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 ...

learn more… | top users | synonyms

-2
votes
1answer
153 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
30answers
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 ...
37
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 ...
47
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: ...
61
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 ...
119
votes
29answers
38k 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, ...
445
votes
187answers
114k 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 ...
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
192 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 ...
61
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, ...
18
votes
18answers
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 ...
15
votes
19answers
18k 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 ...
93
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 ...
160
votes
89answers
27k 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 ...
2
votes
1answer
134 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 ...
94
votes
81answers
71k 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.)
183
votes
65answers
92k 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
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 ...
103
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 ...
132
votes
36answers
35k 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 ...
111
votes
64answers
20k 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 ...
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
163 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, ...
72
votes
10answers
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 ...
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)?
5
votes
0answers
76 views

Which known theorems of Lie algebras are still valid for Leibniz algebras?

Leibniz algebras can be seen as a non-commutative generalization of Lie algebras. Thus, it is common to see a lot of papers which topic is about a generalization of a classic theorem of Lie algebras ...
91
votes
33answers
54k 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 ...
23
votes
3answers
891 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
46answers
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 ...
106
votes
64answers
15k 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 ...
10
votes
18answers
18k 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?
16
votes
5answers
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 ...
94
votes
97answers
56k 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 ...
5
votes
7answers
8k views

Pseudo-random number generation algorithms

What algorithms are used in modern and good-quality random number generators?
7
votes
3answers
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
19answers
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
15answers
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 ...
-3
votes
1answer
125 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 ...
82
votes
28answers
15k 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
1answer
336 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 ...