Motivic Cohomology, Motivic Homotopy Theory, and K-Theory

这个项目主要研究代数、代数几何和拓扑学之间的联系。代数几何研究几何对象在多项式函数给出的变换下不变的性质，而拓扑学研究在光滑、连续变形下不变的性质。其目的是利用代数和拓扑的混合方法，更好地理解代数几何中的结构是如何通过拓扑中的结构来反映的。这位调查人员的目标是解决这一领域几个长期存在的重要猜想。该项目的一部分是为最近得到证实的布洛赫-加藤猜想找到一个更清晰的证据。该项目的一个相关部分是了解沃沃茨基的切片过滤是如何与代数余边图和其他动机谱相关的，并验证几个切片猜想。研究人员还将使用最近发展的上同调技术来研究簇的奇点、它的K理论和它的cdh上同调之间的关系。这个项目的另一个部分是将真实曲面的代数Witt群与最近开发的更容易计算的拓扑不变量联系起来。

This project concerns the investigation of connections between algebra, algebraic geometry, and topology. Algebraic geometry studies properties of geometric objects that are invariant under transformations given by polynomial functions, while topology studies properties invariant under smooth, continuous deformations. The goal is to gain a better understanding of the way that structures in algebraic geometry are reflected by structures in topology, using a blend of algebraic and topological methods. The investigator aims to settle several important longstanding conjectures in this area. One part of the project is to find a cleaner proof of the recently-verified Bloch-Kato conjecture. A related part of the project is to understand how Voevodsky's slice filtration is related to algebraic cobordism and other motivic spectra and to verify several of the slice conjectures. The investigator will also study the relationship between the singularities of a variety, its K-theory, and its cdh-cohomology, using recently-developed cohomological techniques. Another part of this project is to relate the algebraic Witt group of a real surface to a recently-developed topological invariant, one that is easier to compute.