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.
106
votes
35answers
7k 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 ...
104
votes
3answers
5k 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 ...
56
votes
3answers
1k 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 ...
55
votes
25answers
5k 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?
51
votes
7answers
2k 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 ...
50
votes
14answers
7k 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?
49
votes
5answers
1k 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 $\mathbb{R}^n$. ...
48
votes
7answers
3k 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 ...
47
votes
24answers
10k views
Best book ever on Number Theory
Which is the single best book for Number Theory that everyone who loves Mathematics should read?
47
votes
21answers
4k 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 ...
43
votes
17answers
2k 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 ...
43
votes
1answer
811 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 ...
41
votes
3answers
630 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:
...
40
votes
5answers
1k 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 ...
40
votes
3answers
895 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 ...
32
votes
7answers
2k 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 ...
32
votes
8answers
6k 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 ...
31
votes
5answers
3k 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?
31
votes
2answers
652 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 ...
30
votes
19answers
5k views
Good book/lecture notes about category theory
what are the best books or lecture notes on category theory?
30
votes
4answers
2k 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 ...
30
votes
2answers
1k 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 $L^r$ ...
29
votes
13answers
1k 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 ...
29
votes
6answers
716 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 ...
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 $k \in (0,1)$.
...
29
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 ...
28
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 ...
28
votes
1answer
684 views
Is there an atlas of Algebraic Groups and corresponding Coordinate rings?
I was wondering if there was a resource that listed known algebraic groups and their corresponding coordinate rings.
Edit: The previous wording was terrible.
Given an algebraic group $G$, with Borel ...
27
votes
4answers
2k 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 ...
27
votes
4answers
1k 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 ...
26
votes
10answers
1k views
Online resources for learning Mathematics
Not sure if this is the place for it, but there are similar posts for podcasts and blogs, so I'll post this one. I'd be interested in seeing a list of online resources for mathematics learning.
As ...
26
votes
1answer
564 views
A Universal Property Defining Connected Sums
I once read (I believe in Ravi Vakil's notes on Algebraic Geometry) that the connected sum of a pair of surfaces can be defined in terms of a universal property. This gives a slick proof that the ...
25
votes
21answers
5k 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 ...
25
votes
14answers
3k 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 ...
25
votes
2answers
564 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 ...
25
votes
1answer
1k 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.
...
24
votes
10answers
4k views
Good books on mathematical logic?
I just started to learn mathematical logic. I'm a graduate student. I need a book with relatively more examples. Any recommendation?
24
votes
5answers
1k 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 ...
24
votes
3answers
1k views
Asymptotic (divergent) series
MOTIVATION. After having read in detail an article by Alf van der Poorten I read a very short paper by Roger Apéry. I am interested in finding a proof of a series expansion in the latter, which is in ...
23
votes
28answers
2k views
Classical texts that should not be missing from any shelf [closed]
It seems to me as if many modern texts are rather streamlined. They are designed not to expect too much from the reader but they often miss the depth of respective classical literature.
The purpose ...
23
votes
13answers
4k 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 ...
23
votes
5answers
1k views
Is $\pi$ more transcendent than $e$?
I’ve often wondered about this, and I conjecture the affirmative, based mainly on that it is so much easier to prove the transcendence of $e$ than that of $\pi$.
I would be surprised if, just as ...
23
votes
3answers
1k views
Ramanujan's First Letter to Hardy and the Number of $3$-Smooth Integers
A positive integer is $B$-smooth if and only if all of its prime divisors are less than or equal to a positive real $B$. For example, the $3$-smooth integers are of the form $2^{a} 3^{b}$ with ...
23
votes
5answers
1k views
Help understanding Algebraic Geometry
I while ago I started reading Hartshorne's Algebraic Geometry and it almost immediately felt like I hit a brick wall. I have some experience with category theory and abstract algebra but not with ...
23
votes
8answers
889 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 ...
23
votes
4answers
2k views
(Theoretical) Multivariable Calculus Textbooks
(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 ...
23
votes
2answers
452 views
English words in written mathematics
I recently marked over $100$ assignments for a multivariable calculus course. One question which a lot of people did poorly was proving a given set was open. Aside from issues relating to rigour and ...
22
votes
1answer
253 views
Is the Galois group associated to a random polynomial solvable with probability 0?
Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$.
Let $r_1,\ldots,r_n$ be the roots of $P$ and consider ...
22
votes
1answer
970 views
History of “Show that $44\dots 88 \dots 9$ is a perfect square”
The problem
Show that the sequence, $49, 4489, 444889, \dots$, gotten by inserting
the digits $48$ in the middle of the
previous number (all in base $10$), consists only of
perfect squares.
...
22
votes
2answers
997 views
When did Fubini's name get applied to the theorem without measures?
Fubini's theorem, from 1907, expresses integration with respect to a product measure in terms of iterated integrals. The simpler version of this theorem for multiple Riemann integrals was used long ...
