Decompositions of Ideals

PI将继续研究理想的分解，以及对理想的不同操作如何与分解相互作用。该方案中的具体操作包括积分闭包、紧闭包、伴随包、Rees值、符号幂、Groebner基。π在交换代数的数学领域中起作用，在代数几何领域也有应用。特别是，有描述几何对象的代数结构，如球体、锥体、圆柱体等。（非）“平滑”以及几何对象的许多其他几何属性都反映在其相应的代数构造的属性中，这很有用，因为后者更容易受到操纵。一个熟悉的例子可能是整数或多项式的质因数分解，比如在密码学中使用的，但在更一般的情况下，相应的“因数分解”更复杂，而且高度非唯一。

The PI will continue her investigations on the decompositions of ideals and how different operations on the ideals interact with the decompositions. The particular operations in the proposal are integral closure, tight closure, adjoints, Rees valuations, symbolic powers, Groebner bases. The PI works in the area of mathematics known as commutative algebra, with applications to the area of algebraic geometry. In particular, there are algebraic constructs that describe geometric objects, such as spheres, cones, cylinders, etc. The (non) "smoothness" as well as many other geometric properties of a geometric object are reflected in the properties of its corresponding algebraic construct, which is useful because the latter is much more susceptible to manipulation. A familiar example might be the prime factorization of integers or polynomials, such as used in cryptography, but in more general situations the corresponding "factorizations" are more complex and highly non-unique.