**152**

votes

**89**answers

26k views

### Not especially famous, long-open problems which anyone can understand

Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list ...

**57**

votes

**13**answers

16k views

### Top specialized journals

In geometry/topology, there are (at least) three specialized journals that end up publishing a large fraction of the best papers in the subject -- Geometry and Topology, JDG, and GAFA.
What journals ...

**22**

votes

**3**answers

732 views

### Graduate program applications that require questionnaires and other non-letter material

In the December 2014 AMS Notices, a letter to the editor (http://www.ams.org/notices/201411/rnoti-p1311.pdf) by Deconinck and Medlock addresses the problem of (math) graduate programs requiring letter ...

**41**

votes

**46**answers

14k views

### An example of a beautiful proof that would be accessible at the high school level?

The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about ...

**100**

votes

**20**answers

22k views

### Examples of unexpected mathematical images

I try to generate a lot of examples in my research to get a better feel for what I am doing. Sometimes, I generate a plot, or a figure, that really surprises me, and makes my research take an ...

**105**

votes

**64**answers

14k views

### Suggestions for good notation

I occasionally come across a new piece of notation so good that it makes life easier by giving a better way to look at something. Some examples:
Iverson introduced the notation [X] to mean 1 if X is ...

**74**

votes

**50**answers

14k views

### Counterexamples in Algebra?

This is certainly related to "What are your favorite instructional counterexamples?", but I thought I would ask a more focused question. We've all seen Counterexamples in Analysis and Counterexamples ...

**9**

votes

**18**answers

17k views

### Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?

**126**

votes

**36**answers

32k views

### Widely accepted mathematical results that were later shown wrong?

I wonder if there are any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time ...

**16**

votes

**5**answers

1k views

### How and how much do the notations and diagrams influence our understanding of mathematical concepts?

How and how much do the notations and diagrams influence
our understanding of mathematical
concepts?
This question was stimulated by the MathOverflow questions Thinking and Explaining and ...

**86**

votes

**33**answers

51k views

### Best Algebraic Geometry text book? (other than Hartshorne)

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, webpage, etc.
One suggestion ...

**116**

votes

**29**answers

36k views

### Real-world applications of mathematics, by arxiv subject area?

What are the most important applications outside of mathematics of each of the major fields of mathematics? For concreteness, let's divide up mathematics according to arxiv mathematics categories, ...

**88**

votes

**97**answers

55k views

### Famous mathematical quotes [closed]

Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of?
Standard community wiki rules apply: one ...

**45**

votes

**13**answers

6k views

### Contest problems with connections to deeper mathematics.

Hi,
I already posted this on math.stackexchange, but I'm also posting it here because I think that it might get more and better answers here! Hope this is okay.
We all know that problems from, for ...

**14**

votes

**19**answers

17k views

### Good books on problem solving / math olympiad

Hello,
I would want all book tips you could think of regarding Problem solving and books in general, in elementary mathematics, with a certain flavour for "advanced problem solving". An example would ...

**5**

votes

**7**answers

8k views

### Pseudo-random number generation algorithms

What algorithms are used in modern and good-quality random number generators?

**7**

votes

**3**answers

1k views

### The resolution of which conjecture/problem would advance Mathematics the most? [closed]

This is an impossibly broad question, and makes the unwarranted assumption that Mathematics is a uniform field. It might make more sense to ask the same question restricted to, say, Mathematical ...

**28**

votes

**19**answers

6k views

### Interesting applications (in pure mathematics) of first-year calculus

What interesting applications are there for theorems or other results studied in first-year calculus courses?
A good example for such an application would be using a calculus theorem to prove a ...

**36**

votes

**15**answers

7k views

### Is there a nice application of category theory to functional/complex/harmonic analysis?

[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.]
I've read looked at the examples in ...

**430**

votes

**186**answers

110k views

### Examples of common false beliefs in mathematics

The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...

**-3**

votes

**1**answer

121 views

### Decidable theorem or result that is not weaker than Tarski's theorem

I am wondering what other decidable theorem or results that is not weaker or stronger than Tarski's theorem.
Could any one give reference or a simple introduction about such result known in their ...

**81**

votes

**28**answers

14k views

### Are there other nice math books close to the style of Tristan Needham?

Hello, I've been very positively impressed by Tristan Needham's book "Visual Complex Analysis", a very original and atypical mathematics book which is more oriented to helping intuition and insight ...

**0**

votes

**1**answer

313 views

### Applications of the natural bilinear forms on the direct sum between a vector space and its dual

As is known, the vector space $V\oplus V^\ast$ admits the natural symmetric and skew-symmetric bilinear forms
$$\langle X+\xi,Y+\eta\rangle|_\pm:=\frac 1 2 (\xi(Y) \pm \eta(X)).$$
I am interested in ...

**13**

votes

**9**answers

1k views

### Combinatorial Databases

At one point, I remember being excited by seeing the website Encyclopedia of Combinatorial Structures as an extension of Sloane's Online Integer Sequence Database site. Unfortunately, the site (ECS) ...

**33**

votes

**21**answers

8k views

### Open source mathematical software.

I want some recomendation on which software I should install on my computer, an open source program for general abstract mathematical purposes (as opposed to applied mathematics).
I would likely use ...

**87**

votes

**68**answers

14k views

### Most helpful math resources on the web

What are really helpful math resources out there on the web?
Please don't only post a link but a short description of what it does and why it is helpful.
Please only one resource per answer and let ...

**91**

votes

**77**answers

20k views

### Examples of great mathematical writing

This question is basically from Ravi Vakil's web page, but modified for Math Overflow.
How do I write mathematics well? Learning by example is more helpful than being told what to do, so let's try to ...

**16**

votes

**16**answers

2k views

### Math books for advanced high school students

I'm working in a program for teaching a group of students selected in a Olympiad competition. The program is aimed to acquaint the students with the diverse aspects of higher mathematics in a way ...

**24**

votes

**13**answers

1k views

### Big list of repositories of mathematical preprints and postprints

I'm looking for a extensive list of online repositories of mathematical preprints and postprints. I'm interested in every type of repository, including small informal and semi-formal collections, like ...

**33**

votes

**9**answers

3k views

### List of Classifying Spaces and Covers

I am looking for a list of classifying spaces $BG$ of groups $G$ (discrete and/or topological) along with associated covers $EG$; there does not seem to be such cataloging on the web. Or if not a ...

**13**

votes

**11**answers

3k views

### Mathematics and cancer research?

What are applications of mathematics in cancer research?
My answer.
Unfortunately I heard quite small about math, but I heard something about
applications of physics. And let me put this story here, ...

**13**

votes

**5**answers

830 views

### Examples of research on how people perceive mathematical objects

What examples are there on research related to human perception and mathematical objects?
For example, the shape of a beer glass influences drinking habits,
since people are bad at integrating.
...

**37**

votes

**30**answers

37k views

### What programming languages do mathematicians use? [closed]

I understand this might be a slightly subjective question, but I am honestly curious what programming languages are used by the mathematics community.
I would imagine that there is a group of ...

**28**

votes

**14**answers

3k views

### Fantastic properties of Z/2Z

Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ...

**35**

votes

**31**answers

6k views

### Trichotomies in mathematics

Added. Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the ...

**3**

votes

**1**answer

59 views

### The link and equivalence between variant definition of computation model and computational complexity over reals

To unify the numerical computation and classic computability theory, or to pave a foundation for the numerical computation, mathematicians present variant computation model and computational ...

**108**

votes

**75**answers

34k views

### Best online mathematics videos?

I know of two good mathematics videos available online, namely:
Sphere inside out (part I and part II)
Moebius transformation revealed
Do you know of any other good math videos? Share.

**53**

votes

**16**answers

5k views

### What makes four dimensions special?

Do you know properties which distinguish four-dimensional spaces among the others?
What makes four-dimensional topological manifolds special?
What makes four-dimensional differentiable manifolds ...

**35**

votes

**36**answers

5k views

### Results true in a dimension and false for higher dimensions

Some theorems are true in vector spaces for a given dimension $n$ but become false in higher dimensions.
Here are two examples:
A positive polynomial not reaching its minimum. Impossible in ...

**66**

votes

**36**answers

25k views

### Most interesting mathematics mistake?

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincare's 3d sphere charaterization or the search to prove ...

**22**

votes

**17**answers

3k views

### What are your favorite puzzles/toys for introducing new mathematical concepts to students?

We all know that the Rubik's Cube provides a nice concrete introduction to group theory. I'm wondering what other similar gadgets are out there that you've found useful for introducing new math to ...

**24**

votes

**53**answers

9k views

### German mathematical terms like “Nullstellensatz”

There are quite a few german mathematical theorems or notions which usually are not translated into other languages. For example,
Nullstellensatz, Hauptvermutung, Freiheitssatz, Eigenvector (the ...

**18**

votes

**21**answers

3k views

### History Question: AUTObiography of Mathematicians

According to Wikipedia, an autobiography is an account of the life of a person, written by that person sometimes with a collaborator.
An autobiography offers the author the ability to recreate ...

**32**

votes

**19**answers

6k views

### Wonderful applications of the Vandermonde determinant

This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has ...

**15**

votes

**4**answers

888 views

### Mathematical research papers in general science journals

I am interested in collecting a list of research papers with a mainly mathematical focus that appeared in high-reputation general science journals without a dedicated mathematics section. This would ...

**7**

votes

**6**answers

2k views

### Discrete Mathematics textbooks for undergraduates

For the first time, I will be teaching a course on Discrete Mathematics for electrical and computer undergraduates students.
I intend to focus on practical applications.
I would be grateful if ...

**13**

votes

**6**answers

2k views

### “Oldest” bug in computer algebra system?

The goal of this question is to find an error in a computation by a computer algebra system where the 'correct' answer (complete with correct reasoning to justify the answer) can be found in the ...

**65**

votes

**33**answers

15k views

### Computer Algebra Errors

In the course of doing mathematics, I make extensive use of computer-based calculations. There's one CAS that I use mostly, even though I occasionally come across out-and-out wrong answers.
After ...

**34**

votes

**24**answers

6k views

### Examples of seemingly elementary problems that are hard to solve?

I'm looking for a list of problems such that
a) any undergraduate student who took multivariable calculus and linear algebra can understand the statements, (Edit: the definition of understanding here ...

**6**

votes

**2**answers

350 views

### Rational points techniques on curves not using their Jacobian

Let $C/K$ be a curve of genus > 2 over a number field $K$ and suppose there exists a $p \in C(K)$. Then a recurring theme in studying $C(K)$ is using the map $C \to J(C)$ normalized by sending $p$ to ...