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x********i
发帖数: 905 | 1 http://arxiv.org/abs/1509.05576
The Riemann hypothesis is, and will hopefully remain for a long time, a
great motivation to uncover and explore new parts of the mathematical world.
After reviewing its impact on the development of algebraic geometry we
discuss three strategies, working concretely at the level of the explicit
formulas. The first strategy is "analytic" and is based on Riemannian spaces
and Selberg's work on the trace formula and its comparison with the
explicit formulas. The second is based on algebraic geometry and the Riemann
-Roch theorem. We establish a framework in which one can transpose many of
the ingredients of the Weil proof as reformulated by Mattuck, Tate and
Grothendieck. This framework is elaborate and involves noncommutative
geometry, Grothendieck toposes and tropical geometry. We point out the
remaining difficulties and show that RH gives a strong motivation to develop
algebraic geometry in the emerging world of characteristic one. Finally we
briefly discuss a third strategy based on the development of a suitable "
Weil cohomology", the role of Segal's Gamma-rings and of topological cyclic
homology as a model for "absolute algebra" and as a cohomological tool. | l******r
发帖数: 18699 | 2 great 这是篇很好的综述文章，谢谢
打算下周让学生报告一下
world.
spaces
Riemann
【在 x********i 的大作中提到】
: http://arxiv.org/abs/1509.05576
: The Riemann hypothesis is, and will hopefully remain for a long time, a
: great motivation to uncover and explore new parts of the mathematical world.
: After reviewing its impact on the development of algebraic geometry we
: discuss three strategies, working concretely at the level of the explicit
: formulas. The first strategy is "analytic" and is based on Riemannian spaces
: and Selberg's work on the trace formula and its comparison with the
: explicit formulas. The second is based on algebraic geometry and the Riemann
: -Roch theorem. We establish a framework in which one can transpose many of
: the ingredients of the Weil proof as reformulated by Mattuck, Tate and
| l******r
发帖数: 18699 | 3 暑假回国开会碰见一个老同学拉着我跟我谈黎曼猜想，说是听说代数几何＋weil猜想的
证明有戏，想知道国际上最新发展。我的兴趣顿时被挑起来。不过国内上网太麻烦，
google个东西费劲。你这个综述正好可以给他看看。另外大家有什么东西及时交流一下
。RH的credit一定是我们中国人的才行！
world.
spaces
Riemann
【在 x********i 的大作中提到】
: http://arxiv.org/abs/1509.05576
: The Riemann hypothesis is, and will hopefully remain for a long time, a
: great motivation to uncover and explore new parts of the mathematical world.
: After reviewing its impact on the development of algebraic geometry we
: discuss three strategies, working concretely at the level of the explicit
: formulas. The first strategy is "analytic" and is based on Riemannian spaces
: and Selberg's work on the trace formula and its comparison with the
: explicit formulas. The second is based on algebraic geometry and the Riemann
: -Roch theorem. We establish a framework in which one can transpose many of
: the ingredients of the Weil proof as reformulated by Mattuck, Tate and
| h********0
发帖数: 12056 | | l******r
发帖数: 18699 | 5 看我开会回来不久发的帖子问这个事，那时这篇文章还没出来。我发这个帖子时9月9，
这篇文章是9月18号出来的。感觉RH又要火了
http://www.mitbbs.com/article/Mathematics/31229789_3.html
world.
spaces
Riemann
【在 x********i 的大作中提到】
: http://arxiv.org/abs/1509.05576
: The Riemann hypothesis is, and will hopefully remain for a long time, a
: great motivation to uncover and explore new parts of the mathematical world.
: After reviewing its impact on the development of algebraic geometry we
: discuss three strategies, working concretely at the level of the explicit
: formulas. The first strategy is "analytic" and is based on Riemannian spaces
: and Selberg's work on the trace formula and its comparison with the
: explicit formulas. The second is based on algebraic geometry and the Riemann
: -Roch theorem. We establish a framework in which one can transpose many of
: the ingredients of the Weil proof as reformulated by Mattuck, Tate and
| l******r
发帖数: 18699 | 6 主编nash刚去世了
world.
spaces
Riemann
【在 x********i 的大作中提到】
: http://arxiv.org/abs/1509.05576
: The Riemann hypothesis is, and will hopefully remain for a long time, a
: great motivation to uncover and explore new parts of the mathematical world.
: After reviewing its impact on the development of algebraic geometry we
: discuss three strategies, working concretely at the level of the explicit
: formulas. The first strategy is "analytic" and is based on Riemannian spaces
: and Selberg's work on the trace formula and its comparison with the
: explicit formulas. The second is based on algebraic geometry and the Riemann
: -Roch theorem. We establish a framework in which one can transpose many of
: the ingredients of the Weil proof as reformulated by Mattuck, Tate and
| i*****e
发帖数: 218 | 7 乘机搭车问个问题，
在 Andrew Granville 2007 年的一篇文章中， 他说：
"we show that an averaged strong form of Goldbach’s conjecture is
equivalent to the Generalized Riemann Hypothesis; "
下面是他的文章：
http://www.dms.umontreal.ca/~andrew/PDF/GoldbachFinal.pdf
我这方面基础比较弱， 不能重复他的证明。
大家能不能帮忙看看，
１. 他说的这个：an averaged strong form of Goldbach’s conjecture 到底是什
么 ？
２. 是不是说如果证明了上面说的， ”an averaged strong form of Goldbach’s
conjecture“， 就证明了 GRH ？
多谢大家。
world.
spaces
Riemann
【在 x********i 的大作中提到】
: http://arxiv.org/abs/1509.05576
: The Riemann hypothesis is, and will hopefully remain for a long time, a
: great motivation to uncover and explore new parts of the mathematical world.
: After reviewing its impact on the development of algebraic geometry we
: discuss three strategies, working concretely at the level of the explicit
: formulas. The first strategy is "analytic" and is based on Riemannian spaces
: and Selberg's work on the trace formula and its comparison with the
: explicit formulas. The second is based on algebraic geometry and the Riemann
: -Roch theorem. We establish a framework in which one can transpose many of
: the ingredients of the Weil proof as reformulated by Mattuck, Tate and
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