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.

在整数或合理数字中找到对多项式的解决方案自然出现，同时计算受各种约束的对象。它也是数学中最古老的问题之一。在复数上相同多项式的解决方案形成了拓扑空间。例如，n fermat方程的复杂解决方案是具有（n-1）（n-2）/2孔的Toru。复杂解决方案空间的形状会影响解决方案超过有理数或整数的事实，这是使用同义理论方法来研究此问题的第一实例。同型理论使机械通过派生版本替换过程，该版本经常可以对问题进行更多控制。例如，考虑采用旋转球并返回不移动的点的过程。可以得出此过程，以产生一个称为同质固定点的空间，该空间不仅记录了不移动的点，而且还记录了点之间的兼容路径和它们所在的位置。在某些假设下，固定点和同质固定点等效。如果固定点和同质固定点是多项式方程复杂溶液空间的某个类似物的等效点，则可以表明，在某些限制下，在复杂溶液的相应空间上的循环确定了这些多项式上这些多项式的溶液。后来的预测是Grothendieck的Anabelian计划的一部分，尚未解决。它对相应股票的解决方案产生了强大的限制。该提案的重点是从这个角度研究对多项式方程的解决方案和Grothendieck的Anabelian计划。该项目还旨在刺激同质理论的研究，并使数学家在非常不同的领域以及其他科学家更普遍地提供该理论的工具。该提案中的项目分享了一种方法，即从Morel-Voevodsky的A1-Homotophy理论中，人们将计划视为一个空间，然后将各种实现函数应用于z/2 equivariant空间，通过将计划的C-Points置于R上，或者以R的行动，或者通过对Schemes的绝对范围的基础领域的绝对范围集团的行动来实现Pro的空间。主要研究者研究了从场k的典型同型类型到投影线的étale同型类型的pro空间图，使用下部中央系列近似近似为后来的三个点。另外，A1-HOMOTOPY理论中的James-HOPF地图用于研究相同的映射空间。两者都有Grothendieck的Anabelian计划的申请。向后运行相同的方法，从有关解决方案到多项式方程式开始的方案的代数拓扑产生结果。例如，首席研究者继续研究与绝对Galois组的群体共同体相关的差异分级代数。在莫雷尔 - 沃夫德斯基（Morel-Voevodsky）的意义上，有关不稳定空间类别的信息。