Representations, modular forms and Galois groups

二次方程是二阶多项式方程。早在公元前1800年，巴比伦人就已经知道了求解二次方程的公式，几乎每个上过数学课的人都对这个公式很熟悉。另一方面，求解三次方程的公式是在16世纪意大利数学家之间的竞赛中发展起来的。更高次方程仍然是个谜。19世纪，法国数学家伽罗瓦（Galois）引入了一种新工具，即多项式方程的根集合的一组排列，这一突破出现了。这个对象，现在被称为伽罗瓦群，以微妙的方式表达了多项式方程的本质。特别地，给出了方程是否可以用根来解的判据。快进到20世纪。经过数百位数学家的努力，得到了有限简单群的一个分类。一个自然的问题是，这些群体中哪些是伽罗瓦群体。在最近的几部作品中，C. Khare， M. Larsen和G. Savin开发了一种使用朗兰兹泛函原理构造伽罗瓦群的方法。Gordan Savin继续他在这个主题上的工作，作为这个项目的主题。本基金支持的活动包括博士后、研究生、国内外合作者的参与。拟议活动的更广泛影响预计将类似于拨款DMS-0551846所产生的影响。包括但不限于本科生的研究机会。其中一项活动是G. Savin和R. Denomme（俄亥俄州立大学的本科生）在使用椭圆曲线进行复数乘法的素数检验方面的工作，得到了先前拨款的支持。近年来，由于在密码学中的应用，大数的因式分解问题以及验证大数是否为素数的相关问题（素数检验）引起了相当大的关注。事实上，互联网安全是基于这样一种观察：数字相乘很容易，但要分解它们却很难。

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.