylor series 31529.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

29.2 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

29.3 Differentiating them . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

29.4 Analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

29.5 A definition of Euler’s constant and exponentiation . . . . . . . . . . . . 320

29.6 This all works over complex numbers as well, except also complex analysisis heaven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321


30 Riemann integrals 32330.1 Uniform continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

30.2 Dense sets and extension . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

30.3 Defining the Riemann integral . . . . . . . . . . . . . . . . . . . . . . . . 325

30.4 Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327


IX Complex Analysis 331

31 Holomorphic functions 33331.1 The nicest functions on earth . . . . . . . . . . . . . . . . . . . . . . . . 333

31.2 Complex differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

31.3 Contour integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

31.4 Cauchy-Goursat theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

31.5 Cauchy’s integral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 339

31.6 Holomorphic functions are analytic . . . . . . . . . . . . . . . . . . . . . 341


Contents 21

32 Meromorphic functions 34532.1 The second nicest functions on earth . . . . . . . . . . . . . . . . . . . . 345

32.2 Meromorphic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

32.3 Winding numbers and the residue theorem . . . . . . . . . . . . . . . . . 348

32.4 Argument principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

32.5 Philosophy: why are holomorphic functions so nice? . . . . . . . . . . . . 351


33 Holomorphic square roots and logarithms 35333.1 Motivation: square root of a complex number . . . . . . . . . . . . . . . 353

33.2 Square roots of holomorphic functions . . . . . . . . . . . . . . . . . . . 355

33.3 Covering projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

33.4 Complex logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

33.5 Some special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357


X Measure Theory 359

34 Measure spaces 36134.1 Motivating measure spaces via random variables . . . . . . . . . . . . . . 361

34.2 Motivating measure spaces geometrically . . . . . . . . . . . . . . . . . . 362

34.3 σ-algebras and measurable spaces . . . . . . . . . . . . . . . . . . . . . . 363

34.4 Measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

34.5 A hint of Banach-Tarski . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

34.6 Measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

34.7 On the word “almost” (TO DO) . . . . . . . . . . . . . . . . . . . . . . 366


35 Constructing the Borel and Lebesgue measure 36735.1 Pre-measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

35.2 Outer measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

35.3 Carathéodory extension for outer measures . . . . . . . . . . . . . . . . . 370

35.4 Defining the Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . 372

35.5 A fourth row: Carathéodory for pre-measures . . . . . . . . . . . . . . . 374

35.6 From now on, we assume the Borel measure . . . . . . . . . . . . . . . . 375


36 Lebesgue integration 37736.1 The definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

36.2 Relation to Riemann integrals (or: actually computing Lebesgue integrals)379


37 Swapping order with Lebesgue integrals 38137.1 Motivating limit interchange . . . . . . . . . . . . . . . . . . . . . . . . . 381

37.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

37.3 Fatou’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

37.4 Everything else . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

37.5 Fubini and Tonelli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384



38 Bonus: A hint of Pontryagin duality 38538.1 LCA groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

38.2 The Pontryagin dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

38.3 The orthonormal basis in the compact case . . . . . . . . . . . . . . . . 387

38.4 The Fourier transform of the non-compact case . . . . . . . . . . . . . . 388

38.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388


XI Probability (TO DO) 391

39 Random variables (TO DO) 39339.1 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

39.2 Distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

39.3 Examples of random variables . . . . . . . . . . . . . . . . . . . . . . . . 394

39.4 Characteristic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

39.5 Independent random variables . . . . . . . . . . . . . . . . . . . . . . . . 394


40 Large number laws (TO DO) 39540.1 Notions of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395


41 Stopped martingales (TO DO) 397

XII Differential Geometry 399

42 Multivariable calculus done correctly 40142.1 The total derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

42.2 The projection principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

42.3 Total and partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 404

42.4 (Optional) A word on higher derivatives . . . . . . . . . . . . . . . . . . 406

42.5 Towards differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . 407


43 Differential forms 40943.1 Pictures of differential forms . . . . . . . . . . . . . . . . . . . . . . . . . 409

43.2 Pictures of exterior derivatives . . . . . . . . . . . . . . . . . . . . . . . . 411

43.3 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

43.4 Exterior derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

43.5 Closed and exact forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415


44 Integrating differential forms 41744.1 Motivation: line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

44.2 Pullbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

44.3 Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

44.4 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

44.5 Stokes’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423


Contents 23

45 A bit of manifolds 42545.1 Topological manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

45.2 Smooth manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

45.3 Regular value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

45.4 Differential forms on manifolds . . . . . . . . . . . . . . . . . . . . . . . 428

45.5 Orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

45.6 Stokes’ theorem for manifolds . . . . . . . . . . . . . . . . . . . . . . . . 430

45.7 (Optional) The tangent and contangent space . . . . . . . . . . . . . . . 430


XIII Algebraic NT I: Rings of Integers 435

46 Algebraic integers 43746.1 Motivation from high school algebra . . . . . . . . . . . . . . . . . . . . 437

46.2 Algebraic numbers and algebraic integers . . . . . . . . . . . . . . . . . . 438

46.3 Number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

46.4 Norms and traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

46.5 The ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

46.6 Primitive element theorem, and monogenic extensions . . . . . . . . . . 447


47 Unique factorization (finally!) 44947.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

47.2 Ideal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

47.3 Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

47.4 Unique factorization works . . . . . . . . . . . . . . . . . . . . . . . . . . 452

47.5 The factoring algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

47.6 Fractional ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

47.7 The ideal norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457


48 Minkowski bound and class groups 45948.1 The class group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

48.2 The discriminant of a number field . . . . . . . . . . . . . . . . . . . . . 459

48.3 The signature of a number field . . . . . . . . . . . . . . . . . . . . . . . 462

48.4 Minkowski’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

48.5 The trap box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

48.6 The Minkowski bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

48.7 The class group is finite . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

48.8 Computation of class numbers . . . . . . . . . . . . . . . . . . . . . . . . 468


49 More properties of the discriminant 47349.1 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 473

50 Bonus: Let’s solve Pell’s equation! 47550.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

50.2 Dirichlet’s unit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

50.3 Finding fundamental units . . . . . . . . . . . . . . . . . . . . . . . . . . 477

50.4 Pell’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478



XIV Algebraic NT II: Galois and Ramification Theory 481

51 Things Galois 483

51.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

51.2 Field extensions, algebraic closures, and splitting fields . . . . . . . . . . 484

51.3 Embeddings into algebraic closures for number fields . . . . . . . . . . . 485

51.4 Everyone hates characteristic 2: separable vs irreducible . . . . . . . . . 486

51.5 Automorphism groups and Galois extensions . . . . . . . . . . . . . . . . 488

51.6 Fundamental theorem of Galois theory . . . . . . . . . . . . . . . . . . . 491


51.8 (Optional) Proof that Galois extensions are splitting . . . . . . . . . . . 493

52 Finite fields 495

52.1 Example of a finite field . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

52.2 Finite fields have prime power order . . . . . . . . . . . . . . . . . . . . 496

52.3 All finite fields are isomorphic . . . . . . . . . . . . . . . . . . . . . . . . 497

52.4 The Galois theory of finite fields . . . . . . . . . . . . . . . . . . . . . . . 498


53 Ramification theory 501

53.1 Ramified / inert / split primes . . . . . . . . . . . . . . . . . . . . . . . . 501

53.2 Primes ramify if and only if they divide ∆K . . . . . . . . . . . . . . . . 502

53.3 Inertial degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

53.4 The magic of Galois extensions . . . . . . . . . . . . . . . . . . . . . . . 503

53.5 (Optional) Decomposition and inertia groups . . . . . . . . . . . . . . . 505

53.6 Tangential remark: more general Galois extensions . . . . . . . . . . . . 507


54 The Frobenius element 509

54.1 Frobenius elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

54.2 Conjugacy classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

54.3 Chebotarev density theorem . . . . . . . . . . . . . . . . . . . . . . . . . 512

54.4 Example: Frobenius elements of cyclotomic fields . . . . . . . . . . . . . 512

54.5 Frobenius elements behave well with restriction . . . . . . . . . . . . . . 513

54.6 Application: Quadratic reciprocity . . . . . . . . . . . . . . . . . . . . . 514

54.7 Frobenius elements control factorization . . . . . . . . . . . . . . . . . . 516

54.8 Example application: IMO 2003 problem 6 . . . . . . . . . . . . . . . . . 519


55 Bonus: A Bit on Artin Reciprocity 521

55.1 Infinite primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

55.2 Modular arithmetic with infinite primes . . . . . . . . . . . . . . . . . . 521

55.3 Infinite primes in extensions . . . . . . . . . . . . . . . . . . . . . . . . . 523

55.4 Frobenius element and Artin symbol . . . . . . . . . . . . . . . . . . . . 524

55.5 Artin reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526


Contents 25

XV Algebraic Topology I: Homotopy 531

56 Some topological constructions 53356.1 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

56.2 Quotient topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

56.3 Product topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534

56.4 Disjoint union and wedge sum . . . . . . . . . . . . . . . . . . . . . . . . 535

56.5 CW complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

56.6 The torus, Klein bottle, RPn, CPn . . . . . . . . . . . . . . . . . . . . . 53756.7 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 543

57 Fundamental groups 54557.1 Fusing paths together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

57.2 Fundamental groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

57.3 Fundamental groups are functorial . . . . . . . . . . . . . . . . . . . . . 550

57.4 Higher homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 551

57.5 Homotopy equivalent spaces . . . . . . . . . . . . . . . . . . . . . . . . . 552

57.6 The pointed homotopy category . . . . . . . . . . . . . . . . . . . . . . . 554


58 Covering projections 55758.1 Even coverings and covering projections . . . . . . . . . . . . . . . . . . 557

58.2 Lifting theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

58.3 Lifting correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

58.4 Regular coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562

58.5 The algebra of fundamental groups . . . . . . . . . . . . . . . . . . . . . 564


XVI Category Theory 567

59 Objects and morphisms 56959.1 Motivation: isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 569

59.2 Categories, and examples thereof . . . . . . . . . . . . . . . . . . . . . . 569

59.3 Special objects in categories . . . . . . . . . . . . . . . . . . . . . . . . . 573

59.4 Binary products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574

59.5 Monic and epic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578


60 Functors and natural transformations 58160.1 Many examples of functors . . . . . . . . . . . . . . . . . . . . . . . . . . 581

60.2 Covariant functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

60.3 Contravariant functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584

60.4 Equivalence of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

60.5 (Optional) Natural transformations . . . . . . . . . . . . . . . . . . . . . 586

60.6 (Optional) The Yoneda lemma . . . . . . . . . . . . . . . . . . . . . . . . 588


61 Limits in categories (TO DO) 59161.1 Equalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

61.2 Pullback squares (TO DO) . . . . . . . . . . . . . . . . . . . . . . . . . . 592


61.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59261.4 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 592

62 Abelian categories 59362.1 Zero objects, kernels, cokernels, and images . . . . . . . . . . . . . . . . 59362.2 Additive and abelian categories . . . . . . . . . . . . . . . . . . . . . . . 59462.3 Exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59662.4 The Freyd-Mitchell embedding theorem . . . . . . . . . . . . . . . . . . 59762.5 Breaking long exact sequences . . . . . . . . . . . . . . . . . . . . . . . . 59862.6 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 599

XVII Algebraic Topology II: Homology 601

63 Singular homology 60363.1 Simplices and boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 60363.2 The singular homology groups . . . . . . . . . . . . . . . . . . . . . . . . 60563.3 The homology functor and chain complexes . . . . . . . . . . . . . . . . 60863.4 More examples of chain complexes . . . . . . . . . . . . . . . . . . . . . 61263.5 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 613

64 The long exact sequence 61564.1 Short exact sequences and four examples . . . . . . . . . . . . . . . . . . 61564.2 The long exact sequence of homology groups . . . . . . . . . . . . . . . . 61764.3 The Mayer-Vietoris sequence . . . . . . . . . . . . . . . . . . . . . . . . 61964.4 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 624

65 Excision and relative homology 62565.1 The long exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 62565.2 The category of pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62665.3 Excision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62765.4 Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62865.5 Invariance of dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62965.6 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 630

66 Bonus: Cellular homology 63166.1 Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63166.2 Cellular chain complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63266.3 The cellular boundary formula . . . . . . . . . . . . . . . . . . . . . . . . 63566.4 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 637

67 Singular cohomology 63967.1 Cochain complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63967.2 Cohomology of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64067.3 Cohomology of spaces is functorial . . . . . . . . . . . . . . . . . . . . . 64167.4 Universal coefficient theorem . . . . . . . . . . . . . . . . . . . . . . . . . 64267.5 Example computation of cohomology groups . . . . . . . . . . . . . . . . 64367.6 Relative cohomology groups . . . . . . . . . . . . . . . . . . . . . . . . . 64467.7 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 645

68 Application of cohomology 64768.1 Poincaré duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647

Contents 27

68.2 de Rham cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647

68.3 Graded rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648

68.4 Cup products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650

68.5 Relative cohomology pseudo-rings . . . . . . . . . . . . . . . . . . . . . . 652

68.6 Wedge sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652

68.7 Künneth formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654


XVIII Algebraic Geometry I: Classical Varieties 657

69 Affine varieties 65969.1 Affine varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659

69.2 Naming affine varieties via ideals . . . . . . . . . . . . . . . . . . . . . . 660

69.3 Radical ideals and Hilbert’s Nullstellensatz . . . . . . . . . . . . . . . . . 661

69.4 Pictures of varieties in A1 . . . . . . . . . . . . . . . . . . . . . . . . . . 66269.5 Prime ideals correspond to irreducible affine varieties . . . . . . . . . . . 663

69.6 Pictures in A2 and A3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66469.7 Maximal ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665

69.8 Motivating schemes with non-radical ideals . . . . . . . . . . . . . . . . . 666


70 Affine varieties as ringed spaces 66770.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667

70.2 The Zariski topology on An . . . . . . . . . . . . . . . . . . . . . . . . . 66770.3 The Zariski topology on affine varieties . . . . . . . . . . . . . . . . . . . 669

70.4 Coordinate rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670

70.5 The sheaf of regular functions . . . . . . . . . . . . . . . . . . . . . . . . 671

70.6 Regular functions on distinguished open sets . . . . . . . . . . . . . . . . 672

70.7 Baby ringed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673


71 Projective varieties 67571.1 Graded rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675

71.2 The ambient space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676

71.3 Homogeneous ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678

71.4 As ringed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679

71.5 Examples of regular functions . . . . . . . . . . . . . . . . . . . . . . . . 680


72 Bonus: Bézout’s theorem 68372.1 Non-radical ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

72.2 Hilbert functions of finitely many points . . . . . . . . . . . . . . . . . . 684

72.3 Hilbert polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686

72.4 Bézout’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688

72.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688


73 Morphisms of varieties 69173.1 Defining morphisms of baby ringed spaces . . . . . . . . . . . . . . . . . 691

73.2 Classifying the simplest examples . . . . . . . . . . . . . . . . . . . . . . 692


73.3 Some more applications and examples . . . . . . . . . . . . . . . . . . . 694

73.4 The hyperbola effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695


XIX Algebraic Geometry II: Affine Schemes 699

74 Sheaves and ringed spaces 70374.1 Motivation and warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . 703

74.2 Pre-sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703

74.3 Stalks and germs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705

74.4 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709

74.5 For sheaves, sections “are” sequences of germs . . . . . . . . . . . . . . . 710

74.6 Sheafification (optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . 712


75 Localization 71575.1 Spoilers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715

75.2 The definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716

75.3 Localization away from an element . . . . . . . . . . . . . . . . . . . . . 717

75.4 Localization at a prime ideal . . . . . . . . . . . . . . . . . . . . . . . . . 718

75.5 Prime ideals of localizations . . . . . . . . . . . . . . . . . . . . . . . . . 720

75.6 Prime ideals of quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . 721

75.7 Localization commute with quotients . . . . . . . . . . . . . . . . . . . . 721


76 Affine schemes: the Zariski topology 72576.1 Some more advertising . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725

76.2 The set of points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725

76.3 The Zariski topology on the spectrum . . . . . . . . . . . . . . . . . . . 727

76.4 On radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730


77 Affine schemes: the sheaf 73377.1 A useless definition of the structure sheaf . . . . . . . . . . . . . . . . . . 733

77.2 The value of distinguished open sets (or: how to actually compute sections)734

77.3 The stalks of the structure sheaf . . . . . . . . . . . . . . . . . . . . . . . 736

77.4 Local rings and residue fields: linking germs to values . . . . . . . . . . . 737

77.5 Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740

77.6 Functions are determined by germs, not values . . . . . . . . . . . . . . . 740


78 Interlude: nineteen examples of affine schemes 74378.1 Example: Spec k, a single point . . . . . . . . . . . . . . . . . . . . . . . 743

78.2 SpecC[x], a one-dimensional line . . . . . . . . . . . . . . . . . . . . . . 74378.3 SpecR[x], a one-dimensional line with complex conjugates glued (no fear

nullstellensatz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744

78.4 Spec k[x], over any ground field . . . . . . . . . . . . . . . . . . . . . . . 745

78.5 SpecZ, a one-dimensional scheme . . . . . . . . . . . . . . . . . . . . . . 74578.6 Spec k[x]/(x2 − x), two points . . . . . . . . . . . . . . . . . . . . . . . . 74678.7 Spec k[x]/(x2), the double point . . . . . . . . . . . . . . . . . . . . . . . 746

Contents 29

78.8 Spec k[x]/(x3 − x), a double point and a single point . . . . . . . . . . . 74778.9 SpecZ/60Z, a scheme with three points . . . . . . . . . . . . . . . . . . 74778.10 Spec k[x, y], the two-dimensional plane . . . . . . . . . . . . . . . . . . . 748

78.11 SpecZ[x], a two-dimensional scheme, and Mumford’s picture . . . . . . . 74978.12 Spec k[x, y]/(y − x2), the parabola . . . . . . . . . . . . . . . . . . . . . 75078.13 SpecZ[i], the Gaussian integers (one-dimensional) . . . . . . . . . . . . . 75178.14 Long example: Spec k[x, y]/(xy), two axes . . . . . . . . . . . . . . . . . 752

78.15 Spec k[x, x−1], the punctured line (or hyperbola) . . . . . . . . . . . . . 75478.16 Spec k[x](x), zooming in to the origin of the line . . . . . . . . . . . . . . 755

78.17 Spec k[x, y](x,y), zooming in to the origin of the plane . . . . . . . . . . . 756

78.18 Spec k[x, y](0) = Spec k(x, y), the stalk above the generic point . . . . . . 756


79 Morphisms of locally ringed spaces 75779.1 Morphisms of ringed spaces via sections . . . . . . . . . . . . . . . . . . 757

79.2 Morphisms of ringed spaces via stalks . . . . . . . . . . . . . . . . . . . . 758

79.3 Morphisms of locally ringed spaces . . . . . . . . . . . . . . . . . . . . . 759

79.4 A few examples of morphisms between affine schemes . . . . . . . . . . . 760

79.5 The big theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763

79.6 More examples of scheme morphisms . . . . . . . . . . . . . . . . . . . . 765

79.7 A little bit on non-affine schemes . . . . . . . . . . . . . . . . . . . . . . 766

79.8 Where to go from here . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768


XX Algebraic Geometry III: Schemes (TO DO) 769

XXI Set Theory I: ZFC, Ordinals, and Cardinals 771

80 Interlude: Cauchy’s functional equation and Zorn’s lemma 77380.1 Let’s construct a monster . . . . . . . . . . . . . . . . . . . . . . . . . . 773

80.2 Review of finite induction . . . . . . . . . . . . . . . . . . . . . . . . . . 774

80.3 Transfinite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774

80.4 Wrapping up functional equations . . . . . . . . . . . . . . . . . . . . . . 776

80.5 Zorn’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777


81 Zermelo-Fraenkel with choice 78181.1 The ultimate functional equation . . . . . . . . . . . . . . . . . . . . . . 781

81.2 Cantor’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781

81.3 The language of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . 782

81.4 The axioms of ZFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78381.5 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785

81.6 Choice and well-ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . 785

81.7 Sets vs classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786


82 Ordinals 78982.1 Counting for preschoolers . . . . . . . . . . . . . . . . . . . . . . . . . . 789

82.2 Counting for set theorists . . . . . . . . . . . . . . . . . . . . . . . . . . 790


82.3 Definition of an ordinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792

82.4 Ordinals are “tall” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793

82.5 Transfinite induction and recursion . . . . . . . . . . . . . . . . . . . . . 794

82.6 Ordinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

82.7 The hierarchy of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796


83 Cardinals 799

83.1 Equinumerous sets and cardinals . . . . . . . . . . . . . . . . . . . . . . 799

83.2 Cardinalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800

83.3 Aleph numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800

83.4 Cardinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801

83.5 Cardinal exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 803

83.6 Cofinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803

83.7 Inaccessible cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805


XXII Set Theory II: Model Theory and Forcing 807

84 Inner model theory 809

84.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809

84.2 Sentences and satisfaction . . . . . . . . . . . . . . . . . . . . . . . . . . 810

84.3 The Levy hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812

84.4 Substructures, and Tarski-Vaught . . . . . . . . . . . . . . . . . . . . . . 813

84.5 Obtaining the axioms of ZFC . . . . . . . . . . . . . . . . . . . . . . . . 814

84.6 Mostowski collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815

84.7 Adding an inaccessible . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815

84.8 FAQ’s on countable models . . . . . . . . . . . . . . . . . . . . . . . . . 817

84.9 Picturing inner models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818


85 Forcing 821

85.1 Setting up posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822

85.2 More properties of posets . . . . . . . . . . . . . . . . . . . . . . . . . . 823

85.3 Names, and the generic extension . . . . . . . . . . . . . . . . . . . . . . 824

85.4 Fundamental theorem of forcing . . . . . . . . . . . . . . . . . . . . . . . 827

85.5 (Optional) Defining the relation . . . . . . . . . . . . . . . . . . . . . . . 827

85.6 The remaining axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829


86 Breaking the continuum hypothesis 831

86.1 Adding in reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831

86.2 The countable chain condition . . . . . . . . . . . . . . . . . . . . . . . . 832

86.3 Preserving cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833

86.4 Infinite combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834


Contents 31

XXIII Backmatter 837

A Pedagogical comments and references 839A.1 Basic algebra and topology . . . . . . . . . . . . . . . . . . . . . . . . . . 839A.2 Second-year topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 840A.3 Advanced topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841A.4 Topics not in Napkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842

B Hints to selected problems 843

C Sketches of selected solutions 853

D Glossary of notations 875D.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875D.2 Functions and sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875D.3 Abstract and linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . 876D.4 Quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877D.5 Topology and real/complex analysis . . . . . . . . . . . . . . . . . . . . . 877D.6 Measure theory and probability . . . . . . . . . . . . . . . . . . . . . . . 878D.7 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878D.8 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879D.9 Differential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 880D.10 Algebraic number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 880D.11 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881D.12 Algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882D.13 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883

E Terminology on sets and functions 885E.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885E.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886E.3 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 888

IStarting Out

Part I: Contents

0 Sales pitches 350.1 The basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350.2 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360.3 Real and complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370.4 Algebraic number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380.5 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390.6 Algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390.7 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1 Groups 411.1 Definition and examples of groups . . . . . . . . . . . . . . . . . . . . . . . . . . 411.2 Properties of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.3 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461.4 Orders of groups, and Lagrange’s theorem . . . . . . . . . . . . . . . . . . . . . . 481.5 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491.6 Groups of small orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501.7 Unimportant long digression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511.8 A few harder problems to think about . . . . . . . . . . . . . . . . . . . . . . . . 51

2 Metric spaces 532.1 Definition and examples of metric spaces . . . . . . . . . . . . . . . . . . . . . . . 532.2 Convergence in metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.3 Continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.4 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.5 Extended example/definition: product metric . . . . . . . . . . . . . . . . . . . . . 582.6 Open sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.7 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.8 A few harder problems to think about . . . . . . . . . . . . . . . . . . . . . . . . 62

0 Sales pitchesThis chapter contains a pitch for each part, to help you decide what you want to read

and to elaborate more on how they are interconnected.

For convenience, here is again the dependency plot that appeared in the frontmatter.

Ch 1,3-5

Abs AlgCh 2,6-8

TopologyCh 9-15,18

Lin Alg

Ch 16

Grp Act

Ch 17

Grp Classif

Ch 19-22

Rep Th

Ch 23-25

Quantum

Ch 26-30

Calc

Ch 31-33

Cmplx Ana

Ch 34-41

Measure/Pr

Ch 42-45

Diff Geo

Ch 46-50

Alg NT 1

Ch 51-55

Alg NT 2

Ch 56-58

Alg Top 1

Ch 59-62

Cat Th

Ch 63-68

Alg Top 2

Ch 69-73

Alg Geo 1

Ch 74-79

Alg Geo 2-3

Ch 80-86

Set Theory

§0.1 The basics

I. Starting Out.

I made a design decision that the first part should have a little bit both of algebraand topology: so this first chapter begins by defining a group, while the secondchapter begins by defining a metric space. The intention is so that newcomersget to see two different examples of “sets with additional structure” in somewhatdifferent contexts, and to have a minimal amount of literacy as these sorts ofdefinitions appear over and over1

II. Basic Abstract Algebra.

The algebraically inclined can then delve into further types of algebraic structures:some more details of groups, and then rings and fields — which will let yougeneralize Z, Q, R, C. So you’ll learn to become familiar with all sorts of othernouns that appear in algebra, unlocking a whole host of objects that one couldn’ttalk about before.

1In particular, I think it’s easier to learn what a homeomorphism is after seeing group isomorphism,and what a homomorphism is after seeing continuous map.

35


We’ll also come into ideals, which generalize the GCD in Z that you might knowof. For example, you know in Z that any integer can be written in the form 3a+ 5bfor a, b ∈ Z, since gcd(3, 5) = 1. We’ll see that this statement is really a statementof ideals: “(3, 5) = 1 in Z”, and thus we’ll understand in what situations it can begeneralized, e.g. to polynomials.

III. Basic Topology.

The more analytically inclined can instead move into topology, learning more aboutspaces. We’ll find out that “metric spaces” are actually too specific, and that it’sbetter to work with topological spaces, which are based on the so-called opensets. You’ll then get to see the buddings of some geometrical ideals, ending withthe really great notion of compactness, a powerful notion that makes real analysistick.

One example of an application of compactness to tempt you now: a continuousfunction f : [0, 1]→ R always achieves a maximum value. (In contrast, f : (0, 1)→ Rby x 7→ 1/x does not.) We’ll see the reason is that [0, 1] is compact.

§0.2 Abstract algebra

IV. Linear Algebra.

In high school, linear algebra is often really unsatisfying. You are given these arraysof numbers, and they’re manipulated in some ways that don’t really make sense.For example, the determinant is defined as this funny-looking sum with a bunch ofproducts that seems to come out of thin air. Where does it come from? Why doesdet(AB) = detAdetB with such a bizarre formula?

Well, it turns out that you can explain all of these things! The trick is to not thinkof linear algebra as the study of matrices, but instead as the study of linear maps.In earlier chapters we saw that we got great generalizations by speaking of “setswith enriched structure” and “maps between them”. This time, our sets are vectorspace and our maps are linear maps. We’ll find out that a matrix is actuallyjust a way of writing down a linear map as an array of numbers, but using the“intrinsic” definitions we’ll de-mystify all the strange formulas from high school andshow you where they all come from.

In particular, we’ll see easy proofs that column rank equals row rank, determinantis multiplicative, trace is the sum of the diagonal entries, how the dot productworks, and all the words starting with “eigen-”. We’ll even have a bonus chapterfor Fourier analysis showing that you can also explain all the big buzz-words byjust being comfortable with vector spaces.

V. More on Groups.

Some of you might be interested in more about groups, and this chapter will giveyou a way to play further. It starts with an exploration of group actions, thengoes into a bit on Sylow theorems, which are the tools that let us try to classifyall groups.

VI. Representation Theory.

If G is a group, we can try to understand it by implementing it as a matrix, i.e.considering embeddings G ↪→ GLn(C). These are called representations of G; itturns out that they can be decomposed into irreducible ones. Astonishingly we

0 Sales pitches 37

will find that we can basically characterize all of them: the results turn out to beshort and completely unexpected.

For example, we will find out that there are finitely many irreducible representationsof a given finite group G; if we label them V1, V2, . . . , Vr, then we will find that ris the number of conjugacy classes of G, and moreover that

|G| = (dimV1)2 + · · ·+ (dimVr)2

which comes out of nowhere!

The last chapter of this part will show you some unexpected corollaries. Here isone of them: let G be a finite group and create variables xg for each g ∈ G. A|G| × |G| matrix M is defined by setting the (g, h)th entry to be the variable xg·h.Then this determinant will turn out to factor, and the factors will correspond tothe Vi we described above: there will be an irreducible factor of degree dimViappearing dimVi times. This result, called the Frobenius determinant, is saidto have given birth to representation theory.

VII. Quantum Algorithms.

If you ever wondered what Shor’s algorithm is, this chapter will use the built-uplinear algebra to tell you!

§0.3 Real and complex analysis

VIII. Calculus 101.

In this part, we’ll use our built-up knowledge of metric and topological spaces togive short, rigorous definitions and theorems typical of high school calculus. Thatis, we’ll really define and prove most everything you’ve seen about limits, series,derivatives, and integrals.

Although this might seem intimidating, it turns out that actually, by the time westart this chapter, the hard work has already been done: the notion of limits, opensets, and compactness will make short work of what was swept under the rug in APcalculus. Most of the proofs will thus actually be quite short, instead, we sit backand watch all the pieces slowly come together, the reward for our careful study oftopology beforehand.

That said, if you are willing to suspend belief, you can actually read most of theother parts without knowing the exact details of all the calculus here, so in somesense this part is “optional”.

IX. Complex Analysis.

It turns out that holomorphic functions (complex-differentiable functions) areclose to the nicest things ever: they turn out to be given by a Taylor series (i.e. arebasically polynomials). This means we’ll be able to prove unreasonably nice resultsabout holomorphic functions C→ C, like

– they are determined by just a few inputs,

– their contour integrals are all zero,

– they can’t be bounded unless they are constant,

– . . . .


We then introduce meromorphic functions, which are like quotients of holomor-phic functions, and find that we can detect their zeros by simply drawing loopsin the plane and integrating over them: the famous residue theorem appears.(In the practice problems, you will see this even gives us a way to evaluate realintegrals that can’t be evaluated otherwise.)

X. Measure Theory.

Measure theory is the upgraded version of integration. The Riemann integration isfor a lot of purposes not really sufficient; for example, if f is the function equalling 1at rational numbers but 0 at irrational numbers, we would hope that

∫ 10 f(x) dx = 0,

but the Riemann integral is not capable of handling this function f .

The Lebesgue integral will handle these mistakes by assigning a measure to ageneric space Ω, making it into a measure space. This will let us develop a richertheory of integration where the above integral does work out to zero because the“rational numbers have measure zero”. Even the development of the measure willbe an achievement, because it means we’ve developed a rigorous, complete way oftalking about what notions like area and volume mean — on any space, not justRn! So for example the Lebesgue integral will let us integrate functions over anymeasure space.

XI. Probability (TO DO).

Using the tools of measure theory, we’ll be able to start giving rigorous definitionsof probability, too. We’ll see that a random variable is actually a functionfrom a measure space of worlds to R, giving us a rigorous way to talk about itsprobabilities. We can then start actually stating results like the law of largenumbers and central limit theorem in ways that make them both easy to stateand straightforward to prove.

XII. Differential Geometry.

Multivariable calculus is often confusing because of all the partial derivatives. Butwe’ll find out that, armed with our good understanding of linear algebra, that we’rereally looking at a total derivative: at every point of a function f : Rn → R wecan associate a linear map Df which captures in one object the notion of partialderivatives. Set up this way, we’ll get to see versions of differential forms andStoke’s theorem, and we finally will know what the notation dx really means. Inthe end, we’ll say a little bit about manifolds in general.

§0.4 Algebraic number theory

XIII. Algebraic NT I: Rings of Integers.

Why is 3 +√

5 the conjugate of 3−√

5? How come the norm∥∥a+ b

√5∥∥ = a2− 5b2

used in Pell equations just happens to be multiplicative? Why is it we can dofactoring into primes in Z[i] but not in Z[

√−5]? All these questions and more will

be answered in this part, when we learn about number fields, a generalizationof Q and Z to things like Q(

√5) and Z[

√5]. We’ll find out that we have unique

factorization into prime ideals, that there is a real multiplicative norm in play here,and so on. We’ll also see that Pell’s equation falls out of this theory.

XIV. Algebraic NT II: Galois and Ramification Theory.

0 Sales pitches 39

All the big buzz-words come out now: Galois groups, the Frob

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