**Presenter:** Kathleen I. Thompson

**Advisor(s):** Dr. Irina Kogan

**Author(s):** Kathleen I. Thompson

**Graduate Program:** Mathematics

**Title:** Polynomial Invariants of Finite
Groups

**Abstract:** Group actions appear in various areas of mathematics, physics, and engineering. A group acts on a set by rearranging the elements of the set. We will focus on group actions that change the variables of a polynomial. The goal is to compute polynomial invariants under such group actions. Group actions and polynomial invariants are important in the area of materials science. Most materials have intrinsic properties, closely related to the symmetry in the material, that remain unchanged when stresses are applied. Oftentimes, such properties can be modeled using polynomial functions. Understanding the polynomial invariants of the material reduces the number of potentially costly experiments needed to determine the conditions under which the given material will break. We will define groups and invariants and describe a known algorithm for calculating the invariants of a given finite group. An example will be used to demonstrate the algorithm. We will discuss applications to mechanics and directions for further research.