**31**

votes

**18**answers

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

**3**answers

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

**52**answers

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

**17**answers

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

**9**answers

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

**1**answer

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

**90**answers

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

**27**answers

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

**1**answer

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

**2**answers

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

**14**answers

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

**4**answers

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

**28**answers

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

**17**answers

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

**7**answers

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

**64**answers

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

**3**answers

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

**25**answers

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

**12**answers

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

**9**answers

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

**3**answers

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

**11**answers

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

**3**answers

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

**4**answers

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

**26**answers

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

**1**answer

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

**8**answers

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

**53**answers

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

**1**answer

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

**6**answers

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

**107**answers

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

**16**answers

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

**12**answers

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

**2**answers

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

**26**answers

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

**26**answers

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

**1**answer

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

**19**answers

3k views

**29**

votes

**17**answers

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

**4**answers

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

**60**answers

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

**5**answers

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

**1**answer

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

**3**answers

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

**9**answers

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

**67**answers

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

**2**answers

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

**6**answers

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

**18**answers

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

**4**answers

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 ...