Best Combinatorics Book List

• Wilf, generatingfunctionology.
  Engaging, fun to read, and filled with all sorts of cool ways to apply generating functions. I’ve read this book cover to cover and have done all of the exercises, so I’m pretty familiar with it. Exercises include lots of applications of generating functions in different areas of number theory (analytic number theory being the most prominent) and in combinatorics. There are also some exercises that develop applications to theoretical computer science. (This would also make good preparation to read Analytic Combinatorics by Flajolet and Sedgewick, which is an amazing book for both math and computer science students.) The first three chapters are accessible to anyone with high school math. Chapter four does require substantially more background (making use of basic complex analysis). Even without getting into that, the first three chapters are great, and anyone who loves counting things will find it a revelation (at least I did). For self study, the answers to all of the exercises are provided in the back (I recall the solutions being a bit too terse for me to always understand, but at least they are there). The third edition is available here while the second edition is available from Wilf’s website which is being maintained by the Penn math department (Prof. Wilf passed away a few years ago).

• Graham, Knuth & Patashnik, Concrete Mathematics.
  I wish someone had handed me this book in high school while I was trudging through whatever terrible AP calculus book we were using and reassured me that math can be challenging, fun, and awesome. Really, everyone should find the time to read this book. You won’t be able to do all of the problems since every chapter contains around 60-70 problems from routine proofs and calculations up to open research problems. Although I’ve put it in the Core Subjects section of the math book list, it is accessible to undergraduates, and even to high school students with appropriate guidance. At some point I should make a sublist of books every math major, grad student, or self studier absolutely must read. This book will be on the top of that list.

Best Set Theory & Logic Book List

Here are the best set theory and logic books I know of at the undergraduate level. There is more on set theory and foundations (model theory, proof theory, etc) in the Advanced Topics section.

Set Theory

• Jech & Hrbacek, Introduction to Set Theory.
  This is the set theory textbook we used in my undergraduate set theory class. I looked through several other set theory books, but this one seemed like the best to me. It covers more material, and at a bit faster pace than some of the books below. It should also be ideal preparation for Jech’s Set Theory, the bible of set theory at the advanced level. This is definitely the set theory book I recommend.

• Halmos, Naive Set Theory.
  Another Halmos book. Elegantly written, and only weighing in at only 104 pages, this covers the bare essentials of naive set theory. From Halmos himself, “the purpose of the book is to tell the beginning student of advanced mathematics the basic set­ theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism. The point of view throughout is that of a prospective mathematician anxious to study groups, or integrals, or manifolds.” For it’s intended purpose it is the best set theory book, but if you want an introductory textbook that prepares you for further study in foundations, then choose Jech’s book above.

• Enderton, Elements of Set Theory.
  This seems like the most common choice for undergraduate set theory classes. The last time I looked it was very expensive, and put me to sleep in the library every time I tried to read it. Maybe you’ll like it more than I do.

Logic

• Rautenberg, A Concise Introduction to Mathematical Logic.
  Mathematical logic can be tedious, but it’s worth learning, and Rautenberg’s book is the only one at the beginning graduate level that I can really recommend. It’s also got a lot of material for those in computer science and interested in computation. There’s a descriptive chapter on the decision problem and automated theorem proving, and even something on non-standard analysis. There are also chapters on logic programming, model theory, recursion theorem, and Godel’s incompleteness theorem. The most comprehensive book I know of on logic, and the best logic book for computer scientists as well as pure mathematicians. This is one I’d really like to read cover to cover one day.

• Kleene, Mathematical Logic.
  Substantially more introductory than Rautenberg’s book, a good textbook if you have less preparation, but still want a “real” logic book. The second half of the book contains surveys of the work of Godel, Church, and Turing. There’s also a chapter on Godel’s completeness theorem, Gentzen’s theorem, Skolem’s paradox and nonstandard models of arithmetic.

• Ebbinghaus, Flum, & Thomas, Mathematical Logic, 2nd Edition.
  Standard logic textbook for undergraduate logic classes. I’ve looked through it and found it unappetizing. A lot of pushing symbols around without a lot of motivation.

Best Abstract Algebra Book List

Introductory
The Introductory section here is a bit of a challenge. I don’t like the standard “easy” undergraduate abstract algebra textbooks (Fraleigh and Gallian are the most widely used) because I think they are a waste of time. That said, Artin or any of the books in the Intermediate section are going to be a challenge. Of course, challenging textbooks are the best way to learn, so here they are.

• Artin, Algebra.
  This blue book, now on its second edition, is a rather idiosyncratic introduction to abstract algebra at the undergraduate level. It focuses heavily on linear algebra and matrix groups. As you might guess from the number of linear algebra books in the best math books list, I think this is a strong feature of the book. The standard material from groups, rings, and fields is covered in the second half of the book, and from what I’ve seen Artin does a good job with it, but I haven’t read this book as closely as some of the others listed below.

• Armstong, Groups & Symmetry.
  A small Springer UTM (about 140 pages, if I remember correctly) that serves as an introduction to abstract algebra. As the title suggests, it only covers groups, but I think it does a really nice job motivating the idea of a group and getting to non-trivial results. (I was the TA/grader for a one-semester undergraduate algebra class taught from this text. The students seemed to like it and get a lot out of it.) If you’re struggling with the other books on this list, I think this one is the most approachable and should give some additional background to start one of the intermediate level texts.

Intermediate

• Aluffi, Algebra: Chapter 0.
  This is my favorite introductory abstract algebra book. Aluffi’s game is to present abstract algebra at the level of Dummit & Foote or Rotman (see below), but emphasizing category theory from the beginning. He does a great job of building up the language of categories and then explaining how each new concept or structure in the abstract algebra sequence fits into this categorical perspective. This book is also just fun to read. Aluffi managed to write an abstract algebra textbook with a sense of humor. These days, a working knowledge of category theory is pretty much essential to every graduate student in mathematics, and I think it’s best to get started learning it right away.

  One warning for this book. It is a bit harder than Dummit & Foote or Rotman. It’s meant to be at the same level, but I think it is a bit more challenging. I can see using Dummit & Foote for undergraduate abstract algebra classes, I have a harder time seeing undergraduates handle Aluffi’s book without some prior exposure to the subject. This is an awesome book, and if you’re looking for a graduate level introduction to abstract algebra this is definitely the one to get.

• Dummit & Foote, Abstract Algebra.
  This is the most popular choice for first year graduate courses right now, but I feel like it was originally intended to be an undergraduate textbook. It is certainly accessible to undergraduates, but contains enough material for a graduate course. In addition to sections on groups, rings, modules, linear algebra, and Galois theory, there are some extra sections in the last couple hundred pages (the book is almost 1000 pages, as seems to be the standard for abstract algebra textbooks) on algebraic geometry and representation theory. The two biggest problems I have with D&F are that starting around the chapter on modules (chapter 10) the exposition really drags. There’s so much talking it’s hard to figure out which things are important and which are fluff. Additionally, they seem to go out of their way to avoid universal properties. I’m okay with explicit constructions, but I’d like to at least see the universal property.

  The strongest point of Dummit & Foote is the exercises. Each chapter has tons of exercises (I think there were almost 60 exercises on some sections) from simple computations up to quite challenging exercises that development examples and extensions to the text. Also, this is the best abstract algebra book for self study since the solutions to all of the exercises in the first twelve chapters are available at the awesome Project Crazy Project. (Crazy Project was one of the primary inspirations for this website.) (Update, 2019. It appears Project Crazy Project is no longer around. You can still find a lot of it on archive.org if you look around. Maybe this will inspire me to get back to work on StumblingRobot. I think I have a massive binder with all the solutions to the first 10 or 11 chapters of D&F in it…. now all I have to do is TeX it all up…)

• Rotman, Advanced Modern Algebra.
  To my mind this book is isomorphic to Dummit & Foote. It’s approximately the same size (which is to say it is a brick) and pitched at approximately the same level. Maybe Rotman starts out a bit more slowly, and avoids some hairy complications in the early part of text (avoiding Zorn’s Lemma and noncommutative rings until much later). I think the choice here really comes down to style. I prefer Dummit & Foote’s. This is another textbook that is common for first year graduate courses in abstract algebra, although I think it is quite suitable for prepared undergraduates.

Advanced

• Lang, Algebra.
  This is the only Serge Lang book you’ll find anywhere on this list because I just don’t like Lang’s books very much. That includes this one, but it simply has to be included since this is the bible for many algebraists. Lang’s book is absolutely modern in its presentation, and I find it painful to read. This book does have exercises (unlike some of Lang’s other books). One of the exercises in an earlier edition was “find a book on homological algebra and do all the exercises.” Lang is not at all gentle, but if you actually get through the book you’ll know a lot of algebra.

• Jacobson, Basic Algebra I, and Basic Algebra II.
  I really, really like this two volume set by Jacobson. It is published by Dover so usually more reasonably priced than most advanced math textbooks. (This two volume set should not be confused with the earlier three volume set Lectures in Abstract Algebra, the second volume of that set is reviewed in the advanced linear algebra books section.) Jacobson’s style takes a bit of getting used to: the writing is very dense, and theorems and proofs are not always separated out from the main line of exposition. Jacobson writes in paragraphs, explaining the ideas to the reader in English. Of course, this is wonderful once you get into the flow of his explanations, but can make the book difficult to use as a reference, and is probably not ideal if you’re looking for bite-sized pieces you can digest.

  The first volume contains the standard abstract algebra material: groups, rings, fields, modules over PIDs in the first four chapters. The second four chapters of volume 1 contain some nonstandard topics, which I thought were great. I especially liked the final chapter on lattices and Boolean Algebras. There is then a bit of disconnect in difficulty level between volumes 1 and 2. While I would say volume 1 is clearly meant as an introduction to abstract algebra, volume 2 is at a substantially higher level of abstraction. It starts with category theory and universal algebra, then gives the best treatment I’ve seen anywhere of the structure theory of rings and fields. It’s a fantastic treatment if you like Jacobson’s style (which, clearly, I do). Also, don’t let the reviews on Amazon or other places fool you. I think at least two of the negative reviews on Amazon were from people who thought Basic Algebra meant it was for middle school students. These are some of my favorite math books and if you want to go further in algebra I highly recommend them.

Best Topology Book List

This list is concerned with point-set topology. Algebraic topology and differential topology are on the Advanced Topics page. The “Introductory,” “Intermediate,” and “Advanced” sections didn’t make as much sense for this section. Not all of the books are at the same level, but the differences between them aren’t so substantial as in some other areas.

• Munkres, Topology.
  The first-half of Munkres is the near-universal choice for undergraduate topology classes. It is extremely readable, and is an excellent choice, especially for students still building up their skills in abstract areas of math. The first 75 pages are devoted to foundational issues in (naive) set theory, and the rest of the first half goes on to cover standard material in point-set topology (countability and separation axioms, the tychonoff theorem, metrization theorems, etc). I’ve only skimmed through parts of the second half on algebraic topology, but didn’t like it as much, but if you want a quick introduction to the fundamental group and Siefert-van Kampen, then you could read it here.

• Willard, General Topology.
  My absolute first choice for an introduction to general topology. Willard is somewhat more sophisticated than Munkres, but covers a lot more ground in less time. (For instance, Munkres 75 page introduction to set theory and logic is covered by Willard in about nine pages.) If you have some exposure to topology and a reasonable level of mathematical maturity then this is the best textbook to go to for a comprehensive introduction to topology. It covers a lot of material and gives you a strong foundation for later work in analysis. Also, as a topology book for self-study, I plan to add solutions to all of the exercises to the blog starting in October of this year (2016) (Update 2019. Hahahaha. I apologize to everyone for my overly ambitious younger self.). (Yes, this is another on the list of books which I have read cover to cover and done all the exercises.) I really cannot recommend this book highly enough if you’ve got sufficient mathematical maturity.

• Lee, Introduction to Topological Manifolds.
  This book is not actually a general topology book, but it didn’t belong anywhere else on this list and it’s a fantastic book so it needed to go somewhere. I’ve read this book from cover to cover and done all of the exercises and think it has been hugely beneficial. This is not a substitute for a general topology book, and I would recommend reading this roughly in conjunction with Willard (or Munkres) since they really cover different things. Describing the content of the book in the preface, Lee says, “a more accurate title for the book would have been Introduction to Topology with an Emphasis on Manifolds… Perhaps the best way to summarize what this book is would be to say that it represents…the ideal amount of topological knowledge that should be possessed by beginning graduate students who are planning to go on to study smooth manifolds and differential geometry.” This seems about accurate to me: it is not a general topology book, but it is also certainly not differential geometry or analysis on manifolds. Anyway, it’s a really great book if you’re already working on general topology and have an interest in differential geometry (covered in Lee’s sequel Introduction to Smooth Manifolds).

• Kelley, General Topology.
  This is the classic point-set topology book. I haven’t read it, but you might want to read it for culture. I don’t know anyone who has successfully gotten through the whole thing, so send me an email if you manage.

• Steen & Seebach, Counterexamples in Topology.
  Like Counterexamples in Analysis below this is a compendium of pathological examples. This one is probably even more useful since the variety of pathologies in topology is so immense. Some of the spaces that blow up theorems you really want to be true are eye-opening. You can learn a lot from understanding all of these things.

Best Real Analysis Book List

Introductory

• Rudin, Principles of Mathematical Analysis.
  The absolutely standard textbook for real analysis at the undergraduate level. Known as “baby Rudin” in contrast to “papa Rudin” (reviewed below), this is one of the books that every undergraduate math major must read at some point. There are easier books to learn real analysis from, but this is the one you should try to use. I love this book. It presents the material cleanly and clearly and doesn’t distract with extraneous details. That said, it isn’t going to hold your hand, and requires serious work. Also, the exercises are extremely difficult, but if you can do them you really master the text. As a real analysis textbook for self study, I think there are at least three full solution sets available on line. If you’ve studied Apostol or Spivak thoroughly then you are ready for baby Rudin. Some linear algebra at the level of Hoffman & Kunze might help, more for mathematical maturity than for any specific material. The first seven chapters are a stream-lined, elegant presentation of the fundamentals of elementary real analysis, chapter 8 gives a quick introduction to Fourier analysis, chapters 9 and 10 introduce real analysis of several variables, and chapter 11 introduces Lebesgue measure and the Lebesgue integral. Most people find chapters 9 and 10 on multivariable calculus the weakest part of the book, and I agree, but they aren’t bad. They are more useful if you’re already quite comfortable with multivariable calculus and linear algebra and can translate what Rudin is saying into more familiar language. Chapter 11 is not great, but gives you exactly the background you need to start in on Chapter 1 of the sequel, Real & Complex Analysis.

• Apostol, Mathematical Analysis.
  This is the primary alternative to baby Rudin for undergraduate real analysis books, and is the one my undergraduate class used. Readers of the blog and of the best calculus book list know that I really like Apostol. As in his other books, this one is comprehensive and rigorous. I think this book is slightly easier than Rudin in that it provides more examples and explanation of most concepts. It also contains more material, with a couple of chapters covering the basics of complex analysis (though not enough to be a full-fledged course in complex anlaysis) and goes deeper on sequences and series. For some reason though, I have to say I didn’t find this book as clear as Apostol’s Calculus volumes, or as clear as baby Rudin. Maybe it was that the class I took out of this book jumped around, so I never quite got into the flow of Apostol’s exposition. Still, it’s the obvious alternative if you really don’t like Rudin’s style.

• Abbott, Understanding Analysis.
  A much more recent real analysis textbook than Rudin or Apostol, this UTM from Springer is somewhat easier going than the previous two books on this list, but still provides good coverage of elementary real analysis. If your background in rigorous mathematics is not quite strong enough for Rudin or Apostol, or if you find the style of those two uncongenial, this book is a nice choice. The downside to this text is that while I think Rudin or Apostol will prepare you to skip straight to graduate level analysis (at the level of Papa Rudin), you will probably have some work to do to prepare for graduate level analysis textbooks if you learn analysis from Abbott. If I were teaching an undergraduate class in real analysis in which the students had varied math backgrounds, this is the text I would use.

• Gelbaum & Olmsted, Counterexamples in Analysis.
  A similar idea to Counterexamples in Topology (reviewed in the topology section above) this book provides a list of all sorts of counterexamples to things you want to be true in real analysis (but aren’t). The coverage roughly corresponds to everything in the introductory real analysis books above, including counterexamples in multiple variables and in Lebesgue integration. Definitely worth having a copy to help clarify why certain theorems have the hypotheses that they do, and what goes wrong when they don’t.

• An Anti-Recommendation. Lay, Analysis with an Introduction to Proof.
  If you’re trying to learn real analysis, and still need an introduction to proof, then something has gone wrong. This book tries to present elementary real analysis with exposition and pictures at the level of a mainstream Calculus textbook. It covers less analysis than Apostol’s or Spivak’s calculus textbooks, and does a worse job with it.

Intermediate

• Whittaker & Watson, A Course of Modern Analysis.
  This is a classic book first published in 1902, so the “Modern Analysis” in the title is a bit out of date. I’m not sure the mathematics here is more difficult than the “Introductory” books listed above, but old books tend to be harder to read today, and this is no exception. Still, this is a great textbook and reference for transcendental functions. The book is divided into two parts. The first is a classical introduction to analysis of both real and complex variables. It’s much more concrete than most of the standard treatments today. The second part is the unique part, and contains a wealth of material on transcendental functions that you don’t really see in typical real analysis books. There are whole chapters on each of the special functions you always hear about: the Gamma function, the Riemann Zeta function, the Hypergeometric function, Legendre functions, Bessel functions, Mathieu functions, Elliptic functions, Theta functions, and several more. I have this book on my shelf and keep trying to find the time to go through some of these chapters thoroughly. The chapters that I have read are wonderfully clear, and you actually learn to compute things, which I find to be a nice change of pace from purely abstract treatments. Highly recommended if you have any interest in transcendental functions. In addition to several versions available on Amazon, this book is in the public domain and there is a free copy available from Archive.org.

• Wheeden & Zygmund, Measure and Integral: An Introduction to Real Analysis.
  This book takes a much more concrete approach than the books in the advanced section below. It starts out with integration on the real line, and then spends time building up the theory in . This is also a rather gentle book (amongst real analysis books) in that it methodically builds in abstraction rather than throwing you in the deep end right from the start. This is a nice book to learn from if you find Papa Rudin a bit much at first. The authors really take the time to explain things in some detail. A lot of people used this as a supplement in my graduate real analysis to get a better intuitive feel for the abstract theory by working in the more familiar case.

• Kolmogrov & Fomin, Introductory Real Analysis.
  This seems to be the standard reference for real analysis at this level. I’ve looked through it from time to time, but haven’t read it carefully. It didn’t excite me, but I’m not a future analyst. The analysts that I know really like this book, and recommend that aspiring analysts do all of the problems in here.

• Aliprantis, Problems in Real Analysis.
  I love books of problems when they are done well. This one is done extremely well, and I absolutely recommend it as a supplement to any real analysis class (or self study program) for books at this level or the advanced level below. This is a companion to the book Principles of Real Analysis also by Aliprantis, but I don’t think you need that book for the problems in this one to be useful. (I also wasn’t particularly enthusiastic by the presentation in that book.) The problem book itself is extremely helpful for making sure you can actually do things in the abstract real analysis books elsewhere on the list. Of course, you can probably guess from my blog that I really like doing exercises and having complete solutions to them.

Advanced

• Rudin, Real & Complex Analysis.
  I love this book for real analysis. It is relentlessly clear, and the first half of this book remains the standard reference for real analysis fifty years after the publication of the first edition (the current edition is the third, published in 1986). Real & Complex Analysis is somewhat legendary for being difficult, but I don’t think that reputation is deserved. It’s extremely clear and well-written, and it is broken down into small sections each containing one result or idea so you can systematically work through each piece of the text. This makes it the best real analysis book for self study. Of course, the book’s reputation for difficulty is not entirely unearned: the exercises are very difficult (or they were for me). Other books have emerged for graduate level analysis (see below), but this is still the best. Rudin covers more material, and I think does so much more clearly than any of the alternatives. I strongly recommend this book as the place to go for real analysis. If you’ve read baby Rudin or Apostol’s Mathematical Analysis thoroughly, then you are sufficiently prepared to read this. If you want to be analyst then you should go ahead and get a copy now because you’re going to need it at some point.

• Royden & Fitzpatrick, Real Analysis.
  This is a big, fat, talky book on real analysis that covers approximately the same material as the first half of Rudin in about three times as many pages. This seems to be the most common choice for graduate real analysis courses these days (based on an extremely non-comprehensive, non-scientific search of several math department websites), but I think everyone would be better off learning from papa Rudin. Not only is the exposition worse, but Royden tends to leave the difficult proofs to the reader/exercises, while spending entirely too much time talking about the easier points. (If the book needs to be three times as long, at least all that extra explanation could go into more details on the difficult topics rather than spending time on the easy points.) If you’re at US graduate program there is a reasonable chance this is the textbook you’ll have to use.

• Folland, Real Analysis: Modern Techniques and Their Applications.
  Another book that tries to be an easier version of papa Rudin, this is the textbook my graduate analysis class used. This one has the advantage of being much slimmer than Royden, but I found the exposition convoluted, and like Royden, Folland tends to spend time talking about the easy points and then waving his hands on the crucial, difficult points. I spent the first month of my real analysis class trying to follow along in Folland before I switched over to Rudin and never looked back. Other people in the class liked Folland more than I did, and if you don’t like Rudin’s style then this is the next book I would look at.

Best Advanced Linear Algebra Book List

A quote from my current abstract algebra professor: “Now, we’ll continue with more linear algebra, and, actually, if you stay in math you’ll be doing linear algebra for the rest of your life.” The importance of linear algebra can’t be overstated. These books are all good for a second course in linear algebra after something like Hoffman & Kunze.

Introductory & Intermediate

• These are covered in the Basics section.

Advanced

• Roman, Advanced Linear Algebra.
  This is my choice for best advanced linear algebra textbook. The other books on this list are closer to reference books rather than books you can learn linear algebra from. The first couple of chapters are review if you’ve already had a solid course in linear algebra, but it quickly picks up in abstraction. The first part of the book proves the classification theorems the way you’ll see it done in abstract algebra, as decompositions of modules over P.I.D.s. The second part of the book is a sequence of topics in linear algebra including chapters on multilinear algebra, Hilbert spaces, applications in affine geometry, and ending with a chapter on the Umbral Calculus. It’s a really nice book if you feel like you need more background in linear algebra.

• Brown, A Second Course in Linear Algebra.
  I’m not as familiar with this book as with Roman’s, and it seems to be out of print (though, as usual, used copies can be had on Amazon through the link above). One of my professors recommended this very highly, and I do recall looking here for some material on normed linear vector spaces and on Banach spaces that I was having trouble with in Roman (above). Anyway, worth a look if your library has it, or if you can find a used copy on Amazon.

• Jacobson, Lectures in Abstract Algebra II – Linear Algebra.
  This is the second volume in a three volume set of books on Abstract Algebra by Jacobson based on his lectures. Ultimately, these three volumes were substantially reworked and became Basic Algebra I and Basic Algebra II (reviewed in the best abstract algebra books section above). However, this second volume has a much expanded coverage of just linear algebra and is a nice standalone reference on the subject from the advanced algebraic viewpoint. I also really like Jacobson’s style, but your mileage may vary. Some people find him too talky, and they have a point, but it’s never bothered me.

• Greub, Linear Algebra, and Multilinear Algebra.
  These are the standard references at the advanced level, and deservedly so. The coverage is comprehensive, but I wouldn’t want to try to learn linear algebra from them.

Best Differential Equations Book List

I don’t have many differential equations books to recommend at this level. (I pushed books devoted solely to PDEs into the Advanced Topics section.) I split them into only introductory and intermediate. If you just need the basics, chapter 8 in Apostol, Calculus – Volume 1 and chapters 6 and 7 of Apostol, Calculus – Volume 2 contain introductory material and lots of problems.

Introductory

• Braun, Differential Equations and Their Applications: An Introduction to Applied Mathematics.
  This is the only computational/applied book on differential equations I’m familiar enough with to recommend. My current university uses this textbook in all of its introductory differential equations classes so I see questions from it in the math help lab on a regular basis. It is pretty good, and far better than the main alternative I know of (Boyce & DiPrima).

Intermediate

• Simmons, Differential Equations with Applications and Historical Notes.
  Fantastic news! This book is finally going to be back in print. CRC Press has indicated they are printing a new, third edition, which they claim will be available in 2016. Everyone always recommends Arnolds’ book (see below), but this textbook by Simmons is great. Maybe people don’t know about it because it has been out of print for a while? Hopefully, that will change with the new edition. This book is ideal for math majors or for applied math students and engineers. It is a compelling read, and it is fascinating/inspiring to learn about the problems people like Gauss and Legendre were working on that lead them to develop the theory of differential equations. Often “applications” in math books end up being contrived examples like “water is in a tank shaped like a right circular cone.” Having the historical context and problems real mathematicians were working on that lead them to these equations is far more enlightening.

• Arnold, Ordinary Differential Equations.
  This is the differential equations book mathematicians always recommend, and it is a good differential equations book. Much less applied than most, the focus is on the geometric theory of differential equations including material on differential equations on manifolds (which you’ll definitely want to know about later when studying PDEs). Obviously, I really like the Simmons book above, but this is still a very nice book. If you want the abstract theory of differential equations on manifolds and the whole bit, then this is your best bet.

Best Number Theory Book List

This is another section that I’m not going to divide into discrete difficulty levels. It was also difficult to decide what belonged in Advanced topics and what belonged here. The division I chose is that the books included below are the ones I thought could be profitably read by advanced undergraduates, while the books on the Advanced topics list are ones requiring a graduate course in algebra or analysis (or complex analysis) as a prerequisite.

• Hardy & Wright, An Introduction to the Theory of Numbers.
  This is the classic book on elementary number theory, now in its sixth edition. Every number theorist I know has a copy of this book. The prerequisites are actually very low: some exposure to calculus is probably helpful, but not strictly required. However, low prerequisites shouldn’t be confused with easy. The book is rather tightly wound and takes some work to read. Once you get into the flow of it, and get the hang of Hardy’s style, it is full of great elementary number theory results. One downside is that there are no exercises, but if you take the time to work out the proofs to all the theorems, you’ll get a ton out of it. Although I highly recommend this book, it really takes some dedication to get anywhere, so probably best for avowed number theorists. A nice feature is that the chapters tend to be fairly self-contained, so you can easily look up and read the chapters on continued fractions without having to worry about the material in the previous chapters.

• Apostol, Introduction to Analytic Number Theory.
  Hey, another Tom Apostol book. Like the others this one is rigorous and comprehensive and contains a lot of nice exercises. Presumably this requires some amount of complex analysis as background, but I read it before I knew much complex analysis and still got a lot out of it. This is a nice primer on arithmetic functions and the Riemann Zeta function. The primary goal of the book is to prove the major analytic number theory results of the 19th century including Dirichlet’s theorem on primes in arithmetic progressions and the Prime Number Theorem of Hadamard and de la Valle Poisson. If you want a gentle introduction to the classical methods of analytic number theory and the build up to the Riemann Hypothesis, this is a great place to start. If you are self studying, I think there are solutions to most (all?) of the exercises online somewhere.

• Silverman, Rational Points on Elliptic Curves.
  This is the undergraduate introduction to elliptic curves which might help prepare you to read Silverman’s two advanced books on elliptic curve, The Arithmetic of Elliptic Curves and Advanced Topics in the Arithmetic of Elliptic Curves which are reviewed in the Advanced Topics section. This is definitely at the undergraduate level and doesn’t require anything you wouldn’t have seen in a standard undergraduate abstract algebra class. The main topics of the book are the Nagell-Lutz theorem and the Mordell-Weil theorem describing the points of finite order and the finite generation of the group of rational points, respectively. There is also a chapter on Lenstra’s elliptic curve factorization algorithm. This is a new edition released in 2015, and it added a chapter on elliptic curve cryptography. I can see this book making a good self study challenge for people in computer science that are interested in the math behind elliptic curves, or wanting to get into cryptography. Of course, you’d have plenty of work to do after finishing this, but it would get you started on the math behind those topics.

• Ireland & Rosen, A Classical Introduction to Modern Number Theory.
  A common and good choice for senior undergraduate reading or seminar courses. The first half of the book is an introduction to elementary number theory, while the second half is mostly more advanced topics in number theory requiring some abstract algebra. I think the first half is completely accessible without much background, but the algebraic prerequisites definitely get higher as the book moves on. There are nice sections on zeta functions, L-functions, algebraic number fields, and elliptic curves. Definitely the place to go for a broad and modern introduction to number theory if you have sufficient maturity to read it (or can develop sufficient maturity as you read it to keep up). It also has a ton of really nice exercises. If I were looking to study number theory at the graduate level (I’m not) I would probably try to read as much of this as possible while in undergraduate.

• Samuel, Algebraic Theory of Numbers.
  A tiny sliver of a book translated from French. There’s a new, very economical edition that has come out. My undergraduate algebraic number theory class used this a few years ago and I had to hunt down and pay a fortune for a used copy of the old edition. Anyway, since it is now readily available, there’s no good reason not to have it. Algebraic number theorists tell me it’s by far the best treatment available which only requires an undergraduate course in abstract algebra as background.

• Serre, A Course in Arithmetic.
  A lovely book on arithmetic by Serre. Apocryphal stories about people mistaking the book for a book on elementary school arithmetic abound. It’s not actually that hard though. There are two parts: the first part gives a classification of quadratic forms over the rationals while the second part gives a proof of Dirichlet’s theorem and contains a study of modular forms. I found the first part easier to digest, but I’m decidedly more inclined to algebra than analysis, so maybe they are at the same level for people with more balanced interests. There are no exercises, and the book is quite terse, but unravelling what Serre is doing takes a lot of work. This is really at the edge of what could profitably be read by undergraduates, but with some supervision there is plenty of accessible material in here. It will certainly stimulate some development in your ability to deal with tightly written books.

Best Complex Analysis Book List

Introductory

• Saff & Snider, Fundamentals of Complex Analysis with Applications to Engineering.
  This is an undergraduate level introduction to complex variables and complex analysis. It’s heavily computational, and the proofs of many of the main theorems of complex analysis are not entirely rigorous. It’s designed for a broad audience and is a good place to start for applied mathematicians, engineers, and physicists who need to know how to use ideas from complex analysis to aid in computation. I think it does an OK job as an introduction, but if you’re a math major you’ll need a lot more rigor at some point.

• Needham, Visual Complex Analysis.
  This is a very popular companion book for complex analysis students. I don’t think you could use it as a primary text, but as a supplement it’s great. Developing the ability to visualize what is going on in complex analysis is a big asset, as your geometric intuition is a powerful tool. This is a really nice book for that purpose, but I actually prefer the (for some reason much less popular) book by Wegert in the intermediate section below.

Intermediate.

• Ablowitz & Fokas, Complex Variables: An Introduction and Applications.
  This book on complex analysis is written by applied mathematicians for applied mathematicians (or applied mathematics students). The proofs are not completely rigorous, but are not complete hand waving. The strengths are in the exercises and explicit developments of how to apply ideas from complex analysis to solving real world problems. Not for those going into a pure math phd, but if you’re in engineering or applied math this is a good complex analysis textbook to choose.

• Wegert, Visual Complex Functions.
  A great book! I really prefer this to Needham’s Visual Complex Analysis above. This book is focused much more on visual representations of complex functions, while Needham’s is a bit more elementary and spends a lot of time on the geometry of the complex plane. This book aims to be a complete course in complex analysis presented largely through visual means, particularly, phase portraits of complex functions. Although it is designed to be a self-contained introduction, I think it serves better as a supplementary text, or as a review of things you’ve already learned in complex analysis. Highly recommended if you want some geometric intuition of how complex functions work.

Advanced.

• Freitag & Busam, Complex Analysis and Complex Analysis 2.
  This two volume set is my top choice for learning complex analysis at the introductory graduate level. The first four chapters of Volume 1 cover the standard complex analysis material everyone is expected to know (though in my experience graduate courses never seem to explicitly cover) on integration and differentiation in , analytic continuation, etc. The coverage is brisk, but thorough. The second half of Volume 2 is devoted to topics in complex analysis. In particular, there are chapters on elliptic functions and elliptic modular forms. I also think this is the best complex analysis book for self study as complete solutions to all of the exercises of Volume 1 are provided. Volume 2 probably belongs on the Advanced Topics list. It contains an introduction to Riemann surfaces, abelian functions, and modular forms of several variables.

  There are some complaints about the translation from the German in some of the reviews I’ve read. I’ve gone almost all the way through Volume 1 and have dipped into Volume 2 and haven’t noticed any problems. Maybe it was corrected in a later edition or printing? I find the whole thing very readable and well organized and highly recommend the whole two volume set.

• Remmert, Theory of Complex Functions and Classical Topics in Complex Function Theory.
  Another two-volume set originally written in German. I think this is a beautifully written book, and contains a rich historical account of the development of complex analysis. The reason I prefer Freitag & Busam above is that I find Remmert’s book harder to learn from. I think it might be a better book if you have the time to systematically read it cover to cover, but the material is rather tightly wound, so it’s difficult to skim through and pick up just the parts you need. If I had an intention of going into something in complex analysis or complex geometry this might be the book I’d choose. More than any other complex analysis book on this list, Remmert seeks to immerse you in the cultural developments of the subject.

• Ahlfors, Complex Analysis.
  This is on every best complex analysis book list I’ve ever seen, and continues to be the standard reference for complex analysis at the introductory graduate level. That said, it’s not a book I love. I tried to read it before I switched over to Freitag’s books, and I can see why people like it, but it seems to me there are better, more modern treatments available now. Anyway, I’m obligated to list it, and if you like classical books with a concrete outlook, then Ahlfors is a good bet.

• Conway, Functions of One Complex Variable 1 and Functions of One Complex Variable 2.
  The second most common graduate introductory complex analysis book after Ahlfors, and another one that I could never get going with. The development in the first few chapters is so slow that I got bored with it. Maybe the pace picks up later on? Anyway, this book still deserves a spot on the list since it’s a thorough and comprehensive treatment, and contains all of the complex analysis most first or second year grad students will need to know. It’s not exciting, and won’t blow you away, but gets the facts across, and might be the best preparation for prelim/qual exams if you have to do that kind of thing.

• Berenstein & Gay, Complex Variables: An Introduction.
  This is the first of a two volume set. (The other volume is in the Advanced Topics section.) Distinctly more demanding than the books above, the aim of this book and its sequel, Complex Analysis and Topics in Harmonic Analysis is to connect students to current research developments in the field (or to current research developments in the field as of the early 90’s). This set is really fast-paced and is absolutely modern, getting into the active research areas as quickly as possible. I haven’t done more than flip through it, but reading a big chunk of it is on my priority list for this summer (2016). The parts that I have read are very engaging, so I definitely recommend taking a look at it if you are interested in complex analysis. Hopefully I’ll update this in the fall after I’ve read more of this one.

Best Differential Geometry Book List

Introductory

• do Carmo, Differential Geometry of Curves and Surfaces.
  I used this textbook as a supplement in my undergraduate curves & surfaces class (the primary textbook was aimed at doing differential geometry of curves & surfaces with maple). It’s nicely written, and easy to learn from. My problem with it was mostly that it’s very concrete, sticking to componentwise computations for everything. Differential geometers whose book recommendations I’ve read always say you need to learn the concrete theory in before moving to the more abstract viewpoint. I’m not so sure, but they know more differential geometry than I do.

Intermediate

• Tu, An Introduction to Manifolds.
  I took a graduate course in differential geometry two years ago and would have failed without this book. Loring Tu wrote this to cover the material a student should know before starting his famed book with Raoul Bott, Differential Forms in Algebraic Topology. This is the most lucid introduction to manifolds I know of, and conveys the material with remarkable clarity. However, it is probably best paired with a heftier book such as Lee’s Smooth Manifolds book (below) since Tu doesn’t quite cover everything a graduate student is expected to know. What is in there is so well explained though, that I wouldn’t want to go anywhere else for those topics.

• Milnor, Topology from the Differentiable Viewpoint.
  Approximately every book Milnor wrote is a classic. One of the truly great mathematical writers (and one of the truly great mathematicians). This isn’t exactly a book on differential geometry, but this is the most sensible location on the list. The book is only 80 pages, and would be a great supplement in any geometry course.

• Gadea, Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers.
  The other book that got me through my graduate differential geometry class. This is exactly what the title says, a workbook of smooth manifold problems. There are 375 problems total, all with complete, detailed solutions (really detailed). Readers of the blog and this list know I’m fond of exercises with complete solutions, so shouldn’t be surprised that I like this book. It’s hard to find detailed solutions to differential geometry problems for some reason, and I found this extremely helpful.

Advanced

• Grinfeld, Introduction to Tensor Analysis and the Calculus of Moving Surfaces.
  The differential geometry book for physicists and students in applied math. A very different flavor from the pure mathematics books elsewhere on the list. The physics majors I know love this book. I’m not a physicist or applied mathematician, so take this recommendation with a grain of salt.

• Lee, An Introduction to Smooth Manifolds.
  Ah, Lee. A monster of a book, packed with information. This was the official textbook for my grad differential geometry class and I really struggled with it my first time through. I’ve since gone back and have been systematically working my way through it with more success. It’s really a great book, but is not easy-going. I wish there were more worked out examples in the text. (Lee leaves a huge number of examples as exercises for the reader, which is fine if you know what you’re doing, but much of the time during my first reading I felt adrift, so leaving an exercise for me to do was useless.) If you take graduate differential geometry in the US there’s a good chance you’ll be using this book. I recommend the books by Tu and Gadea in the Intermediate section above as supplements. I’ve complained about this book, but still like it enough to keep working through it, and it has taught me a TON of differential geometry in the process. So, if you want to bang your head against a wall and maybe learn some differential geometry in the process, look no further. (Also, be sure to get the second edition as there were pretty big changes from the first.)

• Spivak, A Comprehensive Introduction to Differential Geometry, Volume 1 and Volume 2.
  The first two volumes of Spivak’s mammoth five volume set on Differential Geometry. This is a classic that everyone recommends and some people truly love. It was checked out of the library every time I tried to look at a copy, so I haven’t really read it.