Spectra of Lie groups and L^{2}-invariants

In this talk, I will explore the Laplace operator and Dirac operator on semisimple and nilpotent Lie groups. While the parallel problem on symmetric spaces has been well-studied in the last century, the corresponding problem is much less understood in general homogeneous spaces. We will examine the obstacles in extending existing techniques and discuss how some of them can be resolved in the case of group manifolds. Towards the end, we will see how the spectra data shall aid in computing certain topological L^{2}-invariants.