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5 votes

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 ...
-1 votes

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 ...
4 votes

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 ...
2 votes

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 ...
0 votes

Examples of eventual counterexamples

The word "eventually" connotes a very long sequence of positive examples before the first counterexample. Gerry Myerson points out that no polynomial of the form $x^n-1,$ when factored over $...
5 votes

Examples of eventual counterexamples

Assertion: Every integer greater than 1 can be written as the sum of a prime number and a perfect power of a nonnegative integer. The smallest (and maybe only?) counterexample to this assertion is $11^...
2 votes

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 ...
Timothy Chow's user avatar
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0 votes

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 $n$, ...
6 votes

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 ...
Mark Lewko's user avatar
  • 11.8k
11 votes

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 ...
16 votes

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 ...
8 votes

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 ...
4 votes

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 ...
7 votes

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 ...
8 votes

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

No, it's not dying at all. If anything, now is the best time to do homotopy theory. Thanks to the recent work of Lurie and others, homotopy theory is easier than ever to get into (advances have ...
7 votes

Nice diophantine equations with large smallest solutions

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 ...

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