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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12
votes
4answers
1k views

What results would follow from or imply “randomness” of the primes?

This question on random versions of deterministic problems reminded me that many conditional results in number theory hold if the primes are in some sense random, and it is common knowledge that the ...
19
votes
11answers
1k views

Random versions of deterministic problems

What are the examples of situations where "randomizing" a problem (or some part of it) and analyzing it using probabilistic techniques yields some insight into its deterministic version? An example ...
16
votes
17answers
3k views

Which book would you like to see “texified”? [closed]

Let's see if we could use MO to put some pressure on certain publishers... Although it is wonderful that it has been put online, I think it would make an even greater read if "Hodge Cycles, Motives ...
10
votes
44answers
3k views

Mathematical ideas named after places [closed]

This question is quite unimportant, so feel free to close if you think it is inappropriate. I've been thinking about how mathematicians come up with names for the ideas/objects they study, and how ...
23
votes
18answers
7k views

Interesting and Accessible Topics in Graph Theory

This summer, I will be teaching an introductory course in graph theory to talented high school seniors. The intent of the course is not to establish proficiency in graph theory, per se. Rather, I hope ...
7
votes
6answers
950 views

Seemingly emergent structures in mathematics

I rather suspect that this must have come up here on MO already, but my handful of searches didn't turn up the thread, so... I'm curious about examples of mathematical structure that seems to arise ...
4
votes
1answer
543 views

Tricks of the Trade [closed]

Can you name a mathematical theorem that is simple to state and relatively simple to prove, was essential to your research or to a work you found interesting and significant, has the potential to be ...
57
votes
26answers
4k views

What would you want on a Lie theory cheat poster?

For some long time now I've thought about making a poster-sized "cheat sheet" with all the data about Lie groups and their representations that I occasionally need to reference. It's a moving target, ...
31
votes
12answers
3k views

Recent Applications of Mathematics

What are the recent and new applications of Mathematics in other Sciences ? Let me try to be more precise about the question: By "recent" I mean the last 15 years. By "new" I want to exclude the ...
20
votes
10answers
2k views

What math institutes offer research in pairs/research in teams?

Some math institutes offer programs in which a small number of researchers are enabled to meet at the institute for a week or more. A list seemed as if it could be useful.
24
votes
53answers
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 ...
4
votes
8answers
3k views

What's the difference between 2 and 3? [closed]

Here are two classical results which depend on whether a parameter is 2 or 3: It is possible to bisect an arbitrary angle with ruler and compass, but impossible to trisect it. While there are ...
58
votes
73answers
11k views

Elementary+Short+Useful

Imagine your-self in front of a class with very good undergraduates who plan to do mathematics (professionally) in the future. You have 30 minutes after that you do not see these students again. You ...
72
votes
17answers
6k views

Occurrences of (co)homology in other disciplines and/or nature

I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive ...
6
votes
10answers
867 views

Examples of “Unusual” Classifications

When one says "classification" in math, usually one of a handful of examples springs to mind: -Classification of Finite Simple Groups with 18 infinite families and 26 sporadic examples (assuming one ...
4
votes
2answers
3k views

Examples of naturally occurring Quadratic forms or quadrics.

I am always fascinated when a quadratic form (or a quadric) arises naturally. I have some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too ...
8
votes
4answers
2k views

Which topics/problems could you show to a bright first year mathematics student?

I am teaching a one semester course (January to June) to first year students pursuing various different degrees. Because there are students studying actuarial science, physics, other sciences, other ...
10
votes
7answers
3k views

Leibnizian calculus textbook

Where can I find a calculus textbook that emphasizes differentials? Is there such a book that I could realistically require my calculus students to use? I want a textbook that supports me when I tell ...
8
votes
9answers
1k views

“Surprising” categorical equivalences

This is inspired by this question about the equivalence between the category of finite sets and non-negative integers. Now this question was (rightly, I guess) closed, but the fact was surprising to ...
11
votes
7answers
2k views

Applications of the notion of of Gromov-Hausdorff distance

I am looking for applications of the notion of Gromov-Hausdorff convergence to prove theorems that a priori have nothing to do with it. Examples that I am aware of (thanks to wikipedia and google): ...
43
votes
29answers
9k views

What notions are used but not clearly defined in modern mathematics?

"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions." Felix Klein What notions are used but not ...
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 ...
33
votes
9answers
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 ...
4
votes
3answers
788 views

Examples of results which were surprising but later shown to be natural. [closed]

After Ramanujan formulated his conjectures on the Tau-function, and after the importance of the function was realized, it took the development of the theory of Modular forms for the complete ...
208
votes
72answers
84k 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 ...
19
votes
3answers
2k views

Locales and Topology.

As someone more used to point-set topology, who is unfamiliar with the inner workings of lattice theory, I am looking to learn about the localic interpretation of topology, of which I only have a ...
15
votes
12answers
2k views

Constructions unique up to non-unique isomorphism

1) Fields have algebraic closures unique up to a non-unique isomorphism. 2) Nice spaces (without base point) have universal covering spaces unique up to a non-unique isomorphism. 3) Modules have ...
95
votes
34answers
14k 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 ...
4
votes
10answers
1k views

Proving theorems by using functions with fixed points.

I am trying to get a better feel for solving questions where creating a function with a unique fixed point is the crux of the proof. In particular, the Inverse Function Theorem as well as the ...
5
votes
9answers
1k views

Examples of two different descriptions of a set that are not obviously equivalent?

I am teaching a course in enumerative combinatorics this semester and one of my students asked for deeper clarification regarding the difference between a "combinatorial" and a "bijective" proof. ...
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 ...
9
votes
3answers
988 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 ...
60
votes
52answers
20k 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 ...
28
votes
17answers
7k 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 ...
16
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 ...
81
votes
90answers
11k 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 ...
23
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
941 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 ...
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) ...
12
votes
4answers
982 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 ...
28
votes
28answers
11k 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; ...
56
votes
7answers
6k 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 ...
142
votes
64answers
24k 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 ...
64
votes
25answers
9k 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 ...
55
votes
16answers
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 ...
7
votes
9answers
442 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 ...