Terence Tao on primes and other stuff
Two interesting videos I came across:
He’s talking to an audience who is presumed (..I assume, from his comments along the way..) not to know what a limit is or what the central limit theorem is about, so of course this is a very ‘unstructured’ review – he has to take a lot of shortcuts… As he points out at some point during the end when dealing briefly with his own work: “There’s absolutely no chance I can tell you about how this is proven” – but it’s still interesting stuff. As some of you will know I’ve covered many of the things covered in this video before here on the blog, and if you’re interested to know more about these things Brit Cruise’ applied math playlist on Khan Academy deals with similar stuff and has a more detailed overview of some of these things. This is probably also a good place to remind you of the existence of Simon Singh’s The Code Book.
The second video is more technical. If you watched the first one you should probably just skip the first ten minutes of this which is mostly just a recap:
A quote from the second lecture: “One funny thing about number theory, as opposed to other fields of mathematics, is that we can’t prove everything we want […] but we can conjecture things really well … so almost any question about the primes we can say very confidently whether it’s true or false, whether something is true about the primes, and we have lots of numerical evidence, lots of heuristic evidence, but we can’t prove most of that. Our ability to conjecture is about one hundred years ahead of our ability to prove things … we have all these conjectures, and we’re pretty much, like 99 percent certain they’re all true, it’s just that we can’t prove any of them.”
If you want to know what he and Ben Green did (and you happen to know a lot of stuff about math/number theory/…), you can have a go at their paper here). Here’s a screencap from the paper, illustrating what kind of stuff’s involved (click to view full size):
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