Rational GAGA and Applications to Field Invariants

几何空间在许多情况下都会出现，研究它们的行为可以解决现实世界中的问题。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，几何学的代数方法和解析方法之间建立了联系，这导致了几何问题的重要进展。pi将把这种联系扩展到仅在几何空间的一块而不是整个空间上给出信息的情况。这将使解决与几何空间的行为有关的目前神秘的数值数据的计算有关的开放问题成为可能。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。私人投资机构还将参与具有更广泛影响的活动。这些措施包括指导、为传统上在数学领域代表性不足的群体扩大进入数学研究的渠道，以及向更广泛的受众传播数学。此外，该奖项支持的研究生将接受培训，为这项研究做出贡献，并在未来从事进一步的数学研究。更准确地说，pi将研究Serre的GAGA定理在变量的函数域背景下的类比，而不是变量本身。这将涉及到包含全纯函数和有理函数的结构层。然后，一个关键的目标将是使用这个结果来计算三个或更多变量的复数上的有理函数域的推测周期索引界。pi还致力于在更多的算术领域证明相关的结果，通过引入形式方案理论的思想，并建立在他们先前在较低维度的工作上。此外，pi将致力于理解实有理函数域的绝对微分伽罗瓦群的结构。这项工作的动机是他们之前在复数上的微分伽罗瓦理论和在实函数场上的经典伽罗瓦理论中取得的结果。所使用的方法将包括局部-全局原理和修补，以及线性代数群的结构理论、伽罗瓦上同调和其他技术。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.