Motivic Cohomology, Motivic Homotopy Theory, and K-Theory

这个项目涉及代数，代数几何和拓扑学之间的联系的调查。代数几何学研究的是在多项式函数变换下不变的几何对象的性质，而拓扑学研究的是在光滑、连续变形下不变的性质。 目标是更好地理解代数几何中的结构如何被拓扑结构所反映，使用代数和拓扑方法的混合。 调查员的目的是解决这一领域的几个重要的长期问题。该项目的一部分是找到最近验证的布洛赫-加藤猜想的更清晰的证明。该项目的一个相关部分是了解Voevodsky的切片过滤是如何与代数配边和其他motivic频谱相关的，并验证几个切片结构。研究人员还将研究各种奇异性之间的关系，它的K-理论，和cdh-上同调，使用最近开发的上同调技术。该项目的另一部分是将真实的曲面的代数Witt群与最近开发的拓扑不变量相关联，该拓扑不变量更容易计算。

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.