**9**

votes

**4**answers

1k views

### Role of applications in modern mathematics [closed]

Older days scientists were universalists and philosophy, physics and mathematics were a part the same question - understanding the world.
Nowadays one may get feeling that the role of applications ...

**9**

votes

**4**answers

702 views

### Connections between topos theory and topology

What are some "applications to" / "connections with" topology that one could hope to reasonably cover in a first course on topos theory (for master students)? I have an idea of what parts of the ...

**0**

votes

**1**answer

227 views

### Applications of the natural bilinear forms on the direct sum between a vector space and its dual

As it is know, 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 ...

**15**

votes

**3**answers

943 views

### Research level applications of “row rank = column rank”?

No less an authority than Gilbert Strang frames "row rank equals column rank" (and a couple of other facts) as "The Fundamental Theorem of Linear Algebra."
I'd simply like to assemble (for teaching ...

**3**

votes

**0**answers

712 views

### “Must read ”papers on analytic number theory

Question: What would be some must-read
papers for an aspiring analytic number
theorist? In other words, what are the papers that any analytic number theorist would have read? (Background: ...

**3**

votes

**1**answer

223 views

### Repertory of the different sorts of operads

Many different types of operads have emerged in recent years (symmetric, shuffle, cyclic, anticyclic, coloured, etc.).
I would like, for any of these, list the following data:
Description of the ...

**49**

votes

**5**answers

1k views

### Math Annotate Platform?

Suppose most mathematical research papers were freely accessible online.
Suppose a well-organized platform existed where responsible users could write comments on any paper (linking to its doi, ...

**6**

votes

**3**answers

746 views

### problems from the scottish book

Which of the problems from the Scottish Book (pdf of English version) by Stefan Banach are still open? I know that one of the problems was solved by Per Enflo for which he got a live goose from ...

**10**

votes

**2**answers

843 views

### Using schemes to prove things about rings

I apologize for asking a big list question, I've tried to avoid doing so for a while. I'll give my justification in a moment.
The question is as follows:
What are examples of strict applications ...

**34**

votes

**19**answers

5k views

### Mathematicians whose works were criticized by contemporaries but became widely accepted later

Gauss famously discarded Abel's proof that an algebraic equation of degree five or more cannot have a general solution (Abel himself had rejected divergent series as the work of the devil). Cantor's ...

**5**

votes

**3**answers

724 views

### group of diffeomorphisms of a manifold

How much has been the group of diffeomorphisms of a manifold " been studied.
I got this information from wiki.
" Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra ...

**18**

votes

**8**answers

2k views

### How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there?

Offhand I can think of two ways in classical homotopy theory:
Show that $\pi_k(S^n)=0$ for $k\lt n$ by deforming a map $S^k\to S^n$ to be non-surjective, then contracting it away from a point not in ...

**1**

vote

**0**answers

246 views

### applications of gauss-bonnet theorem

In wikipedia,I was pretty amazed to find a proof of fundamental theorem of algebra
using gauss bonnet theorem.I think given how central it is to mathematics with its far reaching generalizations like ...

**32**

votes

**31**answers

5k views

### Trichotomies in mathematics

Added. Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the ...

**6**

votes

**4**answers

842 views

### fourier analytic proofs

While searching through Mathoverflow, I found out a fourier analytic proof of the Isoperimetric Inequality.Also, by google search I found a fourier analytic proof of Quadratic Reciprocity theorem.I ...

**19**

votes

**10**answers

1k views

### Learning through guided discovery

I have been working through Kenneth P. Bogart's "Combinatorics Through Guided Discovery". You can download it from this page: http://www.math.dartmouth.edu/news-resources/electronic/kpbogart/
I've ...

**5**

votes

**4**answers

1k views

### What are conjectures that are true for primes but then turned out to be false for some composite number?

Note: This is an update formulation since many people misunderstood the question before.
Of course it is easy to make a statement like "Every n is a prime or at most 1000", which is true for every ...

**9**

votes

**2**answers

1k views

### Math Zeitgeist 2012 [closed]

Dear MO-fellows, Happy New Year !
Taking google as an example: https://www.google.com/zeitgeist/2012/
Why not to make a list of noted math events in 2012 ?
E.g. 15 December 2012 Museum of Math was ...

**51**

votes

**61**answers

7k views

### Old books still used

It's a commonplace to state that while other sciences (like biology) may always need the newest books, we mathematicians also use to use older books. While this is a qualitative remark, I would like ...

**23**

votes

**7**answers

2k views

### Excellent mathematical explanations

In the Stanford Encyclopedia of Philosophy there is an entry on mathematical explanation.
The basic philosophical question is: What makes a proof explanatory?
Two main "models" of mathematical ...

**12**

votes

**4**answers

1k views

### Statements which were given as axioms, which later turned out to be false.

I know that early axiomatizations of real arithmetic (in the first half of the nineteenth century) were often inadequate. For example, the earliest axiomatizations did not include a completeness ...

**5**

votes

**4**answers

1k views

### On similar concepts in mathematics whose similarity is a non-trivial fact.

Recently, while undertaking a study of commutative algebra, I learned three concepts: (i) a local ring, (ii) a regular local ring and (iii) a regular ring.
At the end, I found myself asking this ...

**6**

votes

**12**answers

4k views

### Useless math that became useful

I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless.
My idea is to amend my article with some theories that seemed useless when they are created but ...

**27**

votes

**9**answers

2k views

### Covering maps in real life that can be demonstrated to students

Edit: I've really enjoyed everyone's examples (especially the pictures!), but I was mostly looking for a general theorem. For instance, a similar statement to mine is, Can the mapping cylinder of ...

**11**

votes

**7**answers

1k views

### What are some Applications of Teichmüller Theory?

I'm trying to collect some specific examples of applications of Teichmüller Theory. Here are some things I have collected thus far:
No-wandering-domain Theorem (Sullivan)
Theorems of Thurston ...

**4**

votes

**3**answers

810 views

### Consequences of Legendre's conjecture

I am looking for a list/reference which explores the consequences of Legendre's conjecture, which states that one can always find a prime number between $n^2$ and $(n+1)^2$.

**7**

votes

**9**answers

981 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 ...

**1**

vote

**0**answers

246 views

### Applications of the class number formula, etc.

This is a big list of applications of the class number formula and its generalizations. I'll start:
The solution to Gauss's class number problem for imaginary quadratic fields, and more generally ...

**19**

votes

**3**answers

650 views

### What classification theorems have been improved by re-categorizing?

Many classification theorems (e.g. of the finite subgroups of $SO(3)$, or the finite-dimensional complex simple Lie algebras, or the finite simple groups) have some infinite lists, plus some ...

**4**

votes

**3**answers

566 views

### Quotations about the power of simple ideas [closed]

I'm looking for quotations about how very simple mathematical ideas can be very powerful. I know of a few, but they're not quite what I'm looking for insofar as they contain criticism of other ...

**4**

votes

**0**answers

187 views

### Quotations about the class number formula, etc.

I'm looking for interesting and/or expressive quotations from mathematicians about the class number formula. I'm interested both in quotations from historical mathematicians and from modern ...

**30**

votes

**11**answers

2k views

### Important open problems that have already been reduced to a finite but infeasible amount of computation

Most open problems, when formalized, naturally involve quantification over infinite sets, thereby obviating the possibility, even in principle, of "just putting it on a computer."
Some questions ...

**30**

votes

**10**answers

3k views

### Believing the Conjectures

In Believing the axioms (I and II), Penelope Maddy proposes five "rules of thumb" that she then uses to justify large cardinal axioms in set theory. These extrinsic rules are modeled after the ...

**17**

votes

**6**answers

1k views

### Pathological Examples of Dimension

I am trying to wrap my head around all the different notions of dimension (and their equivalences). To get a sense of this, it would be nice to know the subtle difficulties that arise. I thus think it ...

**10**

votes

**3**answers

407 views

### Applications of idempotent ultrafilters

Recently Justin Moore has posted a solution to the amenability of Thompson's group F. A key(?) step exploits the existence of idempotent ultrafilters on $\mathbb N$ to construct an idempotent measure ...

**6**

votes

**9**answers

1k views

### Beautiful theorems with short proof [closed]

I like to ask for beautiful mathematical theorems with short proof. A proof is short in my sense if it is at most one page assuming basic notations and very basic results a second year student will ...

**8**

votes

**12**answers

937 views

### Continuous notions with compelling discrete analogues

Following up on the previous MO question "Are there any important mathematical concepts without discrete analogue?", I'd like to ask the opposite: what are examples of notions in math that were not ...

**15**

votes

**12**answers

1k views

### Instances where an existence result precedes the constructive version

The basic motivation here is to encourage and inspire - via examples - the pursuit of alternate proofs of existing results that might be more accessible and intuitive by cataloging success stories. ...

**6**

votes

**4**answers

1k views

### Discrete Mathematics textbooks for undergraduates

For the first time, I will be teaching a course on Discrete Mathematics for electrical and computer undergraduates students.
I intend to focus on practical applications.
I would be grateful if ...

**5**

votes

**3**answers

1k views

### Battle of the brains; cultural mathematics

Firstly, I apologize if my question is long.
Three years ago, I watched a video with the name Battle of the Brains. It was a wonderful video about challenging some famous peoples to solve some ...

**16**

votes

**19**answers

2k views

### History Question: AUTObiography of Mathematicians

According to Wikipedia, an autobiography is an account of the life of a person, written by that person sometimes with a collaborator.
An autobiography offers the author the ability to recreate ...

**28**

votes

**11**answers

5k views

### “Must read” papers in numerical analysis

In 1993, Prof. L.N. Trefethen published a NA-net posting with a list of thirteen paper he used for teaching the seminar Classic Papers in Numerical Analysis.
In Trefethen's words, ... this course ...

**25**

votes

**65**answers

7k views

### Fiction books about mathematicians? [closed]

What are some fiction books about mathematicians?
It seems to me rather difficult for writers to create good books on this subject.
Some years ago I thought there were no such books at all.
There ...

**10**

votes

**10**answers

956 views

### Properties of natural numbers such that there is a “very large largest” number with that property

I'm looking for properties (P) such that you would assume that there are infinitely many natural numbers with property (P) (for example because there are very large numbers with that property) but ...

**7**

votes

**10**answers

3k views

### Music: mathematical point of view (revised) [closed]

Mathematical analysis of music started when Pythagoras made his observations about consonant intervals and ratios of string lengths.
ADDED:
In the paper Mathematical Music Theory -- Status Quo 2000, ...

**3**

votes

**11**answers

516 views

### A list of symmetric statistics

I would like to have a list of pairs (or tuples) of combinatorial statistics that are (known or conjectured) to have symmetric distribution. Ideally, something like this has already been compiled, ...

**10**

votes

**1**answer

351 views

### List of Whitehead-like Problems

Whitehead problem is a rather well known problem:
Suppose that $G$ is an abelian group and $\mathrm{Ext}^1(G,\Bbb Z)=0$, is $G$ free?
It wasn't long before it was proved that if $G$ is ...

**7**

votes

**9**answers

2k views

### Not especially famous, long-open problems which higher mathematics beginners can understand

This is a pair to
Not especially famous, long-open problems which anyone can understand
So this time I'm asking for open questions so easy to state for students of subjects such as undergraduate ...

**34**

votes

**30**answers

4k 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 ...

**22**

votes

**2**answers

710 views

### Properties of functors and their adjoints

I am interested in collecting in this question a list of properties a functor $F$ may have and what those properties imply for left and right adjoints, $F^L$ and $F^R$, assuming they exist. There are ...