# Bounding the invariant spectrum of the Laplacian

#### Stuart Hall, Newcastle University. Part of the pure mathematics, geometry seminar series.

I will report on an ongoing project with Tommy Murphy to understand spectrum of the Laplacian  of a Riemannian metric invariant under the action of a compact Lie Group G. We are interested in questions such as can you bound the kth invariant eigenvalue independent of the metric?  Are such bounds sharp and are there smooth metrics that achieve them? The only large family of manifolds that have a substantial set of answers to such questions are toric Kahler manifolds. Ill try to detail what is known here and how techniques might be generalised to other settings.

Stuart Hall, Newcastle University

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