Higher Function Field Arithmetic

对称性的研究在数学及其应用中经常有用。考虑系统的对称性可以洞察相关数学对象的行为。这个项目试图回答关于某些高度对称的物体的以下问题：仅仅通过局部检查就可以在多大程度上区分它们？这些物体本身的对称性形成了一个几何空间，研究人员的目标是理解这个空间的行为如何决定这个关键问题的答案。特别是，该项目将探索在任何两个对称可以由一条线连接的情况下，对称空间是否可以完全区分。获得该奖项支持的研究生将接受为研究做出贡献的培训，调查人员将每年举办一次活动，鼓励来自代表性不足群体的本科生申请数学博士课程。更准确地说，调查人员研究线性代数群及其扭子，研究定义在完全离散值域上的函数域。他们将研究给定线性代数群的三个性质之间的关系：群是否满足扭量的局部-整体原理，群是否R-平凡，群是否满足弱逼近性质。这些性质之间的影响已经显示在数字字段上的组的情况下，本项目的目的是将这些影响传递到所考虑的功能域上的组。其他目标包括从几何学的角度描述和限制局部-全球原则的障碍，以及将局部-全球原则扩展到更高的维度，以获得关于布劳尔小组的结果。这一奖项反映了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。

The study of symmetry is frequently useful in mathematics and its applications. Consideration of the symmetries of a system can yield insight into the behavior of related mathematical objects. This project seeks to answer the following question about certain highly symmetric objects: to what extent can they be distinguished from each other simply by examining them locally? The symmetries of such objects themselves form a geometric space, and the investigators aim to understand how the behavior of that space governs the answer to this overarching question. In particular, the project will explore whether symmetric spaces can be completely distinguished in the situation when any two symmetries can be connected by a line. Graduate students supported by the award will receive training to contribute towards the research, and the investigators will host a yearly event to encourage undergraduates from underrepresented groups to apply to mathematics Ph.D. programs.More precisely, the investigators study linear algebraic groups and their torsors, over function fields that are defined over a complete discretely valued field. They will study the relationships among three properties of a given linear algebraic group: whether the group satisfies the local-global principle for torsors, whether the group is R-trivial, and whether the group satisfies the weak approximation property. Implications among these properties have been shown in the case of groups over number fields, and this project aims to carry over those implications to groups over the function fields under consideration. Other goals include describing and bounding the obstruction to a local-global principle in terms of geometry and extending local-global principles to higher dimensions in order to obtain results about Brauer groups.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.