Robert Boltje's research centers around the theory of finite groups, their representations, and applications to algebraic number theory.
Within the theory of finite group representations he has been working on natural induction formulae for many years. A very useful tool in this theory is the language of Mackey functors. This structure occurs surprisingly often in different fields of mathematics when group actions on mathematical objects (sets, vector spaces, topological spaces, fiber bundles) are present. Also, the presence of a Mackey functor structure on the ideal class groups of number fields in a fixed Galois extension provides relations between these class groups. The ideal class group is an invariant which measures how close the ring of integers in a number field is to having unique factorization into primes.
Presently, Robert Boltje is interested in the conjectures of Alperin, Dade, and Broué in the representation theory of finite groups. These conjectures link blocks of representations of a finite group G to blocks of representations of various subgroups arising as normalizers of chains of p-subgroups of G. It seems that the topology of the simplicial complex of p-subgroups together with its G-action plays an important role, and that ideas from other fields of mathematics like geometry or algebraic topology are needed to prove the conjectures.