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[Book recommendation] questions are quite popular on this website, which is, at least for me, one of the best places to get useful and insightful suggestions because it gathers a great number of experts in different areas of mathematics.

In my opinion, it would be very beneficial for the organization of the site (and for the many undergraduate students searching for guidance when facing a new course or looking for a good learning roadmap -- like myself) to make one thread that collects a big "Mathematics Stack Exchange Undergraduate Mathematics Bibliography" divided by categories just like the nice one proposed here, but surely more comprehensive (given the much larger number of contributors).

This thread should collect sparse material already available on the website but hard to find among the numerous questions and also new inputs. Ideally each entry should be briefly commented with matter-of-fact remarks.

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  • $\begingroup$ Is it a good idea to collect all recommendations in a single answer? This short circuits some of the most useful stack exchange functionality, such as upvoting good answers. $\endgroup$
    – littleO
    Commented Dec 27, 2014 at 14:36
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    $\begingroup$ @littleO I know, but having 100 answers would make the list much less organized. What do you propose to do? $\endgroup$
    – Dal
    Commented Dec 27, 2014 at 14:37
  • $\begingroup$ @littleO For example, it is always possible to add a little remark on an entry which is, for some reasons, considered by common agreement "no good" pointing out the exact objective reasons. $\endgroup$
    – Dal
    Commented Dec 27, 2014 at 14:41
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    $\begingroup$ Personally I suspect that it's more useful to have separate threads for separate subjects. We can find the relevant threads easily by googling. Perhaps we should have one master thread that collects links to threads for different subjects. However it's quite possible that many people will find this list useful and there's no harm in making it. $\endgroup$
    – littleO
    Commented Dec 27, 2014 at 15:07
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    $\begingroup$ @JackD'Aurizio I don't think so, because, even though it is a little bit broader than the average book recommendation question, it is still a question on finding a suitable learning roadmap and bibliography. $\endgroup$
    – Dal
    Commented Dec 28, 2014 at 12:18

1 Answer 1

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Please, add your contributions to this answer (which I made community wiki). In case you wish to add some personal comments about your entries, add a tag like [Your_Username].


Foundations

Daniel J. Velleman, How to Prove It: A Structured Approach

Problem solving

  1. G. Polya, How to Solve It: A New Aspect of Mathematical Method

Recreational

Algebra

  • Roger Godement , Algebra (Translation of Cours d' Algebre)
  • Lang - Algebra
  • Jacobson - Basic Algebra I, Basic Algebra II
  • Rotman - Introduction to the Theory of Groups
  • Dummit and Foote - Algebra
  • Hungerford - Abstract Algebra
  • Serre - Linear Representations of Finite Groups
  • Fulton, Harris - Representation Theory - A First Course
  • Humphreys - Introduction to Lie Algebras and Representation Theory
  • Atiyah, MacDonald - Introduction to Commutative Algebra
  • Eisenbud - Commutative Algebra: with a View Toward Algebraic Geometry
  • Weibel - An Introduction to Homological Algebra

Linear algebra

  • Sergei Treil, *Linear Algebra Done Wrong*
  • Roman - Advanced Linear Algebra

Geometry

  • Mumford - The Red Book of Varieties and Schemes

Point-set topology

Differential geometry

  • John Lee, *Introduction to Smooth Manifolds *

Number theory

  • Andre Weil, *Basic Number Theory*
  • Ireland and Rosen - A Classical Introduction to Moder Number Theory
  • Edmund Landau - Introduction to Number Theory
  • Burton - Elementary Number Theory

Combinatorics and discrete mathematics

Probability and Statistics

Real analysis

  • Gerald B. Folland, *Real Analysis: Modern Techniques and Their Applications*
  • Shilov and Gurevich: Integral, Measure and Derivative.

Multivariable calculus

Complex analysis

Differential equations

Functional analysis

Mathematical and theoretical physics

  • Arnold - Mathematical Methods of Classical Mechanics
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    $\begingroup$ for undergraduate real analysis. Folland ? $\endgroup$
    – Airbag
    Commented Dec 28, 2014 at 12:06
  • $\begingroup$ Some schools in China, Japan, Germany, Russia and UK use Folland, adult Rudin or Royden in undergrad as far as I know. $\endgroup$ Commented Dec 28, 2014 at 12:18
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    $\begingroup$ @AranKomatsuzaki In addition, such topics are covered in your 2nd and 3rd year of university in my country. So that book is totally fine (and, by the way, I like it). $\endgroup$
    – Dal
    Commented Dec 28, 2014 at 12:19

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