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.
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}...
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 ...
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 ...
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. ...
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 ...
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 ...
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 $...
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 $$ R=F[x_i,y_i,r_i\ (i\geq 0)] $$ ...
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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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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.
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 ...
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 $SO(3)$ ...
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 ...
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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 ...
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 ...
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 ...
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
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 ...
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,...
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.
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 ...
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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