Topics in algorithmic and computational number theory

在过去的几十年里，数论，这一处理整数性质的数学分支，在加拿大已经成为一个非常重要的研究领域。事实上，这是这个国家最活跃的纯数学研究领域之一。我的研究涉及到方法的发现及其计算机实现，以计算某些数论对象。这些问题包括看似简单的确定整数是否为素数的问题，以及计算数字字段的类号等更复杂的问题。后一个量对相应的数字字段非常重要；非常粗略地说，它可以被认为是数字字段的整数与我们都熟悉的整数：整个数字在属性上有多大不同的度量。我的主要兴趣是找到非常有效的方法来解决这类问题。

Over the last several decades number theory, that branch of mathematics which deals with the properties of integers, has become a very important area of study in Canada. Indeed, this is one of the most active areas of pure mathematical research in this country. My research involves the discovery of methods, and their computer implementation, to compute certain number theoretic objects. These problems include the seemingly simple problem of determining whether or not an integer is a prime, to more complex problems such as computing the class number of a number field. This latter quantity is of great importance to the corresponding number field; very roughly speaking, it can be considered as a measure of how much the integers of the number field differ in their properties from the integers with which we are all familiar: the whole numbers. My main interest is in finding very efficient ways of solving these kinds of problems.