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A math quartet's breakthrough bridges number theory and geometry

75 points by espeed 11y ago

jsprogrammer 11y ago
Almost no discussion of the breakthrough itself?
- qmalzp 11y ago
  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 11y ago
    (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 + Ax + 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).
anusburger 11y ago
It's like a combined eharmony profile for nerds.
evincarofautumn 11y ago
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 11y ago
  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.
  - schoen 11y ago
    I think the parent commenter's complaint is about this journalistic article (from Quanta), which isn't by any of these mathematicians, just about them. Although I sympathize with this concern, it looks like two of these mathematicians write their own names (solely) in Romanization without tone markings in their own academic publications, for example http://arxiv.org/abs/1512.02683 http://arxiv.org/abs/1512.02683 linked from the article. So it's not necessarily something that other people are unilaterally doing to them. (Although Prof. Yun has put his Chinese name at the top of his homepage, but seemingly not in any of his papers.) Maybe other people are pressuring academics with non-Latin names indirectly not to use their names in native scripts in their publications because of editorial policies in journals that would discourage or forbid it, or, as you suggested, because of journals' software tools that would have compatibility problems.
  - yen223 11y ago
    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.
impendia 11y ago
(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.
paulpauper 11y ago
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 11y ago
  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.
CurtMonash 11y ago
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.

75 points by espeed 11y ago

jsprogrammer 11y ago

Almost no discussion of the breakthrough itself?

qmalzp 11y ago
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 11y ago
  (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 + Ax + 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).

anusburger 11y ago

It's like a combined eharmony profile for nerds.

evincarofautumn 11y ago

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 11y ago
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.
- schoen 11y ago
  I think the parent commenter's complaint is about this journalistic article (from Quanta), which isn't by any of these mathematicians, just about them. Although I sympathize with this concern, it looks like two of these mathematicians write their own names (solely) in Romanization without tone markings in their own academic publications, for example http://arxiv.org/abs/1512.02683 http://arxiv.org/abs/1512.02683 linked from the article. So it's not necessarily something that other people are unilaterally doing to them. (Although Prof. Yun has put his Chinese name at the top of his homepage, but seemingly not in any of his papers.) Maybe other people are pressuring academics with non-Latin names indirectly not to use their names in native scripts in their publications because of editorial policies in journals that would discourage or forbid it, or, as you suggested, because of journals' software tools that would have compatibility problems.
- yen223 11y ago
  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.

impendia 11y ago

(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.

paulpauper 11y ago

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 11y ago
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.

CurtMonash 11y ago

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.