MathOverflow

Να:
The standard proof of Fermat's Last Theorem (Wiles, 1995) establishes that no positive integers a, b, c satisfy aⁿ + bⁿ = cⁿ for n≥3. However, the structural reason why n=2 is the unique exponent admitting non-trivial solutions remains underexplored...

Principal Component Analysis is used to reduced the dimensions of atmospheric pressure grids (lat X long X time) into their most important modes of behaviour (e.g, the North Atlantic Oscillation is the PC1 of the North Atlantic pressure field).
I would like to know if there is a way to find the modes/splits of variance within a dataset (such as pressure grids) that are most explained by a seperate dataset. I suppose the analogy would be the cross-correlation compared to the autocorrelation...

It is a well-known conjecture in group theory stated that every non-abelian $p$-group has a non-inner automorphism of order $p$.
Last December, someone uploaded a paper Some results of cohomological properties of $p$-group and non-inner automorphism with order $p$ on non-abelian finite $p$-group to arXiv claiming to have proved this conjecture. However, the paper contains numerous...

Lee and Li conjectured the following.
$10/8$-conjecture. If $X$ is a closed, oriented, smooth $4$-manifold with evenintersection lattice $aU \oplus bE_8$, where $U$ is the hyperbolic lattice, then $a \geq |b|$.
This is equivalent to $b_2(X) \geq \frac{10}{8}|\sigma(X)|$, where $\sigma$ is the signature, and hence the name. (Note that if $X$ is spin, Matsumoto's $11/8$-conjecture proposes a small improvement by replacing $10/8$ with $11/8$.)...