Homotopy theory of schemes, Grothendieck's anabelian program, rational points

图式的同伦论、格洛腾迪克的阿贝尔纲领、有理点

• 批准号: 1406380
• 负责人: Kirsten Wickelgren
• 金额: $ 14.3万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-15 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406380&HistoricalAwards=false
• 关键词: Homotopy; theory; schemes; Grothendieck; anabelian

Finding solutions to polynomials in the integers or rational numbers arises naturally while counting objects subject to various constraints. It is also one of the oldest problems in mathematics. The solutions to the same polynomials over the complex numbers form topological spaces. For example, the complex solutions to the degree n Fermat equation is a torus with (n-1)(n-2)/2 holes. The fact that the shape of the space of complex solutions influences the solutions over rational numbers or integers can be viewed as a first instance of the utility of using methods of homotopy theory to study this problem. Homotopy theory gives machinery to replace a procedure by a derived version which frequently gives more control over the problem. For example, consider the procedure of taking a rotating sphere and returning the points which do not move. This procedure can be derived to produce a space called the homotopy fixed points, which records not only the points which do not move, but also compatible paths between points and where they have traveled. Fixed points and homotopy fixed points are equivalent under some hypotheses. If fixed points and homotopy fixed points are equivalent for a certain analogue of the space of complex solutions of polynomial equations, one can show that the solutions to these polynomials over the rationals are then determined by the loops on the corresponding space of complex solutions, under certain restrictions. This latter prediction is part of Grothendieck's anabelian program, and is unsolved. It produces strong restrictions on the solutions of the corresponding equations. The focus of this proposal is to study solutions to polynomial equations and Grothendieck's anabelian program from this perspective. The project also aims to stimulate research in homotopy theory, and make the tools of this theory available to mathematicians in very different areas, and other scientists more generally.The projects in this proposal share the approach wherein one views a scheme as a space in the sense of Morel-Voevodsky's A1-homotopy theory, and then applies various realization functors, for instance to Z/2-equivariant spaces by taking C-points of a scheme over R, or to pro-spaces with an action of the absolute Galois group of the base field for schemes over more general fields. The Principal Investigator studies the pro-space maps from the étale homotopy type of a field k to the étale homotopy type of the projective line minus three points using lower central series approximations to the latter. Additionally James-Hopf maps in A1-homotopy theory are used to study the same mapping space. Both have applications to Grothendieck's anabelian program. Running the same methods backwards, produces results on the algebraic topology of schemes starting from information about solutions to polynomial equations. For instance, the Principal Investigator continues a study of the differential graded algebra associated to group cohomology of absolute Galois groups. Information about the unstable category of spaces in the sense of Morel-Voevodsky is sought in conjunction.

在计算受各种约束的对象时自然会出现多项式的解，这也是数学中最古老的问题之一。例如，复数上的相同多项式的解。 n 次费马方程的解是一个有 (n-1)(n-2)/2 个孔的环面 复数解的空间形状影响有理数或整数的解这一事实可以被视为第一个。的实例使用同伦理论方法来研究这个问题的实用性提供了用派生版本代替过程的机制，这通常可以更好地控制问题，例如，考虑采用旋转球体并返回旋转点的过程。这个过程可以推导产生一个称为同伦不动点的空间，它不仅记录了不动的点，而且还记录了点之间的兼容路径以及它们所经过的位置。如果不动点和同伦。不动点对于多项式方程的复解空间的某个类似物是等效的，可以证明这些有理数上的多项式的解是由复解的相应空间上的循环在一定限制下确定的。预测是格洛腾迪克阿纳贝尔程序的一部分，并且尚未解决。它对相应方程的解产生了很强的限制。本提案的重点是研究多项式方程和格罗腾迪克阿贝尔程序的解。该项目还旨在促进同伦理论的研究，并使该理论的工具可供不同领域的数学家以及更广泛的其他科学家使用。该提案中的项目共享该方法，一个视图一个方案。作为 Morel-Voevodsky 的 A1 同伦理论意义上的空间，然后应用各种实现函子，例如通过在 R 上采用方案的 C 点来应用到 Z/2 等变空间，或者对于更一般域上的方案，具有基域绝对伽罗瓦群作用的亲空间 首席研究员研究了从域 k 的 étale 同伦类型到射影线减的 étale 同伦类型的亲空间映射。三个点使用后者的下中心级数近似。此外，A1 同伦理论中的 James-Hopf 映射也用于研究相同的映射空间。运行相同的阿纳贝尔程序。方法向后，从有关多项式方程解的信息开始产生方案的代数拓扑结果。例如，首席研究员继续研究与绝对伽罗瓦群的群上同调相关的微分分级代数有关空间不稳定类别的信息。在 Morel-Voevodsky 的意义上，寻求结合。