Rational GAGA and Applications to Field Invariants

几何空间在许多环境中产生，研究它们的行为可以导致现实世界问题的解决。对这些空间的研究既使用了代数的方法，也使用了微积分(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建立了联系，随后导致了几何问题的重要进展。PI将把这种联系扩展到只在几何空间的一部分而不是整个空间给出信息的情况。这将使解决有关当前神秘的数字数据的计算的公开问题成为可能，这些数据与几何空间的行为有关。这种方法将涉及到对空间进行局部研究，以便更好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拓宽进入数学研究的渠道，以及向更广泛的受众交流数学。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将在变种的函数域的背景下学习类似于Serre的Gaga定理，而不是针对变种本身。这将涉及一个既包含全纯函数又包含有理函数的结构层。然后，一个关键的目标将是使用这个结果来计算关于三个或更多变量的复数的有理函数域的猜想的周期指数界。PI还旨在通过引入形式方案理论的思想并在其先前的低维工作的基础上，在更多的算术基础上证明相关结果。此外，PI将致力于了解实有理函数域的绝对可微伽罗华群的结构。这项工作的动机是他们以前在复数上的Galois微分理论和在实函数域上的经典Galois理论中所取得的结果。所使用的方法将包括局部-全局原理和补丁，以及线性代数群的结构理论、伽罗华上同调和其他技术。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.