# From Weyl's group to Weyl's law: The spectral counting function of a compact Lie group

The study of periodic geodesics on Lie groups has deep and intimate connections with partial differential equations. A well known example is the Euler-Arnold equation in fluid mechanics. In the particular case of the special orthogonal group $\mathbf{SO}(N)$ the periodic geodesics have recently given rise to multiple twist solutions to certain geometric problems in the Calculus of Variations. In an attempt to study the nature and size of periodic geodesic on a compact Lie group and the intimately related spectral counting function of the Laplace-Beltrami operator I shall begin by proving that the well-known Avakumovic-Hormander asymptotics for the spectral counting function is not sharp. Moreover I will then proceed to establish a sharp estimate on the remainder term related to the rank and dimension of the group. The talk will also draw connections with the old and challenging Gauss circle problem from analytic number theory and its higher dimensional counterparts. This talk is based on joint work with Dr Ali Taheri.

This talk is part of the Analysis Seminar series.