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One of the aspects of math that I find beautiful is the deep connection between often seemingly unrelated topics. Connecting PDEs, which is a branch of analysis
by vecter 7y ago
One of the aspects of math that I find beautiful is the deep connection between often seemingly unrelated topics. Connecting PDEs, which is a branch of analysis and what I consider "continuous" mathematics to something that seems to me like a discrete number theory problem is beautiful. The bit of intuition for why this might be the case in the article is nice:
> With a PDE, you plug in some values, get other values out, and repeat the process — all to understand that future state of the system. For any given PDE, mathematicians want to know if some starting values eventually lead to infinite values as an output or whether an equation always yields finite values, regardless of the values you start with.
> For Tao, this goal had the same flavor as investigating whether you always eventually get the same number (1) from the Collatz process no matter what number you feed in. As a result, he recognized that techniques for studying PDEs could apply to the Collatz conjecture.
- oefrha 7y agoThe connection is actually very well known. See https://en.wikipedia.org/wiki/Dynamical_system https://en.wikipedia.org/wiki/Dynamical_system.
- vecter 7y agoI don't doubt that. I'm not a mathematician, just a guy who took some classes in college. For a noob like me, this is all cool to see for the first time.
- xdavidliu 7y agoI don't actually think it is that well-known, you were completely right to point out. Also, I suspect the "dynamical systems" wikipedia page is completely irrelevant.
- oefrha 7y agoThe Collatz map defines a discrete dynamical system on N^+. I don't know about your background, but the connection isn't hard to see for anyone with a tangentially related background. I have studied https://en.wikipedia.org/wiki/Arithmetic_dynamics https://en.wikipedia.org/wiki/Arithmetic_dynamics (that page does mention Collatz incidentally) myself in a past research project, so the connection is known to me. The dynamical systems Wikipedia page serves as an entry point to the broad topic, but it's a huge field and intersects with many others so apparently a single short page isn't gonna cover everything. What I meant was that "Collatz's got something in common with PDEs" isn't the unique insight of Terence. Of course the connection isn't obvious to, say, the average coder, but it's universally known among mathematicians who have at least studied related fields for a little bit. Terence's magic is on a much higher level.
- xdavidliu 7y agoI did my PhD in theoretical physics, and part of my thesis was on nonlinear eigenvalue problems in PDEs. I wasn't aware of the connection; though I guess it really depends on what you mean by "it's well known".
- xdavidliu 7y agowhere on that page does it discuss the connection?
- emmelaich 7y agoMy favourite story on this is Feynman's use of PDEs in the Connection Machine design. http://longnow.org/essays/richard-feynman-connection-machine/ http://longnow.org/essays/richard-feynman-connection-machine...
- BlueTemplar 7y agoThank you so much for sharing this!
- edflsafoiewq 7y agoI've always wanted to see his analysis of the behaviour of the router in terms of PDEs.
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