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。 这个项目试图回答以下关于某些高度对称的物体的问题：在多大程度上可以通过局部检查来区分它们？这些物体本身的对称性形成了一个几何空间，研究人员的目标是了解该空间的行为如何决定这个首要问题的答案。特别是，该项目将探讨对称空间是否可以完全区分的情况下，任何两个对称可以连接一条线。该奖项支持的研究生将接受培训，为研究做出贡献，研究人员将举办年度活动，鼓励来自代表性不足群体的本科生申请数学博士学位。程序。更确切地说，研究人员研究线性代数群和他们的torsors，在功能领域，定义在一个完整的离散价值的领域。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torsors的局部-全局原理，群是否是R-平凡的，以及群是否满足弱近似性质。这些属性之间的影响已被证明在数字字段的情况下，组，本项目的目的是进行这些影响的功能域的组正在考虑中。其他目标包括描述和限制的障碍，以局部-全球原则的几何和扩展局部-全球原则，以获得更高的维度，以获得有关Brauer groups的结果。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.