{\huge Halloween Math Style}
The top scariest possible results \vspace{.5in}
Washington Irving was a famous writer of the early 1800's who is best known for his short stories. ``The Legend of Sleepy Hallow'' was based on the folklore that each Halloween a decapitated Hessian soldier, killed in the American Revolution, rises as a ghost, a nasty ghost, who searches for his lost head.
Today is Halloween and while Ken and I are not searching for any lost heads, we do believe it is a good day to think about scary stories.
It's Halloween---variously Allhalloween, All Hallows' Eve, or All Saints' Eve---and we thought we share some really scary results with you. It is the beginning of Allhallowtide, which includes All Saints' Day on Nov. 1 and All Souls' Day on Nov. 2.
Results
Here are some of the top scariest results we can imagine happening on Halloween. May they not happen---may you and we get treats, not tricks.
A New Simple Group
A group of physicists at CERN have been working on a string theory in 1,729 dimensions. In using higher-order amplituhedra to remove infinities, they discovered a new large symmetric structure. They noticed that the group $latex {\cal H}&fg=000000$ of this structure seemed interesting. And it was. Group theorists at CalTech have verified that it is not in the current list of simple groups.
Group theorists have divided into two ``groups'': those looking for where the error occurred in the current proof, and those checking which applications of the classification still are correct theorems.
A Special Even Number
A teen in high school for her science project studied the curious family of numbers
$latex \displaystyle p + q + 100, &fg=000000$
where $latex {q=p+2}&fg=000000$ and both $latex {p}&fg=000000$ and $latex {q}&fg=000000$ are primes. She gave an analytical proof that if there are infinitely many such numbers then not all of them can be the sum of two primes. The proof is not elementary but is clever and seems correct. Number theorists are of course upset because this implies that either the Twin Prime conjecture or the Goldbach conjecture is false---and the proof doesn't tell which.
Quasi-Gems
An infinite sequence $latex {[C_n]}&fg=000000$ of integers has been found such that the polynomials defined by $latex {f_0 = x}&fg=000000$ and recursively for $latex {n \geq 1}&fg=000000$,
$latex \displaystyle f_n = f_{n-1}^2 + C_n, &fg=000000$
have at least $latex {\frac{2^n}{n}}&fg=000000$ distinct integer roots in the range $latex {[-2^{n^2},+2^{n^2}]}&fg=000000$. For the slight disruption this could cause see this post.Sam Bonwit
SIGACT just accepted a paper for the 2017 STOC conference. The sole author is named Sam Bonwit. It extends several recent FOCS papers on theoretical aspects of machine learning (ML), beginning with a short proof of a previous very difficult theorem in ML. The committee has just discovered that the paper was created completely by a deep learning algorithm, with no human intervention.
A Complexity Result
The class $latex {L}&fg=000000$, logspace, has just been shown to be equal to $latex {\oplus\mathsf{P}}&fg=000000$. The proof seems right and of course solves the P=NP question. Besides the shock the theory community is trying to see what can be saved from the past, since many conditional theorems are now gone. A very bad trick.
Nobel Less Oblige?
A group of physicists not at CERN have been working on a non-string theory in 4 spacetime dimensions. They have proved that for any universe is which the cosmological constant Lambda is not exactly zero, spacetime explodes with the intensity of quadrillions of hydrogen bombs per cubic nanometer per nanosecond. It is a consequence of the wave-particle duality for inflation. This has led other scientists to consider revising the statistical confidence for dark energy.
Open Problems
What are your scary results that you could imagine?