# Tag Info

## Hot answers tagged examples

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

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

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
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 ...
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 ...

### 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 ...

### 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 ...

### 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 ...

### 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 ...

### 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.

### 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 ...

### 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 $SO(3)$ ...
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

### 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 ...

### 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&...

### 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 ...

### 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 ...
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 ...

### 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
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 ...