Rational GAGA and Applications to Field Invariants

几何空间出现在许多背景下，研究它们的行为可以导致解决真实的世界问题。 对这些空间的研究使用了代数和微积分（也称为分析）的方法。 几十年前，几何的代数方法和解析方法之间的联系建立起来，这导致了几何问题的重要进展。 PI将把这种联系扩展到只在一片几何空间上给出信息而不是在整个空间上给出信息的情况。 这将使解决与几何空间行为相关的当前神秘数值数据的计算有关的公开问题成为可能。 该方法将涉及局部研究空间，以便更深入地了解它们的整体行为。 参与者还将参与具有更广泛影响的活动。 这些措施包括指导，扩大管道到数学研究的人从群体传统上在数学代表性不足，并沟通数学更广泛的观众。 此外，研究生也将接受培训，为今后的数学研究做出贡献。更确切地说，PI们将在品种的函数场的背景下研究Serre GAGA定理的模拟，而不是品种本身。 这将涉及到一个结构层，它既包含全纯函数又包含有理函数。一个关键的目标，然后将使用这个结果来计算的约束周期指数界的有理函数字段在复杂的数字在三个或更多的变量。PI还旨在通过引入形式方案理论的思想并建立在他们先前在低维方面的工作基础上，在更多的算术基础领域上证明相关结果。 此外，PI将致力于理解真实的有理函数域的绝对微分伽罗瓦群的结构。 这项工作的动机，他们以前取得的成果，在复杂的数字和经典伽罗瓦理论在真实的功能领域的微分伽罗瓦理论。所使用的方法将包括局部-全局原则和修补，以及线性代数群的结构理论，伽罗瓦上同调，和其他techniques.This award reflects NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Geometric spaces arise in many contexts, and studying their behavior can lead to the solution of real world problems. The study of these spaces has used methods both from algebra and from calculus (also called analysis). Decades ago, a linkage between the algebraic and analytic approaches to geometry was established, which then led to important progress on geometric problems. The PIs will extend this linkage to situations in which information is given only on a piece of a geometric space, rather than on the entire space. This will make it possible to solve open problems concerning the computation of currently mysterious numerical data that relate to the behavior of geometric spaces. The approach will involve studying spaces locally in order to gain a greater insight into their overall behavior. The PIs will also engage in activities that have broader impacts. These include mentoring, widening the pipeline into mathematical research for people from groups traditionally underrepresented in mathematics, and communicating mathematics to a broader audience. In addition, graduate students supported by the award will receive training to contribute toward this research as well as to engage in further mathematical research in the future.More precisely, the PIs will study an analog of Serre's GAGA theorem in the context of function fields of varieties, rather than for the varieties themselves. This will involve a structure sheaf that contains both holomorphic functions and rational functions. A key goal will then be to use this result to compute the conjectured period-index bound for rational function fields over the complex numbers in three or more variables. The PIs also aim to prove related results over more arithmetic ground fields, by bringing in ideas from the theory of formal schemes and building on their prior work in lower dimensions. In addition, the PIs will work to understand the structure of the absolute differential Galois group of real rational function fields. This work is motivated by results that they previously achieved in differential Galois theory over the complex numbers and in classical Galois theory over real function fields. The methods used will include local-global principles and patching, as well as the structure theory of linear algebraic groups, Galois cohomology, and other techniques.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.