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MathPrinceps's profile
Laurens Gunnarsen
Laurens Gunnarsen
Laurens Gunnarsen
@MathPrinceps

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Laurens Gunnarsen

@MathPrinceps

Mathematical physicist and mentor to mathematically talented youth. Talent is that which bridges the gap between what can be taught and what must be learned.

Joined June 2012

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    1. Akiva Weinberger‏ @akivaw 14 Mar 2019
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      Replying to @MathPrinceps @divbyzero

      That's pi*cot(pi*x), you can prove it using complex analysis

      2 replies 0 retweets 4 likes
    2. Akiva Weinberger‏ @akivaw 14 Mar 2019
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      Replying to @akivaw @MathPrinceps @divbyzero

      Some Googling says this was first discovered by Euler (presumably using methods other than complex analysis)

      1 reply 0 retweets 1 like
    3. Laurens Gunnarsen‏ @MathPrinceps 14 Mar 2019
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      Replying to @akivaw @divbyzero

      Eisenstein's methods reach extremely far, and indeed inaugurated a profound sequence of developments in analytic number theory that continue to this day. Yet they relied essentially on "only high school algebra." It's not the tool that matters; it's the touch of a master's hand.

      2 replies 0 retweets 1 like
    4. Akiva Weinberger‏ @akivaw 14 Mar 2019
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      Replying to @MathPrinceps @divbyzero

      I see. I oughta look this stuff up. (Just noticed that if you plug in x=¼ you get the Leibniz series for pi)

      1 reply 0 retweets 1 like
    5. Laurens Gunnarsen‏ @MathPrinceps 14 Mar 2019
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      Replying to @akivaw @divbyzero

      Indeed, this approach also provides the most natural elementary explanation of Euler's answer to the Basel problem -- that is, to why the sum of the reciprocal squares is [(pi)^2]/6.

      1 reply 0 retweets 0 likes
    6. Akiva Weinberger‏ @akivaw 14 Mar 2019
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      Replying to @MathPrinceps @divbyzero

      This is the log-derivative of the product formula for the sine, right? Which is connected to both the Basel problem and the Wallis product. (Basel, Wallis, and Leibniz united under one formula, woot)

      1 reply 0 retweets 1 like
    7. Laurens Gunnarsen‏ @MathPrinceps 14 Mar 2019
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      Replying to @akivaw @divbyzero

      Yeah. And the key here is that Eisenstein unites all these threads by stressing periodicity, which he enforces via what is now called "averaging over the group" (in this case, the additive group of the integers.) Hard to exaggerate how far these ideas reach.

      1 reply 0 retweets 0 likes
    8. Akiva Weinberger‏ @akivaw 14 Mar 2019
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      Replying to @MathPrinceps @divbyzero

      So what does this look like for other groups?

      1 reply 0 retweets 1 like
    9. Laurens Gunnarsen‏ @MathPrinceps 14 Mar 2019
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      Replying to @akivaw @divbyzero

      Exactly the right question! The next group Eisenstein studied was the translation group in two dimensions, and its discrete subgroup of "integral translations." That gave him his theory of elliptic functions. Then you look at the modular group, and get automorphic functions.

      1 reply 0 retweets 1 like
    10. Akiva Weinberger‏ @akivaw 14 Mar 2019
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      Replying to @MathPrinceps @divbyzero

      Ah, yeah. Weierstrass p and all that. I started reading a book about this stuff. It's way over my head, but as far as I can tell, it's just magic. The fact that holomorphic functions on compact Riemann surfaces are constant is so overpowered.

      1 reply 0 retweets 1 like
      Laurens Gunnarsen‏ @MathPrinceps 14 Mar 2019
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      Replying to @akivaw @divbyzero

      In some sense, the Liouville theorem and its generalizations are at the heart of everything; it is on their account that all sorts of crucial vector spaces are provably finite-dimensional -- and from this a nearly limitless set of astounding (number-theoretical) things follow.

      1:19 PM - 14 Mar 2019
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