Representations, modular forms and Galois groups

表示、模形式和伽罗瓦群

• 批准号: 0852429
• 负责人: Gordan Savin
• 金额: $ 30.43万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0852429&HistoricalAwards=false
• 关键词: Representations; modular; forms; Galois; groups

A quadratic equation is a polynomial equation of degree two.A formula for solving quadratic equations was already known to Babylonians in 1800BC, and is familiar to just about everybody who has taken some mathematics classes. On the other hand, a formula for solving cubic (degree three equations) was developed much later, in the 16th century, during contests between Italian mathematicians. Higher degree equations remained a mystery. A breakthrough came in 19th century when Galois, a French mathematician, introduced a new tool, a group of permutations of the set of roots of a polynomial equation. This object, now called Galois group, expresses the nature of the polynomial equation in subtle ways. In particular, it gives a criterion whether the equation can be solved in terms of radicals. Fast forward to 20th century. Due to efforts of hundreds of mathematicians, a classification of finite simple groups has been obtained. A natural question is which of these groups appear as Galois groups. In a couple of recent works, C. Khare, M. Larsen and G. Savin have developed a method of constructing Galois groups using Langlands functoriality principles. Gordan Savin is continuing his work on this subject, as the main theme of this project.The activities supported by this grant will include participation of post-docs, graduate students, domestic and international collaborators.The broader impacts of proposed activities are expected to be similar to those resulting from the grant DMS-0551846.They include, but are not limited to,research opportunities for undergraduate students. One such activity, supported by the previous grant, is the work of G. Savin and R. Denomme (undergraduate from the Ohio State University) on primality testing using elliptic curves with complex multiplication. The problem of factoring of large numbers and related problem of verifying that a large number is a prime number (primality testing) has attracted considerable attention in recent years due to applications in cryptography. Indeed, internet security is based on an observation that it is very easy to multiply numbers, but very hard to factor them.

二次方程式是二级程度的多项式方程。一个用于解决二次方程式的公式在1800年公元前已经知道了巴比伦人，并且几乎每个参加了一些数学课程的人都很熟悉。 另一方面，在意大利数学家之间的比赛中，在16世纪很久以后开发了一个用于解决立方体的公式（三级方程式）。更高的方程仍然是一个谜。 19世纪，法国数学家加洛伊斯（Galois）引入了一种新工具，这是一组多项式方程的根源。该对象现在称为Galois组，以微妙的方式表达多项式方程的性质。特别是，它给出了一个标准，是否可以根据自由基来求解方程。快进到20世纪。由于数百名数学家的努力，已经获得了有限简单群体的分类。一个自然的问题是，这些群体中的哪个出现为Galois群体。在最近的几部作品中，C。Khare，M。Larsen和G. Savin开发了一种使用Langlands功能性原理构建Galois组的方法。 Gordan Savin正在继续他的这一主题，这是该项目的主题。该赠款支持的活动将包括参与后DOC，研究生，国内和国际合作者的参与。预计拟议活动的更广泛影响将是与赠款DMS-0551846所产生的类似。它们包括但不限于本科生的研究机会。 先前赠款支持的其中之一是G. Savin和R. Denomme（来自俄亥俄州立大学的本科生）对使用具有复杂乘法的椭圆曲线进行原始测试的工作。由于密码学中的应用，近年来，大量验证大量是质量数字（原始测试）的问题引起了很大的关注。实际上，互联网安全是基于观察到的，即乘以数字非常容易，但很难对其进行分解。