The parameterization of algebraic structures, and applications

这个项目是一个正在进行的研究项目的一部分。它围绕着理解数学中出现的各种基本代数对象——如环、类群、代数曲线和变量，以及与这些结构相关的映射——是如何参数化的。我们理解这些参数化的目标至少有四个方面：1)描述这些基本代数对象是如何根据它们最基本的不变量分布的；2)发现这些对象的新不变量及其应用；3)开发高效实用的算法，对这些代数对象进行计算；也许最重要的是，4)发现和理解各种看似不同的代数结构实际上是如何彼此密切相关的。在过去的几年中，PI已经使用了这种参数化来获得关于诸如判别式和类数可除性等基本不变量的数域分布的精确信息。将精细计数方法应用于这些参数化已经导致，例如，证明了Cohen-Lenstra-Martinet类数启发式的第一个已知案例，并且这种性质的其他定理即将出现。在不久的将来，我们期望类似的定理用于椭圆曲线的秩，以及代数曲线和曲面的其他此类数据。

This project is part of an ongoing research program. It revolves around understanding how various fundamental algebraic objects occurring in mathematics---such as rings, class groups, algebraic curves and varieties, and maps relating such structures---are parametrized. Our goals in understanding these parametrizations are at least fourfold: 1) to describe how these fundamental algebraic objects are distributed with respect to their most basic invariants; 2) to discover new invariants of these objects, and their applications; 3) to develop efficient and practical algorithms for performing computations with these algebraic objects; and, perhaps most importantly, 4) to discover and understand how various seemingly different algebraic structures are in fact closely related to each other.Such parametrizations have been already used by the PI over the past few years to obtain precise information on the distribution of number fields with respect to basic invariants such as discriminant and class number divisibility. Applying refined counting methods to these parametrizations has led, for example, to a proof of the first known case of the Cohen-Lenstra-Martinet class number heuristics for higher degree number fields, and other theorems of this nature are forthcoming. We expect similar theorems for ranks of elliptic curves, and other data of this kind for algebraic curves and surfaces, in the near future.