This project is about representations of braid groups. Viewing a braid group as the mapping class group of a punctured disk, one obtains representations by taking the induced action on homology modules of configuration spaces of the disk. Lawrence has shown that all Specht module representations can be obtained in this way. This project will use topological methods to study these representations. Of particular interest is their behavior at non-generic values of the parameter. There are applications to invariants of knots and three-manifolds.
A braid is a certain kind of arrangement of pieces of string in three-dimensional space. One can join some of the ends of the strings of two braids to form a new braid. A representation is a way to encode this operation using matrices and matrix multiplication. One application is in defining knot invariants, which are numbers or polynomials that can be used to tell different knots apart. The representations studied in this project include the Specht modules, whose rich structure makes them extremely important to representation theory in general. Representation theory is the area of mathematics most useful for studying symmetry whether in nature or in abstract mathematics.
