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

在计算受各种约束的对象时，自然会找到整数或有理数中的多项式的解。它也是数学中最古老的问题之一。复数上相同多项式的解构成拓扑空间。例如，n次Fermat方程的复解是一个有(n-1)(n-2)/2个洞的环面。复解空间的形状影响有理数或整数上的解，这一事实可以看作是使用同伦理论方法研究这一问题的第一个有用的例子。同伦理论给出了用派生版本取代程序的机制，这往往会给问题提供更多的控制。例如，考虑这样一个过程：取一个旋转的球体，然后返回不移动的点。这一过程可以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相容路径和它们所走的地方。在某些假设下，不动点和同伦不动点是等价的。如果多项式方程复解空间的某一类不动点与同伦不动点等价，则在一定的限制条件下，这些多项式在有理数域上的解是由相应复解空间上的环决定的。后一种预测是Grothendieck的Anabelian计划的一部分，尚未解决。它对相应方程的解产生了很强的限制。这一建议的重点是从这个角度研究多项式方程的解和Grothendieck的Anabelian程序。该项目还旨在促进同伦理论的研究，并使该理论的工具可用于非常不同领域的数学家，以及更广泛的其他科学家。该建议中的项目共享这样的方法：将方案视为Morel-Voevodsky的A1-同伦理论意义下的空间，然后应用各种实现函子，例如通过取R上的方案的C点来应用于Z/2-等变空间，或应用于具有基域的绝对Galois群作用的Pro空间以用于更一般的域上的方案。主要研究者研究了从域k的étal型同伦型到射影直线减三点的étal型同伦型的亲空间映射，使用下中心级数逼近后者。此外，还利用A1-同伦理论中的James-Hopf映射研究了相同的映射空间。两者都适用于Grothendieck的Anabelian计划。反向运行相同的方法，从多项式方程的解的信息开始，产生关于方案的代数拓扑的结果。例如，首席研究者继续研究与绝对Galois群的群上同调有关的微分分次代数。同时寻找关于Morel-Voevodsky意义下的不稳定空间范畴的信息。

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.