FRG: Obstructions to Local-Global Principles and Applications to Algebraic Structures

数论和代数几何之间的相互作用一直是现代数学的灵感来源。它解决了许多杰出的猜想，如费马大定理和莫德尔猜想，它继续在代数中产生深刻而重要的问题。局部-全局原则是这些学科相互作用的中心主题，许多重要的突出问题可以用这些原则来表达。这个项目的目标是在比数论更广泛的背景下理解局部-全局原理及其障碍。该项目还将通过研讨会、会议和讲习班以及指导活动，支持和加强对研究生和博士后研究人员的培训。重点研究小组将重点研究在基域（如p进域）上曲线的函数场上定义的代数结构的局部-全局原理，其长期目标是处理全局域上曲线的函数场的情况。这种局部-全局原则的障碍通常可以用上同性来表述。我们的项目旨在研究这些障碍的有限性，并确定它们消失的标准。由此产生的理解将应用于证明猜想和解决有关代数结构的开放问题，如二次型和结合代数。这将包括许多研究人员已经研究过但以前似乎无法解决的情况。研究方法将包括场补片，包括残数和对偶在内的上同调方法，以及来自几何的方法。

The interplay between number theory and algebraic geometry has been a source of inspiration in modern mathematics. Having led to the solution of a number of outstanding conjectures, such as Fermat's Last Theorem and the Mordell Conjecture, it continues to give rise to deep and important problems in algebra. Local-global principles are a central theme in this interplay of subjects, and many important outstanding problems can be expressed in terms of such principles. This project has the objective of understanding local-global principles and their obstructions, in contexts that are broader than those considered in number theory. The project will also support and enhance the training of graduate students and postdoctoral researchers through seminars, conferences and workshops, and mentoring activities.The Focused Research Group will focus on local-global principles for algebraic structures defined over function fields of curves over base fields such as p-adic fields, with a longer term goal of treating the case of function fields of curves over global fields. The obstructions to such local-global principles can often be formulated in terms of cohomology. Our project aims to study the finiteness of these obstructions and determine criteria for them to vanish. The resulting understanding will be applied to proving conjectures and solving open problems concerning algebraic structures such as quadratic forms and associative algebras. This will include situations that have been studied by many researchers but where solutions had previously seemed out of reach. Research methods will include field patching, cohomological methods including residues and duality, and approaches from geometry.