# Tag Info

### Examples of eventual counterexamples

One could reasonably conjecture that there are no positive integers $a,b,c$ satisfying $$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4.$$ I say "reasonably", because the smallest ...

### No starter "accessible" well known open problems

How about: OPEN PROBLEM. Let $k$ be a field. Describe the group of automorphisms of the polynomial ring $k[x_1,\ldots,x_n]$, where $n \geq 3$. This is a seemingly basic problem in algebra. The ...

### No starter "accessible" well known open problems

Since I think the question has a reasonable interpretation, let me get the ball rolling with an open problem where I am not aware of any partial progress or proposals for a proof/counterexample. OPEN ...

### Examples of eventual counterexamples

$n$ is sufficiently large for $P(a)=T$ $\forall a\in\mathbb N$ to be a 'reasonable' conjecture to make. $\ldots$ where 'reasonable' is open to interpretation I won't be too surprised if a ...

### Proofs of the Chevalley-Warning Theorem

Whether the following proof is different from the other ones already mentioned could be debated, but to me it feels different enough to be worth mentioning separately. As noted in Gjergji Zaimi's ...
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### Errata for Atiyah–Macdonald

The statement of Corollary 11.2 should be replaced by If each $k_i=0$ and $P(M,t)\neq 0$, then for all sufficiently large $n$, ...

### High dimensional Lusin conjecture

See Michael Lacey's exposition on Carleson's theorem for some discussion of this, particularly Section 9 there. In higher dimensions one needs to specify the order of summation. If one sums over the ...
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### Where do root systems arise in mathematics?

They arise in the representation theory of quivers: Gabriel's theorem says that a connected quiver has finite representation theory type if and only if it is of type ADE, and then the indecomposable ...
1 vote

### Classic applications of Baire category theorem

Here's a good one, which essentially shows that every closed $2$-dimensional subset of the plane contains a region of the plane. Theorem. Let $X\subset \mathbb R^2$ be closed and $x\in X$. Suppose ...

### Where do root systems arise in mathematics?

I first came across root systems in the classification of finite reflection groups. A point group $\Gamma\subseteq\mathrm O(\Bbb R^n)$ is a reflection group if it is generated by reflections at ...

### Where do root systems arise in mathematics?

The eigenvalue distribution functions of random matrices in different universality classes are determined by the multiplicities of the restricted roots of the corresponding symmetric spaces, see ...

### How common is it for universities to create new positions for dual hires?

This is an old question that lingers on the unanswered queue. Nowadays, it would be migrated to Academia.SE, but that site didn't exist when this was asked. Anyway, I'll try to answer, in case it ...

### Homotopy theory and algebraic topology last 10 years. Is it a dying field?

If the criterion is “results using algebraic topology which shocked the mathematical community in the last 10 years”, then how about Abouzaid and Blumberg’s proof of the Arnol’d conjecture using ...
A better example: I believe conjecturally the equation $(a-N)^2 + (b-a^2)^2 + (c-b^2)^2 + (d-c^2)^2 + (x^2 - (d+a-1) y^2+1)^2$ in the variables $a,b,c,d,x,y$, with $N$ fixed, should do the trick for ...