Algebraic K-theory and algebraic cycles

本课题旨在运用几何、拓扑和代数方法研究代数变异的代数k理论和代数循环。为了更好地理解奇点的k理论和Hochschild同源性，研究者建议进一步应用在早期工作中发展起来的下降方法。这项工作与交换代数同调理论中著名的猜想有关，例如berger猜想具有无扭微分的曲线是光滑的。在一个单独的项目中，研究者计划研究代数环上的新的高度型函数，希望在环的可整除性问题上取得进展模代数等价。本课题研究代数变量的不变量，即可以实现为多项式方程解的集合的对象。特别地，它提出了研究奇点（黑洞是奇点的一个例子）通过一个不变量称为k理论，编码非常大的多项式函数矩阵在奇点上的行为；并研究子变量的形式和族（甚至更多方程的解的集合）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).