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.
