Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the ...

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64
votes
24answers
7k views

Modern Mathematical Achievements Accessible to Undergraduates

While there is tremendous progress happening in mathematics, most of it is just accessible to specialists. In many cases, the proofs of great results are both long and use difficult techniques. Even ...
167
votes
91answers
28k 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 ...
54
votes
10answers
6k views

Applications of mathematics

All of us have probably been exposed to questions such as: "What are the applications of group theory...". This is not the subject of this MO question. Here is a little newspaper article that I found ...
115
votes
66answers
21k views

Your favorite surprising connections in Mathematics

There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the connection between the ...
20
votes
15answers
5k views

Applications of finite continued fractions

I know some applications of finite continued fractions. Probably you know more. Can you add anything? (For for Applications of periodic continued fractions I have made a special topic.) 1) (Trivial) ...
34
votes
20answers
6k views

Fun applications of representations of finite groups

Are there some fun applications of the theory of representations of finite groups? I would like to have some examples that could be explained to a student who knows what is a finite group but does not ...
40
votes
20answers
10k views

Interesting Calculus Questions/Exercises

I am in the process of redesigning the calculus course that I have taught five or six times. What I would like to know is if anyone has some really good examples or exercises that I could either do ...
121
votes
30answers
39k 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, ...
14
votes
2answers
677 views

What are the applications of operator algebras to other areas?

Question: What are the applications of operator algebras to other areas? More precisely, I would like to know the results in mathematical areas outside of operator algebras which were proved by ...
1
vote
1answer
1k views

Famous examples of PhD advisors younger than their student [closed]

What are the most famous examples of PhD advisors in mathematics, younger than their student? (if possible put the date of birth and/or the difference in age).
77
votes
11answers
8k views

Why are modular forms interesting?

Well, I'm aware that this question may seem very naive to the several experts on this topic that populate this site: feel free to add the "soft question" tag if you want... So, knowing nothing about ...
7
votes
4answers
1k views

Proofs of the Chevalley-Warning Theorem

A well known proof of the Chevally-Warning Theorem is the one listed on wikipedia: http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem Are there any other proofs of this, or ...
91
votes
107answers
31k views

Most memorable titles [closed]

Apparently, for a large number of readers, the choice whether they select to read a paper or not is often strongly influenced by the title. I was wondering if the MO-users would be willing to share ...
37
votes
32answers
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 ...
123
votes
132answers
28k views

Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please) I'd love to learn about ...
6
votes
13answers
5k views

Useless math that became useful

I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless. My idea is to amend my article with some theories that seemed useless when they are created but ...
10
votes
19answers
19k 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?
17
votes
19answers
3k 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 ...
84
votes
41answers
16k views

Examples of eventual counterexamples

Define an "eventual counterexample" to be $P(a) = T $ for $a < n$ $P(n) = F$ $n$ is sufficiently large for $P(n) = T\ \ \forall n \in \mathbb{N}$ to be a 'reasonable' conjecture to make. where ...
10
votes
3answers
3k views

Two questions about combinatorics journals

Hello, I have two questions regarding combinatorics journals. I hope that this is the right place for such questions. Which combinatorics/DM journals would you consider as the "top tier"? I tried ...
452
votes
188answers
115k 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 ...
-2
votes
1answer
168 views

Degree of a rational function [closed]

I would like to have a simple proof for the following result: Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). ...
59
votes
34answers
9k views

Experimental Mathematics

I would like to ask about examples where experimentation by computers have led to major mathematical advances. A new look Now as the question is five years old and there are certainly more examples ...
51
votes
29answers
5k views

books well-motivated with explicit examples

It is ultimately a matter of personal taste, but I prefer to see a long explicit example, before jumping into the usual definition-theorem path (hopefully I am not the only one here). My problem is ...
38
votes
18answers
5k views

What are some deep theorems, and why are they considered deep?

All mathematicians are used to thinking that certain theorems are deep, and we would probably all point to examples such as Dirichlet's theorem on primes in arithmetic progressions, the prime number ...
48
votes
51answers
9k views

Books you would like to see translated into English.

I have recently been told of a proposal to produce an English translation of Landau's Handbuch der Lehre von der Verteilung der Primzahlen, and this prompts me to ask a more general question: ...
62
votes
61answers
24k views

Favorite popular math book [closed]

Christmas is almost here, so imagine you want to buy a good popular math book for your aunt (or whoever you want). Which book would you buy or recommend? It would be nice if you could answer in the ...
18
votes
10answers
3k views

Research Experience for Undergraduates: Summer Programs

Some time ago, I found this list of REU programs held in 2009. The main aspects that characterize such programs are: (a) a great deal of lectures on specific topics; and, admittedly more ...
4
votes
2answers
202 views

Proving results about complete Boolean algebras in ZFC using Boolean valued models

I want to know what non-trivial ZFC theorems (not consistency results) about complete Boolean algebras (or more generally of partially ordered sets) one can prove using forcing. I am mainly ...
62
votes
14answers
17k 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 ...
32
votes
16answers
3k views

Counterexamples in universal algebra

Universal algebra - roughly - is the study, construed broadly, of classes of algebraic structures (in a given language) defined by equations. Of course, it is really much more than that, but that's ...
14
votes
12answers
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, ...
15
votes
19answers
19k 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 ...
94
votes
55answers
21k views

What are your favorite instructional counterexamples?

Related: question #879, Most interesting mathematics mistake. But the intent of this question is more pedagogical. In many branches of mathematics, it seems to me that a good counterexample can be ...
8
votes
9answers
1k views

Examples where adding complexity made a problem simpler

I can think of a few situations in math where a problem becomes easier or an object becomes simpler when some complexity is added. Examples: $S^n$ is never contractible, but $S^{\infty}$ is. The ...
2
votes
1answer
169 views

Statistical distance between discrete and continuous distributions

Are there any statistical distance functions that are capable of comparing a continuous and a discrete distribution? From reading this list http://en.wikipedia.org/wiki/Statistical_distance the only ...
95
votes
81answers
73k views

Do good math jokes exist? [closed]

Have a good joke? Share. I know this is subjective, but the principle "should be of interest to mathematicians" trumps. (I hope.)
185
votes
65answers
93k views

Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results? (One could ask if this is of interest to mathematicians, and I ...
77
votes
52answers
15k 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 ...
105
votes
83answers
22k 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 ...
134
votes
36answers
36k 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 ...
46
votes
13answers
6k views

Contest problems with connections to deeper mathematics

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 ...
106
votes
22answers
24k 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 ...
41
votes
30answers
5k views

Fundamental problems whose solution seems completely out of reach [closed]

In many areas of mathematics there are fundamental problems that are embarrasingly natural or simple to state, but whose solution seem so out of reach that they are barely mentioned in the literature ...
110
votes
75answers
36k 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.
30
votes
20answers
3k views

Generalizations of Planar Graphs

This is a follow up to Harrison's question: why planar graphs are so exceptional. I would like to ask about (and collect answers to) various notions, in graph theory and beyond graph theory (topology; ...
4
votes
0answers
170 views

What will be the consequences if second Hardy-Littlewood conjecture turns out to be true?

It is generally believed that the Second Hardy-Littlewood Conjecture is false. But it has not been proved (or disproved) yet. My question is, What would be the consequences if Second ...
12
votes
2answers
1k views

New research and re-discovering classic results in “basic” real analysis

Sometimes, it happens that researchers publish a new proof of an old well-known result in "basic real analysis" (I'm referring to what some American people may call "honors calculus"). For instance, ...
11
votes
12answers
2k views

On proving that a certain set is not empty by proving that it is actually large

It happens occasionally that one can prove that a given set is not empty by proving that it is actually large. The word "large" here may refer to different properties. For example, one can prove that ...
18
votes
5answers
1k views

Collection of conjectures and open problems in graph theory

Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?