Algebraic Varieties and Valuation Theory

这个项目在代数几何领域的关注方面的估值理论。赋值在数学的许多分支中自然出现。赋值空间，作为一个整体，捕捉重要的信息，例如与奇异性的解决问题。有许多自然的方法可以观察赋值并将它们打包到空间中，每一种方法都为研究代数簇提供了有用的工具。建议的研究重点是两个估值空间：空间的弧和代数簇的Berkovich分析。这些空间配备了有趣的结构，可以用来回答重要的数学问题。纳什研究了空间的弧连接到奇点;相同的空间作为潜在的空间motivic整合，并已被应用于研究不变量的奇点的最小模型程序。伯科维奇的非阿基米德几何已被用于各种场合，从p-adic几何到动力学、几何群论、镜像对称和热带几何，以及最近的双有理几何。关于这些空间结构的许多有趣的问题仍然是开放的，更好地理解它们的几何形状将导致新的应用。该项目的一个具体目标是研究各种弧空间上的局部环，这是一个由Shokurov关于最小对数差异的连续性猜想激发的问题，因此，间接地，由关于翻转终止的猜想，这是最小模型程序中缺少的步骤之一。另一个目标提出了一个新的观点，在motivic整合的Berkovich空间的弧空间的地方使用。该项目还解决了一些问题，孤立奇点的联系，他们的接触结构，和他们的CR结构。虽然这些问题显然是不相关的，但链接的接触结构和估值理论之间存在潜在的联系。

This project in the field of algebraic geometry concerns aspects of valuation theory. Valuations appear naturally in many branches of mathematics. Spaces of valuations, as a whole, capture important information, for example in connection with the problem of resolution of singularities. There are many natural ways of looking at valuations and packaging them into spaces, and each provides useful tools for studying algebraic varieties. The focus of the proposed research is on two valuation spaces: the space of arcs and the Berkovich analytification of an algebraic variety. These spaces come equipped with interesting structure and can be used to answer important mathematical questions.Nash investigated the space of arcs in connection to singularities; the same space serves as the underlying space in motivic integration, and has been applied to study invariants of singularities in the minimal model program. Berkovich's non-Archimedean geometry has been used in a variety of contexts, from p-adic geometry to dynamics, geometric group theory, mirror symmetry, and tropical geometry, and most recently in birational geometry. Many interesting questions about the structure of these spaces remain open, and a better understanding of their geometry will lead to new applications. One specific goal of this project is to study the local rings on the arc space of a variety, a problem motivated by Shokurov's semicontinuity conjecture on minimal log discrepancies and therefore, indirectly, by the conjecture on termination of flips, one of the missing steps in the minimal model program. Another objective proposes a new point of view on motivic integration where Berkovich spaces are used in place of arc spaces. The project also addresses some questions about links of isolated singularities, their contact structure, and their CR structure. While these are apparently unrelated questions, there is an underlying connection between the contact structure of a link and valuation theory.