
A math quartet's breakthrough bridges number theory and geometry - espeed
https://www.quantamagazine.org/20151208-four-mathematicians/
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impendia
(Working number theorist here.)

What these guys are doing is definitely awesome. (A bit removed from my
specialty, so I can't answer detailed questions about it unfortunately.)

But let me add that there are already a LOT of bridges between number theory
and algebraic geometry. To get a sense of the scope, google "Arithmetic
geometry" or "Diophantine geometry" and you will find much, including (large)
books on these topics.

Meanwhile, here is a classical one. How do you find all integer solutions to
the equation a^2 + b^2 = c^2?

First of all, it is fairly easy to see that this is the same problem as
finding points on the circle x^2 + y^2 = 1 where both x and y are _rational_
numbers. Now, to find these use "stereographic projection". See here (scroll
down to "stereographic approach"):

[https://en.wikipedia.org/wiki/Pythagorean_triple](https://en.wikipedia.org/wiki/Pythagorean_triple)

The bottom line is that there is a picture you can draw which graphically
gives you an easy solution to this number theory problem -- and once you draw
the right picture, it is easy to use it to write down the algebraic solution
as well.

These four mathematicians are continuing a long and wonderful tradition of
finding applications of geometry to other areas of mathematics.

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evincarofautumn
Their Chinese names are Zhāng Wěi (张伟), Yùn Zhīwěi (恽之玮), and Yuán Xīnyì
(袁新意); I can’t find one for Zhu Xinwen.

I wish English-language publications would treat Chinese names better. Without
the Chinese characters, you can’t tell what their name actually is, and
without accent marks on the transliteration, you can’t tell how to pronounce
it.

Granted, it doesn’t matter to most English-speaking readers, but it _should_
matter, because _those are their names_. It’s no wonder so many Chinese feel
the need to adopt an extra name when moving to the West.

~~~
nabla9
Do English-language publications change the way authors write their names?

I wonder if this is somehow related to Bibliography and Bibliographical
software, indexes and references. If you have started your career with certain
name in BibTeX without accent marks, changing it later may screw up
references.

~~~
yen223
The biggest source of confusion is with the surname/given-name ordering.
Chinese folks put their surname first and their given name last.

E.g. Wei Zhang in the story should be Zhang Wei - his surname is Zhang.

I won't be surprised if he goes by Wei Zhang because that's how his name
appears on official documents.

~~~
schoen
That's definitely a big source of confusion, but in the case of Wei Zhang, he
uses the Western name order on his own homepage and for his e-mail address.

[https://www.math.columbia.edu/~wzhang/](https://www.math.columbia.edu/~wzhang/)

(Yes, it's possible that someone else originally chose this order for him.)

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jsprogrammer
Almost no discussion of the breakthrough itself?

~~~
qmalzp
From the paper:

"Another noteworthy feature of our work is that we need not restrict ourselves
to the leading coefficient in the Taylor expansion of the L-functions: our
formula is about the r-th Taylor coefficient of the L-function regardless
whether r is the central vanishing order or not. This leads us to speculate
that, contrary to the usual belief, central derivatives of arbitrary order of
motivic L-functions (for instance, those associated to elliptic curves) should
bear some geometric meaning in the number field case."

A one-line Clay Prize motivation: Geometric interpretations of higher order
derivatives of L-functions could potentially be leveraged (or more likely,
illuminate a path) to make progress on conjectures about order of vanishing of
L-functions, e.g. Birch Swinnerton-Dyer.

~~~
williamstein
(I'm a number theorist, and this is very close to my research area.) The
Gross-Zagier formula (along with work of Kolyvagin) from the 1980s proved the
Birch and Swinnerton-Dyer conjecture when "r_an <= 1"; this conjecture is an
amazing link between analysis and arithmetic, and also one of the Clay
Problems. This number "r_an" is the order of vanishing of a certain
"generating function" that counts the number of solution to y^2 = x^3 + A _x +
B modulo all prime numbers. For about 30 years now, people have been
completely stumped at coming up with even a wishy-washy conjectural half_
guess* as to what to do when r_an is 2 or larger. In number theory a standard
approach is to replace the integers Z = {... -2, -1, 0, 1, 2, ...} with a
polynomial ring F_p[X] over finite field, and try to solve analogous problems
(this is called working "over a functional field"). The paper that this
article is about has come up with exactly what to do in case r_an is 2 or
larger in the function field setting. So far they haven't figured out how to
do anything over the rational numbers yet. Sometimes solving a problem over
function fields provides incredible and valuable insight into how to solve the
analogous problem over the rational numbers, and sometimes it doesn't (e.g.,
the function field analogue of Fermant's Last Theorem and the ABC conjecture
are both basically trivial to deal with, whereas the same problem over the
rational numbers is ridiculously hard).

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CurtMonash
When I was at Harvard in the 1970s, I just assumed (algebraic) number theory
and algebraic geometry had heavy overlap. I'm surprised that anybody would
still argue this is surprising.

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paulpauper
the pdf to the paper:
[http://arxiv.org/pdf/1512.02683v1.pdf](http://arxiv.org/pdf/1512.02683v1.pdf)

The amount of background research required to even try to understand this is
immense

It's just nuts. How does one begin to put the pieces together

~~~
jjaredsimpson
You go to school for years and become an expert in a field. Understanding
isn't out of the reach of any graduate student in the relevant field I would
assume.

