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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31
votes
18answers
8k 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 ...
9
votes
3answers
818 views

Easier induction proofs by changing the parameter

When performing induction on say a graph $G=(V,E)$, one has many choices for the induction parameter (e.g. $|V|, |E|$, or $|V|+|E|$). Often, it does not matter what choice one makes because the proof ...
58
votes
52answers
18k views

Theorems that are 'obvious' but hard to prove

There are several well-known mathematical statements that are 'obvious' but false (such as the negation of the Banach--Tarski theorem). There are plenty more that are 'obvious' and true. One would ...
27
votes
17answers
6k views

Computer Science for Mathematicians

This is a big-list community question, so I'm sorry in advance if it is deemed too soft but I haven't seen anything similar yet. I've seen computer scienctists post questions looking to learn things ...
15
votes
9answers
4k views

Open Questions in Riemannian Geometry

What are some major open problems in Riemannian Geometry? I tried googling it, but couldn't find any resources.
-1
votes
1answer
1k views

Unpopular “elementary” theorems/identities to impress an audience of mathematicians. [closed]

This question grew out of my recent job interview. Since the interviewers were math professors, I had a hard time searching for interesting elementary theorems in case I got asked for one. I thought ...
78
votes
90answers
10k views

What would you want to see at the Museum of Mathematics?

EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ...
21
votes
27answers
4k views

Problems where we can't make a canonical choice, solved by looking at all choices at once

It's a common theme in mathematics that, if there's no canonical choice (of basis, for example), then we shouldn't make a choice at all. This helps us focus on the heart of the matter without giving ...
7
votes
1answer
1k views

Technical trends quietly aimed at big open problems? [closed]

When I was an undergraduate 35 years ago, I made the mistake of asking some of my mathematics professors what well-known open problems they liked to think about. I got the message that this was ...
6
votes
2answers
855 views

Applications of periodic continued fractions

Some answers from Applications of finite continued fractions in fact are Applications of periodic continued fractions. I think that it should be separate question. What can you add to the following ...
19
votes
14answers
4k 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) ...
12
votes
4answers
971 views

What formal properties should resolution of singularities have?

If I were going to propose a new construction as a "replacement for resolution of singularities", what properties would my replacement have to have? [I am going to do no such thing -- this is purely ...
25
votes
28answers
10k views

Most intricate and most beautiful structures in mathematics

In the December 2010 issue of Scientific American, an article "A Geometric Theory of Everything" by A. G. Lisi and J. O. Weatherall states "... what is arguably the most intricate structure known to ...
12
votes
17answers
2k views

Individual mathematical objects whose study amounts to a (sub)discipline? [closed]

Certain mathematical objects have a theory so rich that their study alone arguably constitutes a distinct (sub)discipline. My own list would begin with 1) the absolute Galois group of the rationals; ...
54
votes
7answers
5k views

Is Grothendieck a computer?

I can't resist asking this companion question to the one of Gowers. There, Tim Dokchitser suggested the idea of Grothendieck topologies as a fundamentally new insight. But Gowers' original motivation ...
133
votes
64answers
22k views

Proofs that require fundamentally new ways of thinking [closed]

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 ...
27
votes
3answers
2k views

Names of finite groups

Question: If you have a finite group, how do you name it? If, for whatever reason, you have to list all subgroups of $GL_2({\mathbb F}_5)$ up to isomorphism in a paper, you are likely to write ...
59
votes
25answers
8k views

More open problems [closed]

Open Problem Garden and Wikipedia are good resources for more or less famous open problems. But many mathematicians will be happy with more specialized problems. They may want to find a research ...
47
votes
12answers
4k 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 ...
7
votes
9answers
429 views

What are some early examples of creation of lists / catalogues of (particularly) combinatorial objects?

A lot of effort in discrete maths / combinatorics is expended in the construction of lists, catalogues or census [sic] of combinatorial objects such as groups, graphs, designs etc. These catalogues ...
6
votes
3answers
802 views

Uses of Divergent Series and their summation-values in mathematics ?

This question was posed originally on MSE, I put it here because I didn't receive the answer(s) I wished to see. Dear MO-Community, When I was trying to find closed-form representations for odd ...
16
votes
11answers
4k views

Examples of using physical intuition to solve math problems

For the purposes of this question let a "physical intuition" be an intuition that is derived from your everyday experience of physical reality. Your intuitions about how the spin of a ball affects ...
5
votes
3answers
5k views

Advanced Math Jokes [closed]

I am looking for jokes which involve some serious mathematics. Sometimes, a totally absurd argument is surprisingly convincing and this makes you laugh. I am looking for jokes which make you laugh and ...
2
votes
4answers
708 views

applications of Plancherel formulae

I've learned a few things about harmonic analysis on semisimple Lie groups recently and the amount of effort that goes into the proof of the Plancherel formula seems overwhelming. Of course it has led ...
45
votes
26answers
6k views

Sexy vacuity …

I included this footnote in a paper in which I mentioned that the number of partitions of the empty set is 1 (every member of any partition is a non-empty set, and of course every member of the empty ...
4
votes
1answer
645 views

Examples and importance of Embedding (and Non-Embedding) Theorems

An embedding is an injective map into a universal, simpler model object. Many embedding theorems are without obstruction, in the sense that every object which you wish to embed can be embedded. ...
20
votes
8answers
1k views

Examples of sequences whose asymptotics can't be described by elementary functions

It is somewhat miraculous to me that even very complicated sequences $a_n$ which arise in various areas of mathematics have the property that there exists an elementary function $f(n)$ such that $a_n ...
51
votes
53answers
11k views

Pseudonyms of famous mathematicians

Many mathematicians know that Lewis Carroll was quite a good mathematician, who wrote about logic (paradoxes) and determinants. He found an expansion formula, which bears his real name (Charles ...
0
votes
1answer
329 views

What is your favorite ADE-style classification? [duplicate]

Possible Duplicate: ADE type Dynkin diagrams What is your favorite ADE-style classification? Here ADE style is to be understood in a very broad sense. A classification which is not ...
46
votes
6answers
6k views

Still Difficult After All These Years

I think we all secretly hope that in the long run mathematics becomes easier, in that with advances of perspective, today's difficult results will seem easier to future mathematicians. If I were ...
86
votes
107answers
27k 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 ...
69
votes
16answers
15k views

What recent discoveries have amateur mathematicians made?

E.T. Bell called Fermat the Prince of Amateurs. One hundred years ago Ramanujan amazed the mathematical world. In between were many important amateurs and mathematicians off the beaten path, but what ...
27
votes
12answers
5k views

What is a good introductory text for moduli theory?

Hi,everyone. I am looking for an introductory textbook on moduli theory,about the background on algebraic geometry,I have read Hartshorne chapter1~4. could you please show some good books or roadmap ...
21
votes
2answers
2k views

Examples where the analogy between number theory and geometry fails

The analogy between $O_K$ ($K$ a number field) and affine curves over a field has been very fruitful. It also knows many variations: the field over which the curve is defined may have positive or zero ...
45
votes
26answers
4k views

Proof synopsis collection

I hate to keep going with the big lists, but the question about one-sentence summaries of topics/areas spurred this question...and I just can't help myself! Definition (Fraleigh): A proof synopsis ...
87
votes
26answers
11k views

Extremely messy proofs

Currently in my undergraduate courses I am being taught how to set up various machinery using slick, short proofs and then how to apply that machinery. What I am not being taught, largely, is what ...
1
vote
1answer
1k views

Classification Problems [closed]

I was thinking about the famous question in philosophy of mathematics: "When are two proofs the same?" and I was wondering if we could somehow "classify" proofs by establishing some sort of functorial ...
18
votes
19answers
3k views

Examples of categorification

What is your favorite example of categorification?
29
votes
17answers
5k 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 ...
12
votes
4answers
989 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 ...
97
votes
60answers
13k 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 ...
6
votes
5answers
2k views

More upper/lower semi-continuous functions in (algebraic) geometry?

The notion of upper/lower semi-continuity is sometimes encountered in algebraic geometry. Here by upper semi-continuity one means a function on a topological space $f:X\rightarrow S$ with value in ...
0
votes
1answer
371 views

Piece of a sequence

Suppose we are given a representation of a finite series of natural numbers: $\sum_{i=0}^N{c_i x^i}$ The representation is essentially an expression that is a rational function of two polynomials. ...
9
votes
3answers
929 views

Concavity of $\det^{1/n}$ over $HPD_n$.

One of my beloved theorems in matrix analysis is the fact that the map $H\mapsto (\det H)^{1/n}$, defined over the convex cone $HPD_n$ of Hermitian positive definite matrices, is concave. This is ...
15
votes
9answers
2k views

Examples of ZFC theorems proved via forcing

This is an old suggestion of Joel David Hamkins at the end of his answer to this question: Forcing as a tool to prove theorems I just noticed it while trying to understand his answer. But indeed it ...
115
votes
67answers
35k 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 ...
14
votes
2answers
846 views

Applications of Stacks

I've been aware of stacks since grad school, and I can usually follow in rough lines a discussion about stacks, but I've often wondered what particular (purely!) scheme-theoretic argument or theorem ...
25
votes
6answers
833 views

Link Repository of International Dissertations

This question (cry for help?) grew out of Colin Tan's question: does anyone have a copy of schmid’s effective work on hilbert 17th? which was a request for a copy of an Habilitationsschrift. We've ...
21
votes
18answers
2k views

PDEs as a tool in other domains in mathematics

According to the large number of paper cited in MathSciNet database, Partial Differential Equations (PDEs) is an important topic of its own. Needless to say, it is an extremely useful tool for natural ...
5
votes
4answers
863 views

Hecke-algebras in your field of mathematics

(How) do Hecke-algebras arise naturally in your field of mathematics and why are they important? How would you define them and how do you think about them? e.g. generators and relations, functions ...