

Forty Minutes with a Fields Medallist - dpflan
http://www.t5eiitm.org/2015/07/forty-minutes-with-a-fields-medallist/

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biot
I can only infer from the picture of a person on the couch two-thirds of the
way through the article that this is about "Prof. Manjul Bhargava":
[https://en.wikipedia.org/wiki/Manjul_Bhargava](https://en.wikipedia.org/wiki/Manjul_Bhargava)

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flipmonk
I was looking around for his name too. Author: Perhaps a quick intro / context
might help at the top of the page.

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JadeNB
Indeed, it seems like the paragraph:

> After the talk about “Poetry, Drumming and Mathematics”, we arrived at the
> appointed time at the Bose–Einstein Guest House to find our man, with his
> mother, standing transfixed by the deer grazing on the lawns. We managed to
> prise him away for a while — here’s what followed.

is meant to provide that intro/context; but, like you and the grandparent, I
was puzzled about who 'our man' was.

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S4M
That man is so down to Earth, very inspiring.

For those who wonder like I did why the probability of picking a square-free
number is 6/Pi^2, it's explained on wikipedia:
[https://en.wikipedia.org/wiki/Square-
free_integer#Distributi...](https://en.wikipedia.org/wiki/Square-
free_integer#Distribution)

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imglorp
I enjoyed how he could explain his work in elementary terms, without invoking
any jargon. This is a sure sign he understands it.

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mrcactu5
Manjul makes it so simple. The odds of two numbers being relatively prime is
6/pi^2 so where's the circle? You can kind of see rotation symmetry if you
draw the relatively prime pairs of numbers in the coordinate plane. However
that symmetry is far from perfect.

[http://ocw.mit.edu/courses/mathematics/18-786-topics-in-
alge...](http://ocw.mit.edu/courses/mathematics/18-786-topics-in-algebraic-
number-theory-spring-2010/18-786s10.jpg)

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impendia
Yeah, but the rotational symmetry doesn't explain why the pi is there. (e.g.
why is it squared?)

If you can explain, convincingly and geometrically, why pi is present in this
formula then I would be VERY VERY interested to read.

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Intermernet
Have a look at
[https://en.wikipedia.org/wiki/Basel_problem](https://en.wikipedia.org/wiki/Basel_problem)
. I'm not a mathematician, and I'm probably wrong, but I think it has to do
with infinite series that "circle" around a value, and asymptotically approach
a value, only accurately expressed in terms of pi.

"Euler's original derivation of the value π2/6 essentially extended
observations about finite polynomials and assumed that these same properties
hold true for infinite series."

Sorry, not very convincing or geometric, and I'm sure someone else can provide
a better answer, but that's how I visualize it.

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impendia
No apologies necessary, I think that's just the state of mathematics.

The sum of the inverse fourth powers is pi^4/90\. Sixth powers, pi^6/915\. The
pattern keeps going for _even_ powers and _not_ for odd.

I don't think there is a geometric proof. But if there were, it would be
incredible.

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darkmighty
A simple explanation I enjoy (not quite visual, but geometric in some sense)
is the on the wiki using Fourier series, understanding it requires just a few
steps:

1) You accept Parseval's identity: by changing between orthogonal basis, the
energy (sum of absolute squared values) remains the same.

2) You find that the energy of the periodic function x mod pi is sum(1/n^2).

3) Evaluating the integral gives pi^2/6!

The square is neatly explained because it involves _energies_. I guess this is
why the pattern works only for even powers.

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journeeman
Wow, how inspiring! Thanks for the post.

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dpflan
Absolutely no problem. Great read for the weekend!

