Algebraic K-theory and algebraic cycles

本项目旨在利用几何、拓扑和代数方法研究代数簇上的代数K-理论和代数圈。为了更好地理解奇点的K理论和Hochschild同调，研究者建议进一步应用早期工作中发展的下降方法。这项工作与交换代数的同调理论中的著名猜想有关，例如Bergers猜想，具有无挠差分的曲线是光滑的。在一个单独的项目中，研究人员计划研究代数循环上的新的高型函数，希望在模代数等价的循环的可除性问题上取得进展。这个项目研究代数簇的不变量，即可以实现为多项式方程的解的集合的对象。特别是，有人建议通过一种称为K-理论的不变量来研究奇点(黑洞是奇点的一个例子)，该不变量编码了非常大的多项式函数矩阵在奇点上的行为；并研究了形式变量族(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).