Algebraic K-theory and algebraic cycles

该项目旨在利用几何、拓扑和代数方法研究代数簇的代数K理论和代数簇上的代数圈。研究者建议进一步应用早期工作中开发的下降方法，以更好地理解奇异点的K理论和Hochschild同调。这项工作涉及到著名的代数在同调理论的交换代数，例如伯格斯猜想，曲线挠自由微分是光滑的。在另一个项目中，研究者计划研究代数圈上的新高度型函数，希望在圈模代数等价的整除性问题上取得进展。该项目研究代数簇的不变量，即可以实现为多项式方程的解集的对象。特别是，它建议通过称为K理论的不变量来研究奇点（黑洞是奇点的一个例子），该理论编码了奇点上多项式函数的非常大的矩阵的行为;并研究子簇的形式和族（甚至更多方程的解集）的行为。

This project aims at studying the algebraic K-theory of, and algebraic cycles on, algebraic varieties, using geometric, topological and algebraic methods. The investigator proposes to further apply the descent methods developed in earlier work in order to better understand the K-theory and Hochschild homology of singularities. This work is related to well-known conjectures in the homology theory of commutative algebras, as for example Bergers conjecture that curves with torsion free differentials are smooth. In a separate project, the investigator plans to study new height-type functions on algebraic cycles in hopes of making progress on question of divisibility of cycles modulo algebraic equivalence.This project studies invariants of algebraic varieties, that is, objects that can be realized as sets of solutions of polynomial equations. In particular, it is proposed to study singularities (a black hole is an example of a singularity) via an invariant called K-theory that encodes the behavior of very large matrices of polynomial functions on the singularity; and to study the behavior of families of formal sums of subvarieties (the sets of solutions of even more equations).