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Think about $\left({\phantom-d\phantom--b\atop-c\phantom{--}a}\right)$ as $tI - A$ where $t=a+d$ is the trace of $A$. Since $A$ satisfies its own characteristic equation (Cayley-Hamilton), we have $A^ …

I would have preferred not to comment seriously on Mochizuki's work before much more thought had gone into the very basics, but judging from the internet activity, there appears to be much interest i …

One problem which I think is mentioned in Guy's book is the integer block problem: does there exist a cuboid (aka "brick") where the width, height, breadth, length of diagonals on each face, and the l …

I'll take a stab at answering this controversial question in a way that might satisfy the OP and benefit the mathematical community. I also want to give some opinions that contrast with or at least c …

I apologize for posting as an answer what should really be a comment, connected to one of Jacques Carette's comments on my earlier answer. Unfortunately, this is way too long for a comment. Jacques …

I find some of this exchange truly depressing. There is a subject of ``brave new algebra''and there are myriads of past and present constructions and calculations that depend on having concrete and …

One can see this effect qualitatively from Newtonian first principles such as $F=ma$ (as opposed to Hamiltonian or Lagrangian principles, such as conservation of energy and angular momentum) by lookin …

The moving sofa problem: What rigid two-dimensional shape has the largest area $A$ that can be maneuvered through an L-shaped planar region with legs of unit width? So far the best results are $2.2195 …

Can we cover a unit square with $\dfrac1k \times \dfrac1{k+1}$ rectangles, where $k \in \mathbb{N}$? (Note that the areas sum to $1$ since $\displaystyle \sum_{k \in \mathbb{N}}\dfrac1{k(k+1)} = 1$) …

This is the second time I've seen this question on MathOverflow and this will be the second time I've posted this answer. Singmaster's conjecture says there is a finite upper bound on the number of ti …

The answer to 1 is yes. For the purpose of this answer, a bipointed space is a topological space $J$ equipped with distinct closed points $e_0$ and $e_1$. As you say, for any bipointed space $J = (J …

The lonely runner conjecture. As Wikipedia puts it: Consider $k + 1$ runners on a circular track of unit length. At $t = 0$, all runners are at the same position and start to run; the runners' sp …

Actually even schoolchildren calculate group co-cycle. (Without knowing that it is called like this). Cohomology occurs in everyday life as soon as one learns to count. 5+7 = 1 2 4 + 5 = 0 9 2 + 8 = …

If $c(G)> 5|G|/8$, then the average character has a dimension-squared of less than $8/5$, so at least $4/5$ of the characters are dimension $1$ (since the next-smallest dimension-squared is $4$), so t …

As others have suggested, your friend is getting it backwards. He's like a hammer asking what a carpenter is useful for. Given a field (of mathematics, say), there are typically some fields that are …