Questions tagged [big-list]

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 question to be made CW.

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1050 votes
292 answers
341k 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 ...
421 votes
91 answers
146k views

Video lectures of mathematics courses available online for free

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...
401 votes
84 answers
185k 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 ...
392 votes
23 answers
65k views

Thinking and Explaining

How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, ...
382 votes
115 answers
105k 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 in ...
382 votes
53 answers
143k views

Widely accepted mathematical results that were later shown to be wrong?

Are there 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 later, possibly ...
355 votes
30 answers
75k views

Geometric interpretation of trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandary; Is there a geometric interpretation of the trace of a matrix? This question ...
325 votes
34 answers
92k views

Why is a topology made up of 'open' sets? [closed]

I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...
291 votes
34 answers
50k views

What are some reasonable-sounding statements that are independent of ZFC?

Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC." For example, suppose $A$ is an abelian group such ...
290 votes
125 answers
90k views

What are some examples of colorful language in serious mathematics papers?

The popular MO question "Famous mathematical quotes" has turned up many examples of witty, insightful, and humorous writing by mathematicians. Yet, with a few exceptions such as Weyl's "angel of ...
290 votes
34 answers
38k views

Which journals publish expository work?

I wonder if anyone else has noticed that the market for expository papers in mathematics is very narrow (more so than it used to be, perhaps). Are there any journals which publish expository work, ...
280 votes
47 answers
109k 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 ...
271 votes
67 answers
136k views

Awfully sophisticated proof for simple facts [closed]

It is sometimes the case that one can produce proofs of simple facts that are of disproportionate sophistication which, however, do not involve any circularity. For example, (I think) I gave an ...
254 votes
41 answers
98k views

A single paper everyone should read? [closed]

Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues. Do ...
249 votes
37 answers
169k views

Best algebraic geometry textbook? (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 ...
249 votes
29 answers
163k views

Intuitive crutches for higher dimensional thinking

I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows- An engineer, a physicist, and a mathematician are discussing how to visualise four ...
233 votes
14 answers
75k views

Have any long-suspected irrational numbers turned out to be rational?

The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...
232 votes
16 answers
55k views

What elementary problems can you solve with schemes?

I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary self-...
230 votes
89 answers
44k 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 ...
229 votes
46 answers
87k 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. Poincaré's 3d sphere characterization or the search to prove ...
228 votes
9 answers
24k views

John Nash's Mathematical Legacy

It would seem that John Nash and his wife Alicia died tragically in a car accident on May 23, 2015 (reference). My condolences to his family and friends. Maybe this is an appropriate time to ask a ...
220 votes
140 answers
48k 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 ...
215 votes
67 answers
45k views

Proofs that require fundamentally new ways of thinking

I do not know exactly how to characterize the class of proofs that interests me, so let me give some examples and say why I would be interested in more. Perhaps what the examples have in common is ...
211 votes
40 answers
37k views

Demonstrating that rigour is important

Any pure mathematician will from time to time discuss, or think about, the question of why we care about proofs, or to put the question in a more precise form, why we seem to be so much happier with ...
209 votes
51 answers
80k views

Ways to prove the fundamental theorem of algebra

This seems to be a favorite question everywhere, including Princeton quals. How many ways are there? Please give a new way in each answer, and if possible give reference. I start by giving two: ...
208 votes
26 answers
50k views

The most outrageous (or ridiculous) conjectures in mathematics

The purpose of this question is to collect the most outrageous (or ridiculous) conjectures in mathematics. An outrageous conjecture is qualified ONLY if: 1) It is most likely false (Being hopeless is ...
202 votes
72 answers
49k 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 ...
199 votes
89 answers
52k 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 ...
194 votes
18 answers
15k views

Great graduate courses that went online recently

In 09.2020 by pure chance I discovered the YouTube channel of Richard Borcherds where he gives graduate courses in Group Theory, Algebraic Geometry, Schemes, Commutative Algebra, Galois Theory, Lie ...
193 votes
30 answers
77k 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, e.g....
193 votes
43 answers
43k views

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

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 than to ...
192 votes
94 answers
105k 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 ...
191 votes
47 answers
27k views

Books you would like to read (if somebody would just write them…)

I think that the title is self-explanatory but I'm thinking about mathematical subjects that have not received a full treatment in book form or if they have, they could benefit from a different ...
191 votes
34 answers
79k views

What is convolution intuitively?

If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
191 votes
12 answers
30k views

Do you know important theorems that remain unknown?

Do you know of any very important theorems that remain unknown? I mean results that could easily make into textbooks or research monographs, but almost nobody knows about them. If you provide an ...
189 votes
79 answers
42k views

Which math paper maximizes the ratio (importance)/(length)?

My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7-manifolds which are ...
187 votes
81 answers
32k 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 ...
186 votes
47 answers
100k views

Magic trick based on deep mathematics

I am interested in magic tricks whose explanation requires deep mathematics. The trick should be one that would actually appeal to a layman. An example is the following: the magician asks Alice to ...
185 votes
62 answers
89k views

Interesting mathematical documentaries

I am looking for mathematical documentaries, both technical and non-technical. They should be "interesting" in that they present either actual mathematics, mathematicians or history of mathematics. I ...
183 votes
127 answers
62k views

Most memorable titles

Given the vast number of new papers / preprints that hit the internet everyday, one factor that may help papers stand out for a broader, though possibly more casual, audience is their title. This view ...
181 votes
60 answers
42k 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 $P(a) = T\ \ \forall a \in \mathbb{N}$ to be a 'reasonable' conjecture to ...
181 votes
33 answers
31k views

What should be learned in a first serious schemes course?

I've just finished teaching a year-long "foundations of algebraic geometry" class. It was my third time teaching it, and my notes are gradually converging. I've enjoyed it for a number of reasons (...
177 votes
80 answers
65k 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.
173 votes
39 answers
42k views

Most harmful heuristic?

What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?
170 votes
36 answers
35k views

Proposals for polymath projects

Background Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by ...
167 votes
50 answers
55k views

17 camels trick

The following popular mathematical parable is well known: A father left 17 camels to his three sons and, according to the will, the eldest son should be given a half of all camels, the middle son ...
163 votes
46 answers
31k views

Every mathematician has only a few tricks

In Gian-Carlo Rota's "Ten lessons I wish I had been taught" he has a section, "Every mathematician has only a few tricks", where he asserts that even mathematicians like Hilbert ...
163 votes
38 answers
27k views

Short exact sequences every mathematician should know

I'd like to have a big-list of "great" short exact sequences that capture some vital phenomena. I'm learning module theory, so I'd like to get a good stock of examples to think about. An ...
160 votes
23 answers
29k views

Do you read the masters?

I often hear the advice, "Read the masters" (i.e., read old, classic texts by great mathematicians). But frankly, I have hardly ever followed it. What I am wondering is, is this a ...
158 votes
48 answers
23k views

Generalizing a problem to make it easier

One of the many articles on the Tricki that was planned but has never been written was about making it easier to solve a problem by generalizing it (which initially seems paradoxical because if you ...

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