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Comment by OmicronCeti | Hacker News Reader

Basically everything quanta writes is so good. They have some of the best science writers around in my opinion, and their efforts to make science and discovery accessible to non-experts and the public is admirable. I highly recommend their podcast that breaks down some of the popular stories, as well as "The Joy of X" which are long-form interviews with leading scientists.

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Funded by billionaire mathematician and hedge fund founder, where the algorithms make all the decisions

https://m.youtube.com/watch?v=QNznD9hMEh0

1976 winner of the Oswald Veblen Prize in Geometry

https://en.m.wikipedia.org/wiki/Oswald_Veblen_Prize_in_Geome...

The person referenced above is “James Simon”.

Simons*

Mostly known for being the founder of Renaissance Technologies which is one of the largest and most successful hedge funds.

Also the Simons Foundation, which funds the Simons Institute, which is just fantastic

Aka, the Simons in Chern-Simons theory.

(Albert Schwarz formulated Chern–Simons theory, a topological quantum field theory, using the Chern-Simons form)

Completely agree! I’ve really enjoyed Erica Klarreich’s writing in particular. I think she may very well be one of the most gifted and capable writers today on math in particular.

She interviewed just the right people too! She is like the equivalent of Ed Yong for math

It helps (and perhaps is no surprise, given her easy facility with the material she covers) that she's a Math PhD as well :)

They're entertaining to read, but I've never come away from one with any sort of understanding of the actual math and how I might use it. It would help if they at least defined the problem and the new theorem.

I figure this is because they're trying to make it accessible. As a somewhat mathematically literate non-mathematician I doubt I'd get much out of the real formulae that stump today's researchers. Quanta do a good job at dumbing it down enough to make sense to someone like me without making it completely empty. More specifically, Erica Klarreich does a good job of it - she seems to write most if not all of the Quanta articles that end up on the HN front page.

Did they define "substantial area" and I just missed it?

No, and in particular by speaking about slices with a lower dimension d-1 it becomes not immediately clear, what their volume has to do with the one in dimension d of the original body, mentioned at the begin of the article.

You missed it.

Bourgain guessed that some of these lower-dimensional slices must have substantial area. In particular, he conjectured that there is some universal constant, independent of the dimension, such that every shape contains at least one slice with area greater than this constant.

In other words there exists C > 0 such that if the n-dimensional hypervolume of an n-dimensional convex shape is 1, then there must be an n-1'th dimensional slice of n-1-dimensional hypervolume at least C.

What was proven is weaker. For any ε > 0 there is an N such that if N < n, then any n-dimensional convex shape of n-dimensional hypervolume 1 must have an n-1'th dimensional slice of n-1-dimensional hypervolume at least 1/n^ε.

It turns out that the exact things that were proven are good enough to improve our bounds on how quickly various machine learning algorithms will converge. Which means we aren't just hoping based on how they worked in a few examples, we have a theory explaining it.

They actually did a better job of that with this problem than I usually see.

But if you don't have a substantial background, it may be hard to track.

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