An In nitely Large Napkin - venhance.github.io · updating the Napkin again. Finally, a huge thanks...

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

Evan Chen

Version: v1.5.20190813

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], or pull-request at https://github.com/vEnhance/napkin.

Last updated August 13, 2019.

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

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 (v1.5.20190813)

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. Pullrequests are quite welcome! I am also happy to receive suggestions and corrections byemail.

Acknowledgements

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

add moreacknowledg-ments

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 copiesof my draft, and pointed out errors and gave other suggestions. Special mentionto Andrej Vuković and Alexander Chua for together catching over a thousanderrors. Thanks also to Brian Gu and Tom Tseng for many corrections. (If you findmistakes or have suggestions yourself, I would love to hear them!)

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 46-51

Alg NT 1

Ch 52-56

Alg NT 2

Ch 57-59

Alg Top 1

Ch 60-63

Cat Th

Ch 64-69

Alg Top 2

Ch 70-74

Alg Geo 1

Ch 75-80

Alg Geo 2-3

Ch 81-87

Set Theory

ix

x Napkin, by Evan Chen (v1.5.20190813)

§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 (v1.5.20190813)

§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 (v1.5.20190813)

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 425

45.1 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 437

46.1 Motivation from high school algebra . . . . . . . . . . . . . . . . . . . . 437

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

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

46.4 Primitive element theorem, and monogenic extensions . . . . . . . . . . 440


47 The ring of integers 443

47.1 Norms and traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

47.2 The ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

47.3 On monogenic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 450


48 Unique factorization (finally!) 451

48.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

48.2 Ideal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

48.3 Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

48.4 Unique factorization works . . . . . . . . . . . . . . . . . . . . . . . . . . 454

48.5 The factoring algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

48.6 Fractional ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

48.7 The ideal norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459


49 Minkowski bound and class groups 461

49.1 The class group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

49.2 The discriminant of a number field . . . . . . . . . . . . . . . . . . . . . 461

49.3 The signature of a number field . . . . . . . . . . . . . . . . . . . . . . . 464

49.4 Minkowski’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

49.5 The trap box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

49.6 The Minkowski bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

49.7 The class group is finite . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

49.8 Computation of class numbers . . . . . . . . . . . . . . . . . . . . . . . . 470


50 More properties of the discriminant 475



51 Bonus: Let’s solve Pell’s equation! 47751.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

51.2 Dirichlet’s unit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

51.3 Finding fundamental units . . . . . . . . . . . . . . . . . . . . . . . . . . 479

51.4 Pell’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480


XIV Algebraic NT II: Galois and Ramification Theory 483

52 Things Galois 48552.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

52.2 Field extensions, algebraic closures, and splitting fields . . . . . . . . . . 486

52.3 Embeddings into algebraic closures for number fields . . . . . . . . . . . 487

52.4 Everyone hates characteristic 2: separable vs irreducible . . . . . . . . . 488

52.5 Automorphism groups and Galois extensions . . . . . . . . . . . . . . . . 490

52.6 Fundamental theorem of Galois theory . . . . . . . . . . . . . . . . . . . 493


52.8 (Optional) Proof that Galois extensions are splitting . . . . . . . . . . . 495

53 Finite fields 49753.1 Example of a finite field . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

53.2 Finite fields have prime power order . . . . . . . . . . . . . . . . . . . . 498

53.3 All finite fields are isomorphic . . . . . . . . . . . . . . . . . . . . . . . . 499

53.4 The Galois theory of finite fields . . . . . . . . . . . . . . . . . . . . . . . 500


54 Ramification theory 50354.1 Ramified / inert / split primes . . . . . . . . . . . . . . . . . . . . . . . . 503

54.2 Primes ramify if and only if they divide ∆K . . . . . . . . . . . . . . . . 504

54.3 Inertial degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

54.4 The magic of Galois extensions . . . . . . . . . . . . . . . . . . . . . . . 505

54.5 (Optional) Decomposition and inertia groups . . . . . . . . . . . . . . . 507

54.6 Tangential remark: more general Galois extensions . . . . . . . . . . . . 509


55 The Frobenius element 51155.1 Frobenius elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

55.2 Conjugacy classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

55.3 Chebotarev density theorem . . . . . . . . . . . . . . . . . . . . . . . . . 514

55.4 Example: Frobenius elements of cyclotomic fields . . . . . . . . . . . . . 514

55.5 Frobenius elements behave well with restriction . . . . . . . . . . . . . . 515

55.6 Application: Quadratic reciprocity . . . . . . . . . . . . . . . . . . . . . 516

55.7 Frobenius elements control factorization . . . . . . . . . . . . . . . . . . 518

55.8 Example application: IMO 2003 problem 6 . . . . . . . . . . . . . . . . . 521


56 Bonus: A Bit on Artin Reciprocity 52356.1 Infinite primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

56.2 Modular arithmetic with infinite primes . . . . . . . . . . . . . . . . . . 523

56.3 Infinite primes in extensions . . . . . . . . . . . . . . . . . . . . . . . . . 525

Contents 25

56.4 Frobenius element and Artin symbol . . . . . . . . . . . . . . . . . . . . 526

56.5 Artin reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528


XV Algebraic Topology I: Homotopy 533

57 Some topological constructions 53557.1 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

57.2 Quotient topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

57.3 Product topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

57.4 Disjoint union and wedge sum . . . . . . . . . . . . . . . . . . . . . . . . 537

57.5 CW complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538

57.6 The torus, Klein bottle, RPn, CPn . . . . . . . . . . . . . . . . . . . . . 53957.7 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 545

58 Fundamental groups 54758.1 Fusing paths together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

58.2 Fundamental groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

58.3 Fundamental groups are functorial . . . . . . . . . . . . . . . . . . . . . 552

58.4 Higher homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

58.5 Homotopy equivalent spaces . . . . . . . . . . . . . . . . . . . . . . . . . 554

58.6 The pointed homotopy category . . . . . . . . . . . . . . . . . . . . . . . 556


59 Covering projections 55959.1 Even coverings and covering projections . . . . . . . . . . . . . . . . . . 559

59.2 Lifting theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

59.3 Lifting correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

59.4 Regular coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564

59.5 The algebra of fundamental groups . . . . . . . . . . . . . . . . . . . . . 566


XVI Category Theory 569

60 Objects and morphisms 57160.1 Motivation: isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 571

60.2 Categories, and examples thereof . . . . . . . . . . . . . . . . . . . . . . 571

60.3 Special objects in categories . . . . . . . . . . . . . . . . . . . . . . . . . 575

60.4 Binary products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576

60.5 Monic and epic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580


61 Functors and natural transformations 58361.1 Many examples of functors . . . . . . . . . . . . . . . . . . . . . . . . . . 583

61.2 Covariant functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584

61.3 Contravariant functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

61.4 Equivalence of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 588

61.5 (Optional) Natural transformations . . . . . . . . . . . . . . . . . . . . . 588

61.6 (Optional) The Yoneda lemma . . . . . . . . . . . . . . . . . . . . . . . . 590



62 Limits in categories (TO DO) 59362.1 Equalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59362.2 Pullback squares (TO DO) . . . . . . . . . . . . . . . . . . . . . . . . . . 59462.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59462.4 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 594

63 Abelian categories 59563.1 Zero objects, kernels, cokernels, and images . . . . . . . . . . . . . . . . 59563.2 Additive and abelian categories . . . . . . . . . . . . . . . . . . . . . . . 59663.3 Exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59863.4 The Freyd-Mitchell embedding theorem . . . . . . . . . . . . . . . . . . 59963.5 Breaking long exact sequences . . . . . . . . . . . . . . . . . . . . . . . . 60063.6 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 601

XVII Algebraic Topology II: Homology 603

64 Singular homology 60564.1 Simplices and boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 60564.2 The singular homology groups . . . . . . . . . . . . . . . . . . . . . . . . 60764.3 The homology functor and chain complexes . . . . . . . . . . . . . . . . 61064.4 More examples of chain complexes . . . . . . . . . . . . . . . . . . . . . 61464.5 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 615

65 The long exact sequence 61765.1 Short exact sequences and four examples . . . . . . . . . . . . . . . . . . 61765.2 The long exact sequence of homology groups . . . . . . . . . . . . . . . . 61965.3 The Mayer-Vietoris sequence . . . . . . . . . . . . . . . . . . . . . . . . 62165.4 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 626

66 Excision and relative homology 62766.1 The long exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 62766.2 The category of pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62866.3 Excision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62966.4 Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63066.5 Invariance of dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63166.6 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 632

67 Bonus: Cellular homology 63367.1 Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63367.2 Cellular chain complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63467.3 The cellular boundary formula . . . . . . . . . . . . . . . . . . . . . . . . 63767.4 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 639

68 Singular cohomology 64168.1 Cochain complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64168.2 Cohomology of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64268.3 Cohomology of spaces is functorial . . . . . . . . . . . . . . . . . . . . . 64368.4 Universal coefficient theorem . . . . . . . . . . . . . . . . . . . . . . . . . 64468.5 Example computation of cohomology groups . . . . . . . . . . . . . . . . 645

Contents 27

68.6 Relative cohomology groups . . . . . . . . . . . . . . . . . . . . . . . . . 646


69 Application of cohomology 64969.1 Poincaré duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

69.2 de Rham cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

69.3 Graded rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650

69.4 Cup products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652

69.5 Relative cohomology pseudo-rings . . . . . . . . . . . . . . . . . . . . . . 654

69.6 Wedge sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654

69.7 Künneth formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656


XVIII Algebraic Geometry I: Classical Varieties 659

70 Affine varieties 66170.1 Affine varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

70.2 Naming affine varieties via ideals . . . . . . . . . . . . . . . . . . . . . . 662

70.3 Radical ideals and Hilbert’s Nullstellensatz . . . . . . . . . . . . . . . . . 663

70.4 Pictures of varieties in A1 . . . . . . . . . . . . . . . . . . . . . . . . . . 66470.5 Prime ideals correspond to irreducible affine varieties . . . . . . . . . . . 665

70.6 Pictures in A2 and A3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66670.7 Maximal ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667

70.8 Motivating schemes with non-radical ideals . . . . . . . . . . . . . . . . . 668


71 Affine varieties as ringed spaces 66971.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669

71.2 The Zariski topology on An . . . . . . . . . . . . . . . . . . . . . . . . . 66971.3 The Zariski topology on affine varieties . . . . . . . . . . . . . . . . . . . 671

71.4 Coordinate rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672

71.5 The sheaf of regular functions . . . . . . . . . . . . . . . . . . . . . . . . 673

71.6 Regular functions on distinguished open sets . . . . . . . . . . . . . . . . 674

71.7 Baby ringed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675


72 Projective varieties 67772.1 Graded rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677

72.2 The ambient space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678

72.3 Homogeneous ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680

72.4 As ringed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

72.5 Examples of regular functions . . . . . . . . . . . . . . . . . . . . . . . . 682


73 Bonus: Bézout’s theorem 68573.1 Non-radical ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

73.2 Hilbert functions of finitely many points . . . . . . . . . . . . . . . . . . 686

73.3 Hilbert polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688

73.4 Bézout’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690

73.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690



74 Morphisms of varieties 69374.1 Defining morphisms of baby ringed spaces . . . . . . . . . . . . . . . . . 693

74.2 Classifying the simplest examples . . . . . . . . . . . . . . . . . . . . . . 694

74.3 Some more applications and examples . . . . . . . . . . . . . . . . . . . 696

74.4 The hyperbola effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697


XIX Algebraic Geometry II: Affine Schemes 701

75 Sheaves and ringed spaces 70575.1 Motivation and warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . 705

75.2 Pre-sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705

75.3 Stalks and germs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707

75.4 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711

75.5 For sheaves, sections “are” sequences of germs . . . . . . . . . . . . . . . 712

75.6 Sheafification (optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . 714


76 Localization 71776.1 Spoilers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

76.2 The definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718

76.3 Localization away from an element . . . . . . . . . . . . . . . . . . . . . 719

76.4 Localization at a prime ideal . . . . . . . . . . . . . . . . . . . . . . . . . 720

76.5 Prime ideals of localizations . . . . . . . . . . . . . . . . . . . . . . . . . 722

76.6 Prime ideals of quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . 723

76.7 Localization commute with quotients . . . . . . . . . . . . . . . . . . . . 723


77 Affine schemes: the Zariski topology 72777.1 Some more advertising . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727

77.2 The set of points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727

77.3 The Zariski topology on the spectrum . . . . . . . . . . . . . . . . . . . 729

77.4 On radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732


78 Affine schemes: the sheaf 73578.1 A useless definition of the structure sheaf . . . . . . . . . . . . . . . . . . 735

78.2 The value of distinguished open sets (or: how to actually compute sections)736

78.3 The stalks of the structure sheaf . . . . . . . . . . . . . . . . . . . . . . . 738

78.4 Local rings and residue fields: linking germs to values . . . . . . . . . . . 739

78.5 Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742

78.6 Functions are determined by germs, not values . . . . . . . . . . . . . . . 742


79 Interlude: eighteen examples of affine schemes 74579.1 Example: Spec k, a single point . . . . . . . . . . . . . . . . . . . . . . . 745

79.2 SpecC[x], a one-dimensional line . . . . . . . . . . . . . . . . . . . . . . 745

Contents 29

79.3 SpecR[x], a one-dimensional line with complex conjugates glued (no fearnullstellensatz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746

79.4 Spec k[x], over any ground field . . . . . . . . . . . . . . . . . . . . . . . 747

79.5 SpecZ, a one-dimensional scheme . . . . . . . . . . . . . . . . . . . . . . 74779.6 Spec k[x]/(x2 − x), two points . . . . . . . . . . . . . . . . . . . . . . . . 74879.7 Spec k[x]/(x2), the double point . . . . . . . . . . . . . . . . . . . . . . . 748

79.8 Spec k[x]/(x3 − x), a double point and a single point . . . . . . . . . . . 74979.9 SpecZ/60Z, a scheme with three points . . . . . . . . . . . . . . . . . . 74979.10 Spec k[x, y], the two-dimensional plane . . . . . . . . . . . . . . . . . . . 750

79.11 SpecZ[x], a two-dimensional scheme, and Mumford’s picture . . . . . . . 75179.12 Spec k[x, y]/(y − x2), the parabola . . . . . . . . . . . . . . . . . . . . . 75279.13 SpecZ[i], the Gaussian integers (one-dimensional) . . . . . . . . . . . . . 75379.14 Long example: Spec k[x, y]/(xy), two axes . . . . . . . . . . . . . . . . . 754

79.15 Spec k[x, x−1], the punctured line (or hyperbola) . . . . . . . . . . . . . 75679.16 Spec k[x](x), zooming in to the origin of the line . . . . . . . . . . . . . . 757

79.17 Spec k[x, y](x,y), zooming in to the origin of the plane . . . . . . . . . . . 758

79.18 Spec k[x, y](0) = Spec k(x, y), the stalk above the generic point . . . . . . 758


80 Morphisms of locally ringed spaces 75980.1 Morphisms of ringed spaces via sections . . . . . . . . . . . . . . . . . . 759

80.2 Morphisms of ringed spaces via stalks . . . . . . . . . . . . . . . . . . . . 760

80.3 Morphisms of locally ringed spaces . . . . . . . . . . . . . . . . . . . . . 761

80.4 A few examples of morphisms between affine schemes . . . . . . . . . . . 762

80.5 The big theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765

80.6 More examples of scheme morphisms . . . . . . . . . . . . . . . . . . . . 767

80.7 A little bit on non-affine schemes . . . . . . . . . . . . . . . . . . . . . . 768

80.8 Where to go from here . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770


XX Algebraic Geometry III: Schemes (TO DO) 771

XXI Set Theory I: ZFC, Ordinals, and Cardinals 773

81 Interlude: Cauchy’s functional equation and Zorn’s lemma 77581.1 Let’s construct a monster . . . . . . . . . . . . . . . . . . . . . . . . . . 775

81.2 Review of finite induction . . . . . . . . . . . . . . . . . . . . . . . . . . 776

81.3 Transfinite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776

81.4 Wrapping up functional equations . . . . . . . . . . . . . . . . . . . . . . 778

81.5 Zorn’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779


82 Zermelo-Fraenkel with choice 78382.1 The ultimate functional equation . . . . . . . . . . . . . . . . . . . . . . 783

82.2 Cantor’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783

82.3 The language of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . 784

82.4 The axioms of ZFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78582.5 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787

82.6 Choice and well-ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . 787


82.7 Sets vs classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788


83 Ordinals 79183.1 Counting for preschoolers . . . . . . . . . . . . . . . . . . . . . . . . . . 791

83.2 Counting for set theorists . . . . . . . . . . . . . . . . . . . . . . . . . . 792

83.3 Definition of an ordinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794

83.4 Ordinals are “tall” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

83.5 Transfinite induction and recursion . . . . . . . . . . . . . . . . . . . . . 796

83.6 Ordinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797

83.7 The hierarchy of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798


84 Cardinals 80184.1 Equinumerous sets and cardinals . . . . . . . . . . . . . . . . . . . . . . 801

84.2 Cardinalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802

84.3 Aleph numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802

84.4 Cardinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803

84.5 Cardinal exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 805

84.6 Cofinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805

84.7 Inaccessible cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807


XXII Set Theory II: Model Theory and Forcing 809

85 Inner model theory 81185.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811

85.2 Sentences and satisfaction . . . . . . . . . . . . . . . . . . . . . . . . . . 812

85.3 The Levy hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814

85.4 Substructures, and Tarski-Vaught . . . . . . . . . . . . . . . . . . . . . . 815

85.5 Obtaining the axioms of ZFC . . . . . . . . . . . . . . . . . . . . . . . . 81685.6 Mostowski collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817

85.7 Adding an inaccessible . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817

85.8 FAQ’s on countable models . . . . . . . . . . . . . . . . . . . . . . . . . 819

85.9 Picturing inner models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 820


86 Forcing 82386.1 Setting up posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824

86.2 More properties of posets . . . . . . . . . . . . . . . . . . . . . . . . . . 825

86.3 Names, and the generic extension . . . . . . . . . . . . . . . . . . . . . . 826

86.4 Fundamental theorem of forcing . . . . . . . . . . . . . . . . . . . . . . . 829

86.5 (Optional) Defining the relation . . . . . . . . . . . . . . . . . . . . . . . 829

86.6 The remaining axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831


87 Breaking the continuum hypothesis 83387.1 Adding in reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833

87.2 The countable chain condition . . . . . . . . . . . . . . . . . . . . . . . . 834

87.3 Preserving cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835

Contents 31

87.4 Infinite combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83687.5 A few harder problems to think about . . . . . . . . . . . . . . . . . . . 837

XXIII Backmatter 839

A Pedagogical comments and references 841A.1 Basic algebra and topology . . . . . . . . . . . . . . . . . . . . . . . . . . 841A.2 Second-year topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842A.3 Advanced topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843A.4 Topics not in Napkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844

B Hints to selected problems 845

C Sketches of selected solutions 855

D Glossary of notations 877D.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877D.2 Functions and sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877D.3 Abstract and linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . 878D.4 Quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879D.5 Topology and real/complex analysis . . . . . . . . . . . . . . . . . . . . . 879D.6 Measure theory and probability . . . . . . . . . . . . . . . . . . . . . . . 880D.7 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 880D.8 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881D.9 Differential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882D.10 Algebraic number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 882D.11 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883D.12 Algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884D.13 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885

E Terminology on sets and functions 887E.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887E.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 888E.3 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 890

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 46-51

Alg NT 1

Ch 52-56

Alg NT 2

Ch 57-59

Alg Top 1

Ch 60-63

Cat Th

Ch 64-69

Alg Top 2

Ch 70-74

Alg Geo 1

Ch 75-80

Alg Geo 2-3

Ch 81-87

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

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