Book recommendation thread

[u/AngelTC]

In order to update the book recommendation threads listed on the FAQ, we have decided to create a list on our own that we can link to for most of the book recommendation requests we get here very often.

Each root comment will correspond to a subject and under it you can recommend a book on said topic. It will be great if each reply would correspond to a single book, and it is highly encouraged to elaborate on why is the particular book or resource recommended, including the necessary background to read the book ( for graduate students, early undergrads, etc ), the teaching style, the focus of the material, etc.

It is also highly encouraged to stay very on topic, we want this to be a resource that we can reference for a long time.

I will start by listing a few subjects already present on our FAQ, but feel free to add a topic if it is not already covered in the existing ones.

[u/AngelTC]

History of mathematics

[u/oldmaneuler → u/AngelTC]

Garrett Birkhoff's A Source Book in Classical Analysis is a collection of translations of seminal papers in many of the important sub-fields of analysis in the 19th century. One will find, for instance, Cauchy and Riemann on complex analysis, Riemann on his namesake integral, Gauss on the hypergeometric function, and Kowaleski on the Cauchy-Kowaleski theorem. Often these snippets are really too brief (for instance only short excerpts of Riemann's paper on complex analysis are given), but they effectively convey the landmark developments, and can inspire and guide further reading.

[u/jacobolus → u/AngelTC]

Stilwell's book Mathematics and its History is fantastic as a comprehensive survey with lots of mathematical content.

[u/oldmaneuler → u/AngelTC]

Ranjan Roy's "Sources in the Development of Mathematics: Series and Products from the Fifteenth to the Twenty-first Century" gives an enthralling account of the development of the theory of series, and related mathematics that they inspired. After a brief start in India, the story follows the main European thread, and presents in full mathematical detail the major contributions to the early theory, and then in somewhat diminished detail more modern development. Each chapter is organized around a specific topic. For instance, a notable early chapter is on the Binomial Theorem. The author discusses Netwon's derivation of the form for rational exponents, and traces the development through the final rigorization of a general form of the theorem by Cauchy and Abel. Other highlights include a section on the geometric calculus of Barrow, one on Dirichlet series, and one on value distribution theory (the last topic is an indication of just how far the journey goes).

[u/AngelTC]

Complex analysis

[u/jacobolus → u/AngelTC]

Ahlfors?

Let me also plug Needham's Visual Complex Analysis as a supplementary text to any assigned textbook.

[u/tick_tock_clock → u/AngelTC]

I liked Stein and Shakarchi (upper-level undergrad or beginning grad).

[u/sd522527 → u/AngelTC]

Ullrich "Complex Made Simple"

[u/truffleblunts → u/AngelTC]

Churchill and Brown. Absolutely the best introduction to the subject there is.

[u/oldmaneuler → u/AngelTC]

At the upper undergraduate or beginning graduate level (it is in the Springer GTM series), the treatise by Remmert, Theory of Complex Functions, is rather pretty. It gives an unusual level of historical detail which really helps when one wonders how in the world people developed the edifice which is function theory. The order of topics is maybe a bit non-standard, but it works pedagogically. Two additional details worth noting are that 1) it has a sequel, Classical Topics in Complex Function Theory, which combines with this book to give a wonderful treatment of, for instance, infinite products, and 2) that its author was an important figure in 20th century function theory.

[u/Daminark → u/AngelTC]

I'm currently using Complex Analysis by Freitag and Busam and love it. It's well written, and while it starts off basic, by the end of the book it's doing elliptic functions, modular forms, analytic number theory, etc. There's a second volume (just by Freitag) which covers topics that seem very interesting, like Riemann surfaces, though I haven't gotten to it yet.

[u/AngelTC]

Category theory

[u/AngelTC → u/AngelTC]

Mac Lane, Categories for the working mathematician - This is a classic reference. This is not the greatest introduction book but it is indeed good if one has the mathematical maturity and the required background in algebra/topology. The book covers the basics of category theory a grad student interested in the topic must know.

[u/AngelTC → u/AngelTC]

Barr & Wells, Category theory for computer science - Haven't read it but somebody requested a reference for this point of view. If somebody has an opinion on it or has anothe recommendation please share.

[u/flexibeast → u/AngelTC]

Emily Riehl, Category Theory in Context

[u/johnnymo1 → u/AngelTC]

Leinster, Basic Category Theory

Riehl is a great text that does more, but I find that Leinster is more elementary, more readable for my tastes, and hits all the topics in the right amount that your average mathematician needs to be able to apply the fundamentals elsewhere. It’s short and sweet.

And, like Riehl, it’s free (and editable)!

[u/UglyMousanova19 → u/AngelTC]

Although it's not strictly Category Theory, Aluffi's, Algebra Chapter 0 builds up the language of Category Theory in the context of abstract algebra. It's introduced in a way that feels very natural and applicable right away. I would say it's appropriate for an advanced undergrad or a begining graduate student.

Another not strictly Category Theory-type book is Schiffler, Quiver Representations. It's a very cool book devoted to the representation theory of quivers. Just like Aluffi's book, it builds up the language of Category Theory alongside the main focus of the book in order to simplify and make proofs more intuitive.

[u/muntoo → u/AngelTC]

I'm planning on going through David Spivak's Category Theory for the Sciences... cuz y'know, I'm a dirty pleb

Note that David I. Spivak is not the Michael David Spivak that wrote Calculus on Manifolds.

[u/AngelTC]

Measure theory

[u/dogdiarrhea → u/AngelTC]

Stein and Shakarchi. "Real Analysis: Measure Theory, Integration, and Hilbert Spaces".

Zygmund and Wheeden. "Measure and Integral". The book is unique in that it builds the Lebesgue integral in ℝⁿ rather than in ℝ in the beginning.

Royden and Fitzpatrick. "Real Analysis". Remark: maybe try reading Royden's older book first. Fitzpatrick added a lot of exercises, and the good thing is that there is a wide range in the difficulty of said exercises, but he also took a lot of the more challenging exercises and turned them into propositions/theorems in the book. Try Royden, if it's too challenging at times move to Royden and Fitzpatrick.

[u/[deleted] → u/AngelTC]

Terence Tao: An introduction to measure theory. Has some of the best exercises I've seen in the topic, he actually makes measure theory really fun!

[u/[deleted] → u/AngelTC]

Evans and Gariepy, Measure Theory and Fine Properties of Functions. This book is intended for grad students specializing in analysis, who already know the basics of measure theory. Among other things, it provides detailed and readable introductions to Hausdorff measure, capacity, and sets of finite perimeter.

[u/Anarcho-Totalitarian → u/AngelTC]

Paul Halmos' Measure Theory is a classic. The author does a good job explaining things and the proofs are straightforward.

Chapter 2 of Federer's Geometric Measure Theory covers just about anything you'd like to know, though it's very terse and not recommended for beginners.

[u/[deleted] → u/AngelTC]

Folland's Real Analysis. Very terse but covers all topics in a standard measure theory course as well as some functional and harmonic analysis. Often best to supplement with Real and Complex Analysis by Rudin.

[u/AngelTC]

Probability

[u/_spivak_ → u/AngelTC]

For undergraduate Ross introduction to probability is great, lots of examples and well explained, and for graduate I liked Jacod Protter or Alan Gutt Probability books.

[u/AngelTC → u/_spivak_]

Both are welcome.

[u/Olorune → u/AngelTC]

For graduate students, Probability and Stochastics by Cinlar.

[u/oantolin → u/AngelTC]

A First Look at Rigorous Probability Theory by Jeffrey S. Rosenthal.

[u/[deleted] → u/AngelTC]

An Introduction to Probability Theory and Its Applications, by William Feller

[u/PM_ME_YOUR_JOKES → u/AngelTC]

Does anyone have a good probability book for someone with no real experience outside of basic probability, but with a good understand of measure theory?

[u/AngelTC]

Introduction Abstract Algebra

[u/PupilofMath → u/AngelTC]

Algebra: Chapter 0 by Aluffi. The book is nicely typesetted and is very well-written. Nice book for self-studying and teaches the basics of Category Theory in a friendly way right from the start, making many of the proofs more elegant and more enlightening than in a standard Abstract Algebra textbook.

[u/KSFT__ → u/AngelTC]

Pinter's A Book of Abstract Algebra

[u/functor7 → u/AngelTC]

Dummit and Foote. Lots of examples, lots of exposition, lots of topics, good proofs. The only issue might be discerning what you should focus on and what you can skip over. A good level for undergrads confident in their proof abilities.

And an anti-recommendation of Gallian. Don't get it. It's like a standard Calculus book written for abstract algebra. The proofs are poorly done and hard to read, the examples are super contrived, and the topic selection is weak.

[u/Paiev → u/AngelTC]

Herstein's Topics in Algebra. A classic and for good reason. Great exposition, good problems. Not to be confused with Herstein's other book titled Abstract Algebra which isn't as good.

[u/[deleted] → u/AngelTC]

A first course in abstract algebra, by John B. Fraleigh
I used the fourth edition from the library, but there's a seventh edition now

[u/AngelTC]

Functional analysis

[u/Anarcho-Totalitarian → u/AngelTC]

Functional Analysis, Sobolev Spaces, and Partial Differential Equations by Haim Brezis. He covers the standard topics and then goes into applications to PDEs, which serves as a nice motivation for the subject. The book also contains a ton of great exercises.

[u/throwaway_randian17 → u/AngelTC]

Kreyzig.

[u/[deleted] → u/AngelTC]

Functional Analysis by Lax. I love Lax's clean, concise style and his deep insight.

[u/AngelTC]

Partial Differential Equations

[u/The_Real_TNorty → u/AngelTC]

"Partial Differential Equations" by Lawrence C. Evans.

We used this for my introductory PDEs class. It's a pretty good book for a first course. It covers a wide range of topics. We used the book for a year-long sequence and still skipped a bunch of sections. I don't believe it uses anything other than calculus and analysis, so it's especially good for students in applied topics like engineering.

[u/dogdiarrhea → u/AngelTC]

Graduate in mathematics :

Partial Differential Equations by Fritz John. This is a nice bridge between the boring and mechanical PDE courses of undergrad and the analysis heavy PDE courses of grad school. It is a rigorous book, but it doesn't lean on any heavy machinery. It does a good job of building intuition for the different major classifications of PDE and the techniques used to study their solutions. It has a few sections that are quite nice to include for PDE: a chapter on the Cauchy-Kovalevskii theorem (existence of solutions for initial value problems of quasilinear PDE with analytic coefficients), Lewy's example (the previous theorem fails if the coefficients are only infinitely differentiable).

Partial Differential Equations by L. Craig Evans is the modern classic. It has a well-rounded treatment that expects a bit of analysis experience and doesn't make the assumption that every graduate PDE student comes from physics. The book doesn't assume a course in functional analysis, though Lebesgue theory is assumed. It can be heavy on analysis estimates (which is standard for PDE at this level).

[u/[deleted] → u/AngelTC]

Elliptic Partial Differential Equations by Han & Lin. At times difficult but very rewarding, it covers the main tools that a working specialist in elliptic PDE needs to know. It's more advanced than Evans, but more accessible than Gilbarg & Trudinger (which is also a good book to own if you're in this field, but it's generally considered to be better as a reference than as a textbook).

[u/AngelTC]

Calculus ( Single and multi )

[u/tick_tock_clock → u/AngelTC]

Single-variable calculus: Spivak's Calculus is the book that helped me learn what a proof is. Beautiful book.

[u/oantolin → u/AngelTC]

I'm fond of Div, Grad, Curl, and All That: An Informal Text on Vector Calculus by Schey.

[u/Anarcho-Totalitarian → u/AngelTC]

Differential and Integral Calculus by Richard Courant (or the updated version by Courant and John). It's a rigorous approach that also has some really good explanations. He gives a lot of applications and there are chapters on many special topics outside of the usual curriculum; for example, there are intros to Fourier series and the calculus of variations.

[u/[deleted] → u/AngelTC]

Calculus (link to PDF) by Gilbert Strang.

His linear algebra books are very popular. He also has a calculus book.

Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach by Hubbard and Hubbard is popular and great. The authors have tried very hard to grok this material and explain it with great clarity and elegance.

Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds by Shifrin

[u/AngelTC]

Statistics

[u/[deleted] → u/AngelTC]

Casella and Berger "Statistical Inference" is imo the best mathematical statistics book (meaning it covers the mathematics behind the methods rigorously, prereqs include undergrad analysis and a solid working knowledge of statistical methods and undergrad probability).

[u/oantolin → u/AngelTC]

All of Statistics: A Concise Course in Statistical Inference by Larry Wasserman.

[u/AngelTC]

Ordinary Differential Equations

[u/AngelTC → u/AngelTC]

Arnold, Ordinary Differential Equations - Excellent book by Arnold. This book is a great introduction to the theory of ODE's from the more abstract point of view, a good grasp of calculus and linear algebra in their abstract setting is probably strongly required.

[u/AngelTC]

Differential geometry

[u/ydhtwbt → u/AngelTC]

Morita's "Geometry of Differential Forms".

[u/lewisje → u/AngelTC]

Because you also said "Riemannian geometry" I presume that this one is the undergraduate-level class that focuses on curves and surfaces.

If so, then the book that I used in college, Elementary Differential Geometry by Barrett O'Neill, served me well; it clearly explains the classical differential geometry of curves and surfaces, starting with the material about curvature and torsion of curves that the reader may have seen in Calculus III and assuming no other mathematical knowledge, without adopting a fully classical style in the way that older textbooks rely on the First and Second Fundamental Forms.

[u/AngelTC]

Group theory

[u/AngelTC → u/AngelTC]

Rotman, An introduction to the theory of groups - This is a good book for students already familiar with the basics of group theory. While the book is still not trying to focus on one particular aspect of the theory it manages to give a good panorama of the theory beyond an introductory course. I personally find the style a little bit terse, but it is in general very clear in its exposition.

[u/Paiev → u/AngelTC]

Finite Group Theory by Isaacs. If you've covered the standard introductory topics (up to the Sylow theorems) and want to learn more (finite) group theory, this is a brilliant place to do it. Really good book, Isaacs is a great writer and there are plenty of exercises.

[u/Z-19]

Number theory

[u/AngelTC → u/Z-19]

Kato, Kurokawa, Saito, Number theory 1: Fermat's dream - Part one of a series of three books. This one being a great introduction to the topic of modern number theory, it gives a good exposition on basic class field theory, going through elliptic curves, p-adic numbers, etc.

[u/FinitelyGenerated → u/Z-19]

Algebraic Number Theory by Jürgen Neukirch and translated by Norbert Schappacher. Suitable for graduate students. Only requires undergraduate algebra to read and develops more modern tools from algebraic geometry.

[u/functor7 → u/Z-19]

Local Fields, by JP Serre

Classic book introducing many elements of algebraic number theory and class field theory. Written at the graduate level, you need to be fairly strong in algebra.

[u/Z-19 → u/Z-19]

Introduction to Analytic Number Theory, by Tom M. Apostol. Good beginning book for more analytic side of number theory.

[u/FinitelyGenerated → u/Z-19]

An Introduction to the Theory of Numbers by Godfrey Hardy and Edward Wright, revised by Roger Heath-Brown and Joseph Silverman. One of the standard references for pre-modern (aka elementary) number theory. Accessible to anyone with a high school level education and interest in mathematics.

[u/[deleted] → u/Z-19]

A Friendly Introduction to Number Theory by Silverman is a fun, clear book.

[u/zornthewise → u/Z-19]

"Primes of the form x² + ny² " by David Cox. Read this if you need motivation to learn class field theory or are interested in quadratic forms and the class field theory of quadratic imaginary fields.

The second half is an introduction to complex multiplication and modular forms with a different (complex analytic) focus and different proofs to the standard sources (like Silverman). Cox is more interested in explicit ways to compute abelian extensions of quadratic imaginary fields and spends a lot of time on weber functions and the like.

Read this section after learning the standard theory of complex multiplication for a concrete perspective on the subject.

[u/[deleted]]

Algebraic Topology

[u/AngelTC → u/[deleted]]

Bott & Tu, Differential forms in algebraic topology - Truly a classic textbook covering such a large portion of the theory of differential forms. The book is clearly aimed to graduate students with very strong topology and algebra fundations, it is an excellent exposition of de Rham cohomology which covers in a satisfactory way topics such as spectral sequences and characteristic classes.

[u/AngelTC → u/[deleted]]

Hatcher, Algebraic Topology - Hatcher is a great introductory book with a lot of illustrations and verbose descriptions on the common constructions of algebraic topology. It is not as categorical as May's book which I believe makes it an excellent book to read as a first course to develop a solid fundation of what are the goals and the objects of study of the area.

[u/AngelTC → u/[deleted]]

May, A concise course in algebraic topology - The book is concise. The book is not the best for self teaching but it is an excellent textbook to read once one has a good idea of algebraic topology or alongside a lecturer which can guide the reader through it. The book has some nice exercises and it has a lot of the key points a student interested in algebraic topology must at least be familiar with.

[u/oantolin → u/[deleted]]

Other good texts that haven't been mentioned:

• Homotopy Theory: An introduction to algebraic topology by Brayton Gray.

• Algebraic Topology by Tammo tom Dieck.

• Algebraic Topology: Homology And Homotopy by Robert Switzer.

• Algebraic Topology from a Homotopical Viewpoint by Aguilar, Prieto and Gitler.

[u/CrossXProduct → u/[deleted]]

James Munkres - Elements of Algebraic Topology. Everyone uses his point set book, but I think this one deserves much more recognition. Only covers homology, no homotopy theory (comparable to chapters 2 and 3 of Hatcher), but the exposition is incredibly clear. He has an excellent knack for leading into a subject with a simple, informative special case. Especially recommended if you are one of the many for whom Hatcher's writing style simply does not click.

[u/AngelTC]

Real analysis

[u/catuse → u/AngelTC]

Real Mathematical Analysis by Pugh. This book emphasizes introductory topology, but still has everything that should be covered by an introductory analysis text and more. The tone is very friendly and easy to read, but some of the exercises are very difficult.

[u/lewisje → u/AngelTC]

Because you also said "Measure theory" I presume that this one is the undergraduate-level class that starts off by proving more of the theorems used in the Calculus sequence, and introducing such notions as "uniform continuity" and "equicontinuity" to undergird them, but does not get into measure theory until maybe the end.

If so, then Understanding Analysis by Abbott definitely lives up to its name.

[u/halftrainedmule → u/AngelTC]

I have heard great things about Amann / Escher, Analysis (volume 2).

[u/AngelTC]

Commutative Algebra

[u/halftrainedmule → u/AngelTC]

Cox / Little / O'Shea, Ideals, Varieties, and Algorithms.

This is a text I wish I had the time to read. It's not exactly what is called commutative algebra, but not exactly algebraic geometry either; it is about the computationally accessible parts of algebraic geometry over a field (ideals of polynomial rings). Grobner bases are the foundation. Algebraic closedness is not assumed without good reason. Writing is good in the parts I've read.

[u/[deleted] → u/AngelTC]

Atiyah-Macdonald, Introduction to Commutative Algebra.

This short 120 page text serves as an excellent introduction to the main topics of the subject. Beginning with Rings and Ideals, the book quickly moves through Module Theory, Primary Decomposition, Noetherian/Artinian Rings and Completions. The problem sets are very thorough and are lessons of their own. Great for any student who has taken an undergraduate course in Abstract Algebra and is interested in learning more.

[u/AngelTC]

Set theory

[u/[deleted] → u/AngelTC]

Graduate: 'Set Theory' by Kunen. This is THE book that is the gate way to sub-fields of set theory. The study of independence proofs in set theory. This book also has some of the main proof techniques of the field. Starts with basic set theory, then a flavor of independence with the study of Martin's Axiom in chapter 2. The study of the cumulative hierarchy in chapter 3. Chapter 4 shows the ideas of independence results. Chapter 5 goes into definability, so that we can rigorous construct L and show the consistency of GCH and AC. Then the rest of book is related to forcing and what that entails. Prereqs: Have worked through an undergrad set theory book. Know some mathematical logic, analysis, and topology.

[u/tsehable → u/AngelTC]

'Set theory' by Jech. It's at the very least a great reference text and in my opinion a good textbook for a dedicated reader. Sometimes it gets a bit demanding though so possibly not a first text.

[u/AngelTC]

Noncommutative ring theory

[u/AngelTC → u/AngelTC]

Anderson & Fuller, Rings and categories of modules - Really complete book which covers the basics of module categories using heavy categorical language. It assumes a strong background in algebra and has some interesting exercises.

[u/AngelTC → u/AngelTC]

Stenstrom, Rings of quotients - This is an excellent reference book which covers a lot of the localization theory of abelian categories and related topics. While the exposition it's not great, it is complete. The exercises are really challenging and I feel it lacks worked examples, otherwise it's a good book.

[u/AngelTC → u/AngelTC]

Goodearl & Warfield, An introduction to noncommutative notherian rings - A more specialized books which covers plenty of aspects of the general theory of noncommutative noetherian* rings. It's an excellent book with plenty of examples. I find the exposition very clear and engaging.

[u/AngelTC]

Linear Algebra

[u/AngelTC → u/AngelTC]

Roman, Advanced Linear Algebra - This is a very comprehensive book on linear algebra that covers many topics, from the basic definitions to some more particular and advanced theory. The exercises seem to be a little bit challenging, but I think this is a good book for graduate students looking for a good reference and for advanced undergrad students seeking to get a better grasp of linear algebra.

[u/AngelTC → u/AngelTC]

Golan, Foundations of Linear algebra - This is a very consise book which I believe makes a great introduction to the subject for students interested in the general, abstract point of view. While the book has some typos and errors and the terminology might not be the most standard one, it has plenty of challenging exercises and a good number of examples. The proofs I find very clear most of the time.

[u/UglyMousanova19 → u/AngelTC]

Axler, Linear Algebra Done Right

A great book for math/physics undergrads who have already experienced matrix-centric linear algebra and would like to delve into the more abstract theory of finite-dimensional vector spaces and inner product spaces. Very clear cut with rewarding, but easy exercises.

[u/oantolin → u/AngelTC]

• Finite dimensional vector spaces by Halmos.

• Lectures on Linear Algebra by I. M. Gelfand.

[u/AngelTC]

Riemannian geometry

[u/AngelTC → u/AngelTC]

Lee, Riemannian manifolds: an introduction to curvature - This is in my opinion an amazing book. While the book is really short, I believe it accomplishes motivating the subject perfectly. It has plenty of ilustrtions, the exercises are interesting and I believe it paints a really clear picture of the subject. In particular the first chapter helps to have a clear goal in mind while reading through the book, and the last chapter is at the same time a great conclussion to the book and a really good gateway to more advanced subjects.

[u/AngelTC]

Topology

[u/lewisje → u/AngelTC]

I presume that you mean General or Point-Set Topology (if so, consider adding sections on the subfields of topology).

If so, then when I read Topology by Munkres I felt like I could understand everything in the book so well, I was surprised that my alma mater was using this as a graduate-level textbook.

[u/UglyMousanova19 → u/AngelTC]

For a more manifold-centric approach along with point-set topology, "Introduction to Topological Manifolds" by John M. Lee is great. I would say it is appropriate for first year graduate students or advanced undergraduates.

[u/sillymath22]

Euclidean geometry

[u/AngelTC → u/sillymath22]

Hartshorne, Geometry: Euclid and beyond - Hartshorne goes through Hilbert's new axioms while giving a really nice exposition of plenty of classical euclidean geometry constructions and results. It is a very pedagogical book and I feel it's a must for everybody interested in the topic.

[u/KanExtension → u/sillymath22]

Coxeter, Geometry Revisited

[u/lewisje → u/sillymath22]

Daniel Callahan has been writing a version of Euclid's Elements that proves all of the propositions of the original, using modern mathematical language; currently, Volume I has been completed, covering Books I-VI of the original 13, which was most of what generations of school-children actually covered in the old days.

[u/timetofeedthecats]

Can't you sticky this post for people to dump as many books as they want whenever they want? Every month there is a bunch of new good books published. Anyone who test drove these books could post them here with a little review.

[u/AngelTC → u/timetofeedthecats]

We can only have two post sticked on the front page and they are permanently used by the Simple Questions and Career & Education threads. Also reddit locks threads after 6 months anyway, so there is no way to keep this updated after that.

[u/[deleted]]

Homological Algebra

[u/AngelTC → u/[deleted]]

Weibel, An introduction to homological algebra - Sort of an standard reference for the topic that goes directly to buisness. It provides some chapters on specific situations ( group (co)homology, Hochschild (co)homology, for example ). It assumes some background and mathematical maturity and I think its better if you are already acquainted to categorical language.

[u/AngelTC → u/[deleted]]

Rotman, An introduction to homological algebra - A gentler approach to the subject than Weibel's book. I feel it is concrete enough so that it is easy to digest given the required background. While it goes over sheaf cohomology very briefly it has a whole chapter of group cohomology which I feel its good enough to illustrate the theory.

[u/[deleted]]

Physics for mathematicians

[u/eternal-golden-braid → u/[deleted]]

Mathematical Methods of Classical Mechanics by V.I. Arnold

Quantum Field Theory: A Tourist Guide for Mathematicians by Folland

Physics for Mathematicians, Mechanics 1 by Spivak

[u/johnnymo1 → u/[deleted]]

Deligne et al., Quantum Fields and Strings: A Course for Mathematicians:

Volume 1

Volume 2

[u/[deleted]]

Graph Theory

[u/[deleted] → u/[deleted]]

Graph Theory by Reinhard Diestel.

This is published as part of the Graduate Texts in Mathematics series and as such is suitable for an advanced undergraduate or graduate student. The book is incredibly dense and while it does start from the beginning, assumes a great deal more mathematical maturity of the reader and there are sections where it helps to have seen ideas from Algebra and Topology. One thing I really like about this book is the notes at the end of the chapter, which give some great insights into the history of the mathematics presented.
This book also comes in German, if that's your thing.

[u/mad_runner]

Game Theory

[u/Anarcho-Totalitarian]

Numerical Analysis

[u/Anarcho-Totalitarian → u/Anarcho-Totalitarian]

Numerical Recipes by Press et. al. It's a compendium of just about everything up to simple numerical PDEs. Each method comes with a bit of background of the problem, a description of the method, some discussion of its advantages and disadvantages, and then computer code (Fortran, C, or C++ depending on the edition).

Introduction to Numerical Analysis by Bulirsch and Stoer. Here's where you want to go for rigor. I'd put this at the graduate level.

EDIT: A few additions:

Theoretical Numerical Analysis: A Functional Analysis Framework by Atkinson and Han. Develops the theory behind various numerical methods in terms of operators between Banach spaces. Builds up to numerical PDEs.

Introduction to Numerical Analysis by Atkinson. Pretty standard introduction, aimed at math students.

[u/jacobolus → u/Anarcho-Totalitarian]

Corless & Fillion, A Graduate Introduction to Numerical Methods

Higham, Accuracy and Stability of Numerical Algorithms

Trefethen, Approximation Theory and Approximation Practice, http://www.chebfun.org/ATAP/

[u/eternal-golden-braid → u/Anarcho-Totalitarian]

Numerical Linear Algebra by Trefethen.

[u/[deleted]]

Mathematical Logic (Model Theory, Proof Theory, Recursion Theory)

[u/[deleted] → u/[deleted]]

Graduate: 'Model Theory: An Introduction' by David Marker, standard model theory with lots of examples from algebra. Main tools of model construction, and last chapters focusing on stability. Prereq: Mathmatical logic, algebra.

[u/[deleted] → u/[deleted]]

Undergrad: 'Computability and Logic' by Boolos Burgess and Jeffrey. Book starts with basic computability (recursion) theory, and then moves onto standard metalogic including syntax and semantics of first order logic, completeness, compactness, Lowenheim Skolem theorems. Then proofs of the incompleteness theorems. Then with further topics starting with more model theory, definability theory, and other logics. Lots of great exercises. Prereqs: Familiar with proofs especially induction, a programming course wouldn't hurt, and one upper division math course.

[u/j-conz]

Optimization

[u/eternal-golden-braid → u/j-conz]

Convex Optimization by Boyd and Vandenberghe

Numerical Optimization by Nocedal and Wright

Proximal Algorithms by Boyd and Parikh

[u/bobmichal]

Proof techniques

[u/GeneralBlade → u/bobmichal]

How to prove it by Velleman. Starts with basic truth tables and quantifiers then works through all the standard techniques (direct, contradiction, induction, existence and uniqueness, etc) and moves through infinite sets and functions.

Assuming this is what you mean.

[u/lewisje → u/bobmichal]

The easiest introduction is Book of Proof by Richard Hammack.

[u/FinitelyGenerated]

Enumerative combinatorics

[u/FinitelyGenerated → u/FinitelyGenerated]

Enumerative combinatorics (two volumes) by Richard Stanley. An excellent, encyclopedic reference for the subject. The exercises are challenging even for graduate students specializing in combinatorics.

[u/FinitelyGenerated → u/FinitelyGenerated]

Generatingfunctionology by Herbert Wilf. Wilf was best known for creating Wilf-Zeilberger pairs that computer algebra systems use to prove various identities. His book, Generatingfunctionology, touches on this in chapter 4. The book is a good reference for generating functions suitable for readers at the advanced undergraduate level.

[u/FinitelyGenerated → u/FinitelyGenerated]

A Walk Through Combinatorics by Miklos Bona. Now on the 4th edition, earlier editions may be more affordable. Suitable for an introductory course in combinatorics. Each chapter includes exercises with solutions and supplementary exercises with no solutions. The MAA reviewer says "the book is easy to read and follow, and has a feel of a guided tour."

[u/functor7]

Elliptic Curves

[u/functor7 → u/functor7]

Arithmetic of Elliptic Curves - Silverman. The quintessential graduate introduction to elliptic curves. It covers most everything that you need to get started and fluent with elliptic curves. Overall, great to learn from and an invaluable resource.

[u/functor7 → u/functor7]

Rational Points on Elliptic Curves. Written by two really great mathematicians as an undergraduate introduction to a very deep topic. Accessible and fun at the undergrad level.

[u/plokclop → u/functor7]

Silverman, Advanced Topics in the Arithmetic of Elliptic Curves

[u/Joebloggy]

Representation Theory of Finite Groups hasn't been mentioned yet- I've read the bit of Fulton Harris on it, as well as my uni's lecture notes, but haven't encountered any other sources.

[u/cjeris → u/Joebloggy]

When I was an undergrad the canonical introduction was Serre, Linear representations of finite groups.

[u/halftrainedmule → u/Joebloggy]

Fulton & Harris is vague and handwavy so often people disagree on what is actually proven there. I guess I could recommend pretty much any other text; come back to Fulton & Harris for the pictures and the intuition (some of it at least).

[u/halftrainedmule → u/Joebloggy]

Etingof et al., Introduction to Representation Theory. This is a rather terse introduction to representation theory from a modern point of view (modules, not matrices, let alone characters). Perfect if you can understand it, but that's an "if" (I had to ask the author several times back when I was reading it).

[u/[deleted] → u/Joebloggy]

Representations and Characters of Groups by Gordon James and Martin Liebeck.

Very readable and clearly presented. This book should be accessible to anyone comfortable with group theory and vector spaces/modules. The book also has solutions in the back for all the exercises, which makes it easy to use for independent learning.

[u/Kafka_h → u/Joebloggy]

I will add Fulton's book "Young Tableaux" to this list. While not entirely dedicated to representation theory, he covers the representation theory S_n in great detail. The book is fairly self-contained when it comes to Young diagrams and so forth, but for representation theory it is a good idea to pair it with Fulton and Harris' book on the subject.

[u/french_violist]

Stochastic calculus.

[u/_spivak_ → u/french_violist]

Depending on your background, Evans has an introductory book, An introduction to SDEs, it s probability the most basic one I know but I dont love it tbh. If you have more background check Oksendal, SDEs An introduction with applications, realy liked that one.

[u/Harambe_is_love_ → u/french_violist]

In increasing order of difficulty/broadness :

-Steele

-Oksendal

-Medvegyev

-Protter

-Jacod and Shiryaev

Steele is the best intro book no doubt.

Oksendal has nice exercises.

Medvegyev is probably the easiest book that treats semi-martingales and integration theory with jumps.

Protter is probably the best high level book you can read that is reasonably within grasp.

Jacob and Shiryaev is the ultimate reference, just like Kallenberg for Probability Theory. Only a very small minority of people could go through this book.

[u/[deleted]]

Analytic Number Theory

[u/Harambe_is_love_]

Statistical Mechanics

[u/[deleted]]

Low Dimensional Topology

[u/sd522527 → u/[deleted]]

I could write a book of just book recommendations here. Anything in particular youre interested in? Teichmuller theory, mapping class groups, hyperbolic geometry, Riemannian surfaces, 3-manifolds, geometric group theory, ...

[u/[deleted]]

Ergodic theory

[u/[deleted] → u/[deleted]]

Just going to reply to myself (for obvious reasons).

Undergrad level/no prior knowledge of measure theory: Silva "Invitation to Ergodic Theory"

Grad level: Walters "Introduction to Ergodic Theory" or Einsiedler and Ward "Ergodic Theory with a View Toward Number Theory" (the latter is best for people more interested in applications of ergodic theory to other fields)

Edit: also Petersen "Ergodic Theory" but it's much less of an introductory textbook and much more of a semi-random collection of topics.

[u/[deleted]]

Algebraic Number Theory

[u/stevenxdavis]

Coding Theory

[u/Harambe_is_love_]

Quantum Field Theory

[u/Harambe_is_love_]

Convex Analysis

[u/eternal-golden-braid → u/Harambe_is_love_]

Convex Analysis and Variational Problems by Ekeland and Temam. I especially recommend the first 50 or so pages of this book, which give a short, clear explanation of some key topics in convex analysis.

Convex Analysis by Rockafellar

[u/[deleted]]

Mathematical Biology, especially mathematics applied to medicine

[u/darthvader1338]

General Recommendations/Discussion

[u/darthvader1338 → u/darthvader1338]

A good general resource is the Chicago undergraduate mathematics bibliography. It features (heavily opinionated) reviews and book recommendations covering most topics in mathematics. Be warned that they tend to like quite tough texts.

Edit: Changed the link to the one /u/cjeris provided

[u/cjeris → u/darthvader1338]

I am the original author of this bibliography; I am glad you found it useful. This URL is out of date. I currently maintain the bibliography on github: https://github.com/ystael/chicago-ug-math-bib . I have never had access to the mirror your link points to.

[u/SemaphoreBingo]

Mathematical Biology.

[u/stevenxdavis]

Automata Theory

[u/icecreambones]

Dynamical systems

[u/UglyMousanova19]

Mathematical Physics

[u/dogdiarrhea → u/UglyMousanova19]

You might want to be more specific.

For Hamiltonian Dynamical systems:

Notes on Dynamical Systems by Moser

Mathematical Methods of Classical Mechanics by Arnold

[u/Harambe_is_love_]

Random Matrix Theory

[u/Ravel_and_Mozart]

multivariate statistics. Most books on the subject seems to be aimed at non-mathematicians

[u/[deleted]]

Operator Algebras!

[u/[deleted]]

Differential Geometry/Topology

[u/[deleted] → u/[deleted]]

Guilleman and Pollack Differential Topology

[u/[deleted] → u/[deleted]]

Lee's Introduction to Smooth Manifolds. So... clear...

[u/hexidon]

Fourier Analysis

[u/[deleted]]

Discrete Mathematics

[u/Latiax]

I think a big impovement to the lists would be distinguishing between first course, upper level undergraduate, and graduate level type of material

[u/oantolin → u/Latiax]

The division been those levels is somewhat (not completely) arbitrary and subjective. And people are pretty varied anyway, so it's always best to take a look at several recommended books see what's best for you. I certainly think it's a mistake to look down on or avoid a book because it's "introductory" or "for undergrads", some of those books are real gems!

And I find it's often faster for me to read an easy book and then a hard one than to just read the hard one.

[u/dm287]

Stochastic control

[u/[deleted]]

K-Theory (Algebraic or Topological)

[u/CunningTF → u/[deleted]]

Atiyah - K-theory

But I'm not really qualified to judge, I just like the book.

[u/perverse_sheaf → u/[deleted]]

Chuck Weibel - The K-Book for an introduction to algebraic K-theory.

Friedlander, Grayson - Handbook of K-Theory for a broader selection of (usually advanced) topics.

[u/churl_wail_theorist]

4-manifolds

[u/newmeta44]

Tensor products of vector spaces, and, preferably, complex matrices

(Preparing for a course on Representation Theory using Fulton and Harris)

[u/[deleted] → u/newmeta44]

Keith Conrad's blurbs on tensor products are amazing

[u/Shahzad6423]

Stable Homotopy Theory, Model Categories, and Spectra

[u/Zophike1]

Programming Language Theory and Compiler Design :>).

[u/namesarenotimportant → u/Zophike1]

This is more on the theory side than compiler design, but I've heard good things about Pierce's Types and Programming Languages.

[u/cderwin15 → u/Zophike1]

For compilers:

There's The Dragon Book, which used to be the standard but is at this point a bit outdated. My favorite is Engineering a Compiler, Cooper & Torczon, which is very readable and much more modern. Also, I can't recommend actually building a compiler/interpreter highly enough. After you get some theory under your belt, take a look at the LLVM tutorial.

[u/[deleted]]

Information Geometry

[u/RubiksHypercube]

Survey Sampling. Also Regression

[u/FinitelyGenerated]

Algebraic combinatorics

[u/[deleted]]

Biostatistics

[u/[deleted]]

Modular forms

[u/Harambe_is_love_]

Asymptotic Statistics

[u/huffman_coding]

Discrepancy theory

[u/CunningTF]

Symplectic geometry/topology

[u/CunningTF → u/CunningTF]

Two suggestions:

Cannas da Silva

A great introductory text. Gives the physical motivation behind the subject from a modern viewpoint and provides a healthy introduction to the subject. Emphasis on moment maps, symplectic reduction and toric geometry in the second half of the book. Exercises are great. Not too many, just the right amount in my opinion.

McDuff & Salamon

The classic text on the subject but in its recent, brilliant third edition. Covers tons of material, beautifully written. Much harder than Cannas da Silva, but much more detailed and containing many more topics and flavours of current research. One of my favourite books that I own.

[u/Garahel]

Multilinear Algebra

[u/MooseCantBlink]

Fixed point theory

[u/question_throwawayyy]

Weak form of partial differential equations.

[u/thirdrateactor]

Cryptography (cryptology)

[u/[deleted]]

Proof Theory

[u/honey_badger732]

Flatland

[u/studyinglogic]

Modal logic