Mathematics Readings

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Recommended titles are denoted by a cute cube . I consider a good textbook to be well-written, motivated, and to contain quality exercises. I consider a book motivated if you can glean the big picture from reading the text. I consider a text well-written if it is rigorous with a good flow, and has good style. I consider exercises to be of high quality if they force the reader to apply the material learned, progress from easy to very difficult, and are realistically solvable by the intended audience. Additionally, a good exercise will make the reader see the material in a new light, and form connections between different areas of the subject.

For non-fiction books, I look for good prose and exposition for the intended audience. General population mathematics books should be accessible to a reader with a high-school mathematics education. Books intended for a more mathematically inclined audience should not spend too much time covering basics, and should explain the concepts in an approachable manner.

Non-Textbooks

Books For Mathematicians

• Krantz, A Mathematician's Survival Guide

  Great guide for graduate school in mathematics and beyond. One thing though, he recommends not studying for the GRE... Ignore this advice and STUDY FOR THE GRE!

General Mathematics

• Abbott, Flatland

• Dunham, Journey Through Genius

• Dunham, The Mathematical Universe

• Hardy, A Mathematician's Apology

• Paulos, Innumeracy

• Schechter, My Brain is Open

• Seife, Zero

  Not really worth the read.

• Singh, Fermat's Enigma

• Strogatz, The Joy of X

• Villani, Birth of a Theorem

  I really wanted to like this book, but it isn't great. The good: the book has really good sections on living mathematicians and their contributions. The bad: The book assumes that you already have your PhD and studied analysis. Furthermore the writing is as eccentric as Villani himself (who I think is quite a cool person), and jumps from topic to topic quite erratically.

Millennium Problems

• Derbyshire, Prime Obsession

• Devlin, The Millennium Problems

• O'Shea, The Poincare Conjecture

Textbooks

Algebra

• Artin, Algebra

  I don't find this book particularly fun to read, but if you need to learn algebra there are worse choices. The book is comprehensive and covers good topics.

• Lang, Algebra

  Lang and Rudin would have been good friends. Incredibly comprehensive. Minimal hand-holding... for better or for worse.

• Pinter, A Book of Abstract Algebra

  This book is well done. A warning though, the book is designed so that you at least read all of the exercises, and preferably do them. A lot of the important results are relegated to the exercises, and the reader is guided through the proofs. I actually like this exposition, since it helps to get you familiar with the proof techniques. If you cannot be bothered to actually do the exercises, reading through them at least gives you a picture of the important theorems. The book is very well written and a fantastic entry level text on abstract algebra.

Analysis (Real and Functional)

• Lieb, Loss, Analysis

  Good for people interested in PDE analysis. The book does a good job covering analysis with a angle towards PDE.

• Rosenlicht, Introduction to Analysis

  There are better, more modern analysis books. For a more comprehensive treatment, consider Thompson, Bruckner, and Bruckner.

• Stein, Shakarchi, Real Analysis

• Thompson, Bruckner, Bruckner, Elementary Real Analysis

  Impressively comprehensive. This book is available for free online for those who like that kind of thing. The book is well written and motivated. The text contains many great exercises and advanced sections are highlighted as such and are omitted with no issue. Many of the advanced sections cover topics of lesser importance, but their inclusion makes this a self contained book on pre-20th century (undergraduate) analysis. This book is not very "difficult" in the sense that Rudin is "difficult".

• Abbott, Understanding Analysis

  Exceptionally average.

• Ross, Elementary Analysis

  There are so many better analysis books, why bother?

• Rudin, Principles of Mathematical Analysis

  Classic book: the so called "baby Rudin". Hard. This is a good second book on undergraduate analysis. If you can read Rudin there isn't much else that can come close to it.

• Rudin, Real and Complex Analysis

  Rudin's follow up to principles. Also hard. Very good exposition and I like it more than his principles. This is an essential book for anyone interested in analysis.

• Stein and Shakarchi, Real Analysis

  The third in the Princeton analysis lecture series. Covers some non-standard real analysis topics including Hausdorff dimensions. This is a very good book in the signature Stein and Shakarchi style.

• Wade, An Introduction to Analysis

  The chapters on integration and the real number system are good, the rest of the book is unmotivated and a slog.

Analysis (Complex)

• Churchill, Brown, Verhey, Complex Variables and Applications

• Marsden, Hoffman, Basic Complex Analysis

  A lot of detail, too much for my taste. Might be good for someone who struggles with analysis concepts.

• Needham, Visual Complex Analysis

• Stein, Shakarchi, Complex Analysis

  Well motivated and comprehensive, the authors do a good job of showing how complex analysis fits into the greater whole of analysis. The material is rigorous and advanced, and leaves the reader feeling satiated. The exercises are very good.

Analysis (Other)

• Stein, Shakarchi, Fourier Analysis

The first in the incredible four part series by Stein and Shakarchi. The book manages to touch on everything from Hilbert spaces to the theory of Abelian groups, and gives a fantastic survey of applications of Fourier analysis, including the applications to PDE and number theory. Exercises are challenging but insightful. I highly recommend this book.

Applied Math (General)

• Logan, Applied Mathematics

  This book has a typo or two every page. I don't know how four editions of this book made it to print with such egregious and frequent typos, but here we are...

  This book is almost really good, but it falls short. The author is knowledgeable, and the order and selection of topics is good. The execution, however, is so poor that I cannot recommend using this book.

  To top it all off, the spine separated from the pages after less than an hour... avoid this one.

• Holmes, Introduction to the Foundations of Applied Mathematics

  I really like this book, but it doesn't cover as much as I would like it to. This book is good for learning about dimensional analysis, perturbation methods, and continuum mechanics. It covers the diffusion equation but I wish it had more PDE applications. Overall it is well written and I like most of the exercises.

Dynamics

• Hale and Kocak Dynamics and Bifurcations

  A far superior book to Strogatz in almost every way. The only drawback it has against Strogatz is no section on Chaos. This book has all the same graphical motivation and compelling examples that Strogatz has, coupled with far more rigor and mathematical intuition as well as geometric intuition. Proof are... proved! Since your class is likely going to require Strogatz, get this book and read along to fill in the (ahem... many) gaps left by Strogatz.

• Strogatz, Nonlinear Dynamics and Chaos

  This book is hailed by many as being one of the best math textbooks of all time, but I disagree with this assertion. The book is decently written at best, and their are glaring ambiguities between the text and the exercises. The book lacks rigor of any sort, and for a mathematics text it is subpar at best.

  This book is good for people who are hardcore applied mathematicians or non-mathematicians using dynamics in their research, but this is hardly a "mathematician's" textbook.

Partial Differential Equations

• Bleecker, Csordas, Basic Partial Differential Equations

  The text focuses on theory with little to know exposition on solving problems. The exercises are predominantly computational. The theory is largely unmotivated and is often overly verbose and difficult to follow. Not the worst PDE textbook, but it won't be winning any prizes.

• Evans, Partial Differential Equations

  This book is a PDE analyst rite of passage. Comprehensive coverage of the topic from a more analytic viewpoint than undergraduate PDE which is often more computation focuses.

• Strauss, Partial Differential Equations

  Another rite of passage for PDEs. This book is really good. The focus is on theory and the underlying relations to physics. Strauss is a really good comprehensive introduction to PDE.

• Haberman, Applied Partial Differential Equations

  This book is very good and very comprehensive. It introduces many advanced topics (including dispersive waves) and is pedagogically sound. This would be a fantastic book for an undergraduate PDE course.

Proof Writing and Logic

• Hammack, Book of Proof

  This is one of the best mathematical books I have ever read, and provides a comprehensive and complete treatment of upper division mathematics. This is a great book for undergraduate mathematics majors transitioning from calculus to upper division, proof based courses. This book should be required reading for math majors. The book also has fantastic exercises covering a good range of difficulties and full solutions. This is a great book for self study.

Numerical Analysis

• Cheney, Kincaid, Numerical Mathematics and Computing

  Nothing special, dated, miles better than Saur.

• Press, Flannery, Teukolsky, Vetterling, Numerical Recipes in C

  Covers lots of practical implementation and the ideas behind the methods. Doesn't offer much of a dive into theory.

• Saur, Numerical Analysis

  Avoid this book at all costs!

Other Textbooks

• Stillwell, Mathematics and its History

• Steele, The Cauchy-Schwartz Masterclass

Topology

• Mendelson, Introduction to Topology

  Not terribly motivated, but covers the basics. Dense.

The Three Greatest Textbooks of All Time

Honorary Mentions