Mathematical Sciences: Results from the Monodromy Method

数学科学：单峰法的结果

• 批准号: 9305590
• 负责人: Michael Fried
• 金额: $ 6万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9305590&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Results; Monodromy; Method

This award supports the research of Professor Fried to work in number theory, specifically, in the application of the monodromy method. The monodromy method takes the following form. A precise statement S about Diophantine equations translates to this: Rational points on certain varieties produce a desired Galois theory situation "g". By forgetting where the Galois theory arises, we get a group theory situation G. The method applies the group theory to solve G. It then eliminates geometric situations in the list of "g" that don't correspond to what remains of G. This leaves the harder problem of deciding if what geometrically remains produces the arithmetic situation under investigation. This is research in the field of number theory. Number theory starts with the whole numbers and questions, such as the divisibility of one whole number by another. It is among the oldest fields of mathematics and it was originally pursued for purely aesthetic reasons. However, within the last half century, it has become an essential tool in developing new algorithms for computer science and new error correcting codes for electronics.

该奖项支持弗里德教授在数论方面的研究工作，特别是在单一性方法的应用方面。单选方法采用以下形式。关于丢番图方程的一个精确表述S可以翻译为：在某些变体上的有理点产生一个理想的伽罗瓦理论情形g。由于忘记伽罗瓦理论的起源，我们得到一个群论情况g，该方法应用群论来解决g，然后它消除了g列表中与g的剩余情况不对应的几何情况，这留下了一个更难的问题，即决定几何上剩余的情况是否会产生正在研究的算术情况。这是数论领域的研究。数论从整数和问题开始，比如一个整数能被另一个整数整除。它是数学中最古老的领域之一，最初纯粹出于审美原因而追求它。然而，在过去的半个世纪里，它已经成为开发计算机科学新算法和电子学新纠错码的重要工具。