Representation theory and combinatorics of symmetric groups

The symmetric group is a rich source of interesting combinatorial phenomena that have been well studied for over a century. One of the most fruitful approaches is to study symmetric groups of different ranks together. This idea turns out to produce beautiful links to geometry, combinatorics and integrable systems. One possible direction for this project is to use the representation theory of the symmetric group to make calculations coming from certain quantum integrable systems. This will involve some representation theory and some combinatorics.