A simplified version of the problem goes like this: Imagine that you are imprisoned in a tunnel that opens out onto a precipice two paces to your left, and a pit of vipers two paces to your right. To torment you, your evil captor forces you to take a series of steps to the left and right. You need to devise a series that will allow you to avoid the hazards — if you take a step to the right, for example, you’ll want your second step to be to the left, to avoid falling off the cliff. You might try alternating right and left steps, but here’s the catch: You have to list your planned steps ahead of time, and your captor might have you take every second step on your list (starting at the second step), or every third step (starting at the third), or some other skip-counting sequence. Is there a list of steps that will keep you alive, no matter what sequence your captor chooses?
Category: Science
Big Data’s Mathematical Mysteries
At a dinner I attended some years ago, the distinguished differential geometer Eugenio Calabi volunteered to me his tongue-in-cheek distinction between pure and applied mathematicians. A pure mathematician, when stuck on the problem under study, often decides to narrow the problem further and so avoid the obstruction. An applied mathematician interprets being stuck as an indication that it is time to learn more mathematics and find better tools.
I have always loved this point of view; it explains how applied mathematicians will always need to make use of the new concepts and structures that are constantly being developed in more foundational mathematics. This is particularly evident today in the ongoing effort to understand “big data” — data sets that are too large or complex to be understood using traditional data-processing techniques.
Our current mathematical understanding of many techniques that are central to the ongoing big-data revolution is inadequate, at best. Consider the simplest case, that of supervised learning, which has been used by companies such as Google, Facebook and Apple to create voice- or image-recognition technologies with a near-human level of accuracy. These systems start with a massive corpus of training samples — millions or billions of images or voice recordings — which are used to train a deep neural network to spot statistical regularities. As in other areas of machine learning, the hope is that computers can churn through enough data to “learn” the task: Instead of being programmed with the detailed steps necessary for the decision process, the computers follow algorithms that gradually lead them to focus on the relevant patterns.
Algorithms Need Managers, Too
In Shakespeare’s Julius Caesar, a soothsayer warns Caesar to “beware the ides of March.” The recommendation was perfectly clear: Caesar had better watch out. Yet at the same time it was completely incomprehensible. Watch out for what? Why? Caesar, frustrated with the mysterious message, dismissed the soothsayer, declaring, “He is a dreamer; let us leave him.” Indeed, the ides of March turned out to be a bad day for the ruler. The problem was that the soothsayer provided incomplete information. And there was no clue to what was missing or how important that information was.
Like Shakespeare’s soothsayer, algorithms often can predict the future with great accuracy but tell you neither what will cause an event nor why. An algorithm can read through every New York Times article and tell you which is most likely to be shared on Twitter without necessarily explaining why people will be moved to tweet about it. An algorithm can tell you which employees are most likely to succeed without identifying which attributes are most important for success.
Sean Parker : Hacker Philanthropist
Not your average pharma bro’ :
Hacker philanthropy is meant to be the antidote to what Parker calls the conservative, incremental work of most charitable foundations; a timidity he says is borne of institutional self-preservation and a need to assuage philanthropists’ “deep-seated anxiety that their capital may not be accomplishing anything”.
Hackers, by contrast, are iconoclasts drawn to fix the holes in big, complex systems, and they are willing to make bold experiments and embrace failure as a learning experience. Since the world’s billionaires lists are increasingly populated by computer programmers who have built insanely large tech companies, it is only a matter of time until their hacker mentality is brought into the world of philanthropy.
Mathematician Solves the Centuries-Old Sphere Problem in Higher Dimensions
It’s possible to build an analogue of the pyramidal orange stacking in every dimension, but as the dimensions get higher, the gaps between the high-dimensional oranges grow. By dimension eight, these gaps are large enough to hold new oranges, and in this dimension only, the added oranges lock tightly into place. The resulting eight-dimensional sphere packing, known as E8, has a much more uniform structure than its two-stage construction might suggest. “Part of the mystery here is this object turns out to be vastly more beautiful and symmetric than it sounds,” Cohn said. “There are tons of extra symmetries.”
Abnormal Capitalist 