Homology and Cohomology over Commutative Rings

阿夫拉莫夫正在研究代数、代数几何和代数拓扑边界上出现的问题。研究的基本对象是反映代数变异几何性质的交换诺ether环的同调不变量。重点讨论Hochschild上同调消失的几何意义，它作为代数的有限代，以及Andre-Quillen同调的消失。用几何方法研究了完全交上经典同调和上同调的渐近消失问题。这是一个在数学领域被称为交换代数和代数几何的项目。交换代数和代数几何可以被认为是研究许多方程在许多未知数中的解，通常情况下，解不是唯一的。然后可以从几何角度来看待解的集合，但它通常最好被编码成在这个集合上定义的一系列函数。这种函数族的抽象形式称为交换环。

Avramov is investigating problems arising at the boundary of algebra, algebraic geometry, and algebraic topology. The basic objects of study are homological invariants of commutative noetherian rings that reflect geometric propertied of algebraic varieties. Special attention will be focused on the geometric meaning of the vanishing of Hochschild cohomology, on it finite generation as an algebra, and on the vanishing of Andre-Quillen homology. Asymptotic vanishing of classical homology and cohomology over complete intersections will be studied by geometric methods.This is a project in the areas of mathematics known as Commutative Algebra and Algebraic Geometry. Commutative algebra and algebraic geometry may be thought of as studying solutions of many equatons in many unknowns when, typically, the solution is not unique. The set of solutions can then be viewed geometrically, but it is often best encoded into a family of functions defined on this set. The abstract version of such families of functions are called commutative rings.