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

在计算受各种约束的对象时，在整数或有理数中寻找多项式的解是很自然的。这也是数学中最古老的问题之一。复数上相同多项式的解形成拓扑空间。例如，n次费马方程的复解是一个有（n-1）(n-2)/2个孔的环面。复解空间的形状会影响有理数或整数上的解，这一事实可以看作是利用同伦理论方法研究这一问题的第一个实例。同伦理论提供了一种机制，可以用派生版本来代替程序，这种派生版本通常可以更好地控制问题。例如，考虑取一个旋转球体并返回不移动的点的过程。这个过程可以推导出一个叫做同伦不动点的空间，它不仅记录了不移动的点，而且记录了点之间的相容路径和它们走过的地方。不动点与同伦不动点在某些假设下是等价的。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种模拟是等价的，那么可以证明在一定的限制条件下，这些多项式在理数上的解是由相应的复解空间上的环决定的。后一种预测是格罗滕迪克的安娜贝尔计划的一部分，尚未得到解决。它对相应方程的解产生了很强的限制。本提案的重点是从这个角度研究多项式方程的解和Grothendieck的安娜贝尔规划。该项目还旨在促进同伦理论的研究，并使这一理论的工具可供不同领域的数学家和其他科学家使用。本提案中的项目分享了一种方法，即将方案视为Morel-Voevodsky的a1 -同伦理论意义上的空间，然后应用各种实现函子，例如通过取R上方案的c点来应用于Z/2等变空间，或者应用于更一般域上方案的基域的绝对伽罗瓦群作用的亲空间。研究了从域k的<s:1>同伦型到投影线减去三点的<s:1> <s:1>同伦型的前空间映射。另外，利用a1 -同伦理论中的James-Hopf映射来研究相同的映射空间。两者都适用于格罗滕迪克的安娜贝尔项目。反向运行相同的方法，从多项式方程的解的信息开始，在方案的代数拓扑上产生结果。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梯度代数。在Morel-Voevodsky的意义上寻找关于空间不稳定范畴的信息。

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.