Algebraic Structures over Fields of Functions

该项目涉及使用几何思想来研究代数问题。 代数和几何通常被认为是数学的两个独立领域。 然而，正如笛卡尔通过使用图表所观察到的那样，将它们结合起来使用可能会非常富有成效。 在几何学中，人们通常可以通过研究空间的局部行为来研究空间的整体性质。 利用代数和几何之间的联系，有时也可以研究代数中的问题，通过将全局问题简化为被视为更局部的问题。 该项目的目标是将这种方法推广到新的情况下，包括高维空间，以解决在代数的几个领域出现的开放问题。第一个目标是将迄今为止只在一维的各种情况下开发的场修补方法推广到高维空间，在一个完整的离散值场上。 第二个目标是利用这种推广来获得高维函数域上代数结构的局部-全局原理。 这样的结构包括中心单代数、线性代数群下的扭子和二次型。 这些局部-全局的原则，反过来又有望导致结果的数值领域的不变量，如周期指数界和u-不变。 该计划的活动还将在指导、加强研究基础设施和学术界以外的数学交流方面产生更广泛的影响。该奖项反映了NSF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。

The project concerns the use of geometric ideas to study problems in algebra. Algebra and geometry are generally regarded as separate areas of mathematics. Nevertheless, it can be very fruitful to use them in combination, as observed by Descartes through the use of graphs. In geometry, one can often study global properties of spaces by examining the local behavior of the spaces. Using the connection between algebra and geometry, it is sometimes possible to study problems in algebra as well, by reducing global questions to ones that are viewed as more local. The goals of the project involve generalizing this approach to new situations, including higher dimensional spaces, in order to solve open problems that arise in several areas of algebra.The first goal is to generalize to higher dimensions the method of field patching that has so far been developed only in the context of varieties of dimension one, over a complete discretely valued field. The second goal is to use this generalization to obtain local-global principles for algebraic structures over higher dimensional function fields. Such structures include central simple algebras, torsors under linear algebraic groups, and quadratic forms. These local-global principles in turn are expected to lead to results on numerical field invariants such as the period-index bound and the u-invariant. The activities of the proposal will also have broader impacts in terms of mentoring, enhancing the research infrastructure, and communicating mathematics outside of academia.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.