123
votes

### What are examples of (collections of) papers which "close" a field?

In this classic article, Steinitz closed not just one, but all fields.

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78
votes

### Theorems that are essentially impossible to guess by empirical observation

One of the most interesting examples that happened recently is the Katz-Sarnak conjecture asserting that the average rank of elliptic curves (ordered by some reasonable height) defined over $\mathbb{Q}...

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75
votes

### Theorems that are essentially impossible to guess by empirical observation

Bootstrap percolation is a two-dimensional two-state cellular automaton with a von Neumann 5-square ("plus") neighborhood where a "white" cell become "black" if it has at ...

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73
votes

Accepted

### $A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$

Let $A$ be the additive group of bounded sequences of elements of $\mathbb{Z}[\sqrt{2}]$. Clearly $A\cong A\oplus\mathbb{Z}[\sqrt{2}]\cong A\oplus\mathbb{Z}^2$ as abelian groups, so we just need to ...

- 30.4k

71
votes

### How would you have answered Richard Feynman's challenge?

There's a certain gaming/sporting aspect to Feynman's challenge that works in his favor. First of all, as phrased, the challenge gives him a 50/50 shot at being right even if he guesses randomly. ...

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60
votes

### The use of computers leading to major mathematical advances II

There is the recent computer-assisted verification of some key statements by Scholze and Clausen about "condensed mathematics". The task has been accomplished by Buzzard, Commelin, and ...

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59
votes

Accepted

### What are examples of (collections of) papers which "close" a field?

Let me preface this by saying that this is just my own account, based on various conversations I've had over the years with many mathematicians, of the following example.
In 1976, William Thurston ...

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59
votes

### Theorems that are essentially impossible to guess by empirical observation

Letting $\pi$ be the prime counting function and $\mathrm{Li}$ the logarithmic integral, Littlewood proved in his 1914 article "Sur la distribution des nombres premiers" that the difference $...

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56
votes

Accepted

### $R$ is isomorphic to $R[X,Y]$, but not to $R[X]$

The answer to this quite beautiful question is that there does exist a commutative ring $R$ with $R\cong R[X,Y]$ but $R\not\cong R[X]$.
Let $F$ be a field, and take
$$
R=F[x_i,y_i,r_i\ (i\geq 0)]
$$
...

- 15.4k

56
votes

Accepted

### Theorems demoted back to conjectures

I just discovered that
Wikipedia maintains a page entitled "List of incomplete proofs."
Each of the more than $60$ entries is marked with these symbols:
"Several of the examples on the list were ...

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53
votes

Accepted

### Examples of algorithms requiring deep mathematics to prove correctness

Group Isomorphism of simple groups. There is a trivial polynomial time algorithm for testing if two (finite) simple groups $G$ and $H$, specified by their multiplication tables, are isomorphic: guess ...

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52
votes

### Request for examples: verifying vs understanding proofs

Don Zagier has a well-known paper, A one-sentence proof that every prime $p\equiv 1\pmod 4$ is a sum of two squares. An undergraduate mathematics major should be able to verify that this proof is ...

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51
votes

### Taking a theorem as a definition and proving the original definition as a theorem

From a conversation I had with Gian-Carlo Rota when I was undergraduate, I know that one simple but important example that he specifically had in mind was the calculus of vector fields (whether ...

Community wiki

49
votes

### The use of computers leading to major mathematical advances II

Here is an example of type A: Stavros Garoufalidis and Don Zagier have extensive work on refinements of Kashaev's Volume Conjecture (which relates the order of growth of the values of Jones ...

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47
votes

### Examples of algorithms requiring deep mathematics to prove correctness

The Miller-Rabin tests determines whether an integer $n$ is prime in time $O_{\epsilon}((\log n)^{4+\epsilon})$. The bound on the running time is conditional on the truth of the Generalized Riemann ...

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46
votes

### How would you have answered Richard Feynman's challenge?

Can you hear the shape of a drum?
Easy to understand problem for a physicist, with a non-trivial answer given by Gordon, Webb and Wolpert in 1990.

Community wiki

44
votes

### What are examples of (collections of) papers which "close" a field?

Index theorem of Atiyah and Singer closed a substantial field of research in the 1960s. I knew people who were working in this field, and had to switch the field of their research
completely.
A more ...

Community wiki

42
votes

### Proofs without words

Late to the party, but David Lehavi and Bob Palais both mentioned the proof that $\pi_1(SO(3))$ has an element of order 2. In fact it is the only nontrivial element, and so the double cover of $SO(3)$ ...

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40
votes

Accepted

### A topology on $\Bbb R$ where the compact sets are precisely the countable sets

There is no such topology.
Suppose there were. Then $\mathbb R$ itself is not compact, so there is an open cover $\mathcal U$ of $\mathbb R$ with no finite subcover. Using recursion, we can construct ...

- 15.2k

39
votes

### Non-homeomorphic spaces that have continuous bijections between them

I know this is super old, but somebody asked the same question again (Non-homeomorphic topological spaces) and so I wanted to share a "proof by picture" that settles the question.
(I came up with ...

Community wiki

38
votes

### Taking a theorem as a definition and proving the original definition as a theorem

Many of the standard abstract mathematical structures were first defined and studied "externally" (in terms of some sort of concrete representation) and only later defined "internally&...

36
votes

### Proofs without words

$$\arctan \frac{1}{3} + \arctan \frac{1}{2} = \arctan 1$$
It's easy to generalize this to
$$ \arctan \frac{1}{n} + \arctan \frac{n-1}{n+1} = \arctan 1, \text{ for } n \in \mathbb{N}$$
which can ...

33
votes

### Proofs without words

This proof-without-words of the Pythagorean Theorem is far from a new one, but it's the first one I've ever seen 'in the wild' (this photo was snapped after finishing dinner at a Mongolian Grill ...

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32
votes

Accepted

### can another topology be given to $\mathbb R$ so it has the same continuous maps $\mathbb R\rightarrow \mathbb R$?

The only topology similar to the Euclidean topology on $\mathbb{R}$ is the Euclidean topology.
Suppose there is such a topology $\tau$. I'll use "open," "continuous," etc. to mean with respect to ...

- 4,104

32
votes

### The use of computers leading to major mathematical advances II

Here is an interesting one. Reinforcement learning to generate counter-examples to several open conjectures in combinatorics and graph theory.
https://arxiv.org/abs/2104.14516

Community wiki

32
votes

### Theorems that are essentially impossible to guess by empirical observation

Recent breakthrough work of Ben Green together with an improvement by Zach Hunter imply that there is a red-blue coloring of $[1,n]$ with no red $3$-term progression and no blue $k$-term progression ...

Community wiki

31
votes

### Theorems demoted back to conjectures

For over $130$ years people have been steadily looking for a resolution to the following problem: what is the maximum number of limit cycles for the system of differential equations $x'=f(x,y), y'=g(x,...

Community wiki

31
votes

### What are examples of (collections of) papers which "close" a field?

Bell's inequality killed the search for the elusive hidden variable theory that was supposed to complete Quantum Mechanics. IMO this qualifies because the result is mathematical.

Community wiki

31
votes

### Request for examples: verifying vs understanding proofs

Ivan Niven has published A simple proof that $\pi$ is irrational. Verifying that the proof is correct requires only elementary calculus. On the other hand, to "understand" it, a professional ...

Community wiki

30
votes

### What are examples of (collections of) papers which "close" a field?

This is not, perhaps, a very large area, nor a complete "ending", but it was an interesting development in early semigroup theory that I think bears writing down.
Some background, first. A semigroup $...

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