Motivic Cohomology, Motivic Homotopy Theory, and K-Theory

该项目涉及代数、代数几何和拓扑之间联系的研究。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属性，而拓扑研究在平滑、连续变形下不变的属性。 目标是使用代数和拓扑方法的混合，更好地理解代数几何中的结构通过拓扑中的结构反映的方式。 研究人员旨在解决该领域几个长期存在的重要猜想。该项目的一部分是为最近验证的布洛赫-加藤猜想找到更清晰的证明。该项目的一个相关部分是了解 Voevodsky 的切片过滤如何与代数共边和其他动机谱相关，并验证几个切片猜想。研究人员还将使用最近开发的上同调技术来研究簇的奇点、其 K 理论及其 cdh 上同调之间的关系。该项目的另一部分是将真实表面的代数维特群与最近开发的拓扑不变量联系起来，这是一个更容易计算的拓扑不变量。

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.