This tag is for questions where the poster seeks references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.

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138
votes
37answers
11k views

Fun but serious mathematics books to gift advanced undergraduates.

I am looking for fun, interesting mathematics textbooks which would make good studious holiday gifts for advanced mathematics undergraduates or beginning graduate students. They should be serious but ...
137
votes
4answers
9k views

The Integral that Stumped Feynman?

In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods ...
128
votes
14answers
22k views

Is computer science a branch of mathematics?

I have been wondering, is computer science a branch of mathematics? No one has ever adequately described it to me. It all seems very math-like to me. My second question is, are there any books about ...
77
votes
29answers
24k views

Best book ever on Number Theory

Which is the single best book for Number Theory that everyone who loves Mathematics should read?
71
votes
4answers
2k views

Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: ...
66
votes
15answers
12k views

Mathematical equivalent of Feynman's Lectures on Physics?

I'm slowly reading through Feynman's Lectures on Physics and I find myself wondering, is there an analogous book (or books) for math?
66
votes
7answers
3k views

A Case Against the “Math Gene”

I'm currently teaching a mathematics course for elementary educators (think of it as math methods, but with less focus on methods and more focus on content). In a student's essay, I encountered the ...
64
votes
28answers
9k views

What is the single most influential book every mathematician should read?

If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?
64
votes
22answers
5k views

The Best of Dover Books (a.k.a the best cheap mathematical texts)

Perhaps this is a repeat question -- let me know if it is -- but I am interested in knowing the best of Dover mathematics books. The reason is because Dover books are very cheap and most other books ...
62
votes
23answers
11k views

Complete course of self-study

I am about $16$ years old and I have just started studying some college mathematics. I may never manage to get into a proper or good university (I do not trust fate) but I want to really study ...
59
votes
3answers
2k views

Is there a definitive guide to speaking mathematics?

Is there a definitive guide to speaking mathematics to avoid ambiguity? I'm writing a program to generate text for a variety of mathematical expressions and would like to code it so that it adheres ...
57
votes
7answers
4k views

Why do books titled “Abstract Algebra” mostly deal with groups/rings/fields?

As a computer science graduate who had only a basic course in abstract algebra, I want to study some abstract algebra in my free time. I've been looking through some books on the topic, and most seem ...
57
votes
5answers
2k views

Defining a manifold without reference to the reals

The standard definition I've seen for a manifold is basically that it's something that's locally the same as $\mathbb{R}^n$, without the metric structure normally associated with $\mathbb{R}^n$. ...
56
votes
5answers
2k views

What are the issues in modern set theory?

This is spurred by the comments to my answer here. I'm unfamiliar with set theory beyond Cohen's proof of the independence of the continuum hypothesis from ZFC. In particular, I haven't witnessed ...
55
votes
8answers
12k views

Teaching myself differential topology and differential geometry

I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology. I have decided to fix this lacuna once for ...
54
votes
4answers
6k views

Books that every student “needs” to go through

I'm still a student, but the same books keep getting named by my tutors (Rudin, Royden). I've read Baby Rudin and begun Royden though I'm unsure if there are other books that I "should" be working on ...
51
votes
17answers
11k views
47
votes
3answers
1k views

Paul Erdos's Two-Line Functional Analysis Proof

Legends hold that once upon a time, some mathematicians were rather pleased about a 30-ish page result in functional analysis. Paul Erdos, upon learning of the problem, spent ten or so minutes ...
46
votes
46answers
4k views

What was the book that opened your mind to the beauty of mathematics?

Of course, I am generalising here. It may have been a teacher, a theorem, self pursuit, discussions with family / friends / colleagues, etc. that opened your mind to the beauty of mathematics. But ...
46
votes
5answers
4k views

Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them ...
45
votes
2answers
2k views

Sheaf cohomology: what is it and where can I learn it?

As I understand it, sheaf cohomology is now an indispensable tool in algebraic geometry, but was originally developed to solve problems in algebraic topology. I have two questions about the matter. ...
45
votes
1answer
852 views

A fleshed-out version of the Noncommutative Geometry proof of the Gauss-Bonnet Theorem?

In Connes's book on noncommutative geometry, he outlines a rather short "algebraic" proof of the Gauss-Bonnet theorem that uses multilinear forms. (Start reading on page 19 of the book) This is given ...
44
votes
2answers
1k views

Reference request for tricky problem in elementary group theory

The following could have shown up as an exercise in a basic Abstract Algebra text, and if anyone can give me a reference, I will be most grateful. Consider a set $X$ with an associative law of ...
43
votes
23answers
15k views

What is a good complex analysis textbook?

I'm out of college, and trying to learn complex analysis on my own. I took out Ahlfors' text from the library, but I'm finding it difficult. Any textbook recommendations? I'm probably at an ...
43
votes
3answers
6k views

phd qualifying exams

Where can I find phd qualifying exams questions.Is there any website that keeps a collection of such problems? I need it for doing some revision of the basic topics.I know of a book but that do not ...
41
votes
13answers
5k views

Interesting math-facts that are visually attractive

To give a talk to 17-18 years old (who have a knack for mathematics) about how interesting mathematics (and more specifically pure mathematics) can be, I wanted to use nice facts accompanied by nice ...
40
votes
4answers
4k views

String Theory: What to do?

This is going to be a relatively broad/open-ended question, so I apologize before hand if it is the wrong place to ask this. Anyways, I'm currently a 3rd year undergraduate starting to more seriously ...
40
votes
0answers
1k views

Theorem that von Neumann proved in five minutes.

In "How To Solve It", George Pólya writes: "There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it ...
38
votes
25answers
14k views

Best Maths books for non-mathematicians

I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often ...
37
votes
4answers
2k views

Why can't we define more elementary functions?

$\newcommand{\lax}{\operatorname{lax}}$ Liouville's theorem is well known and it asserts that: The antiderivatives of certain elementary functions cannot themselves be expressed as elementary ...
36
votes
7answers
3k views

Open math problems which high school students can understand

I request people to list some moderately and/or very famous open problems which high school students,perhaps with enough contest math background, can understand, classified by categories as on ...
35
votes
5answers
5k views

Completion of rational numbers via Cauchy sequences

Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences?
35
votes
5answers
1k views

Understanding the Laplace operator conceptually

The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it? Any good essays (combining both history and ...
35
votes
13answers
2k views

Is there a good “bridge” between high school math and the more advanced topics?

I love math and would like to know more of it. However, whenever I try to pick up a book on what I consider to be "advanced" mathematical topics, I often have a hard time understanding some of the ...
34
votes
2answers
1k views

Small primes attract large primes

$$ \begin{align} 1100 & = 2\times2\times5\times5\times11 \\ 1101 & =3\times 367 \\ 1102 & =2\times19\times29 \\ 1103 & =1103 \\ 1104 & = 2\times2\times2\times2\times ...
33
votes
10answers
7k views

Best Algebraic Geometry text book? (other than Hartshorne)

Lifted from Mathoverflow: I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, ...
33
votes
2answers
2k views

Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$?

Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$? In some sense, this is a follow-up to my answer to this question where the non-isomorphism between the spaces $L^r$ ...
32
votes
18answers
4k views

What should be in every grad student's library?

Now I'm not a graduate student, but I was hoping to compile a list of what may be considered definitive texts in various branches of mathematics. I'm curious to what books are considered good and ...
32
votes
6answers
1k views

Original works of great mathematicians

In almost every mathematical text there is a line as This was first proved by Gauss or This formula first appeared in a work of Riemann, but for me it's more like My friend told me once that... For ...
32
votes
0answers
2k views

Continuous projections in $\ell_1$ with norm $>1$

I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case ...
31
votes
15answers
9k views

Good book for self study of functional analysis

I am a EE grad. student who has had one undergraduate course in real analysis (which was pretty much the only pure math course that I have ever done). I would like to do a self study of some basic ...
31
votes
5answers
5k views

(Theoretical) Multivariable Calculus Textbooks [duplicate]

(Note that I have used bold text frequently simply to highlight the key points of my question for those who do not have the time to read through it thoroughly (it is not very long, however); I hope ...
31
votes
4answers
5k views

The Langlands program for beginners

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things he must know before he can understand the Langlands program and its ...
30
votes
14answers
14k views

Good book for self study of a First Course in Real Analysis

Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction ...
30
votes
2answers
706 views

Fekete's conjecture on repeated applications of the tangent function

A high-school student named Erna Fekete made a conjecture to me via email three years ago, which I could not answer. I've since lost touch with her. I repeat her interesting conjecture here, in case ...
29
votes
15answers
5k views

Good Book On Combinatorics

What is your recommendation for an in-depth introductory combinatoric book? A book that doesn't just tell you about the multiplication principle, but rather shows the whole logic behind the questions ...
29
votes
10answers
4k views

A really complicated calculus book

I've been studying math as a hobby, just for fun for years, and I had my goal to understand nearly every good undergraduate textbook and I think, I finally reached it. So now I need an another goal. ...
29
votes
8answers
2k views

How to use math textbooks

I'm a higher schooler who was recently gifted a book by my teacher (Schaum's outline of advanced calculus) which is really awesome and I've started working my way through it. I have run into a ...
29
votes
2answers
1k views

Image of a math problem that was stated in Cuneiform, Arabic, Latin and Finally in modern math notation

Many years ago a lecturer of mine had a photocopy of a page from a book containing a math problem ( I think it was a simple quadradic equation ) that was stated/solved in Cuneiform, Arabic, Latin ...
29
votes
2answers
1k views

Is this function decreasing on $(0,1)$?

While doing some research I got stuck trying to prove that the following function is decreasing $$f(k):= k K(k) \sinh \left(\frac{\pi}{2} \frac{K(\sqrt{1-k^2})}{K(k)}\right)$$ for $k \in (0,1)$. ...