Melissa Tacy (Northwestern) 

Asymptotic $L^{p}$ eigenfunction estimates: concentration and geometry   

Abstract: Asymptotic $L^{p}$ estimates for Laplacian eigenfunctions give us 
some  insight into the behaviour of quantum systems in the semiclassical  
(high energy) limit. As such eigenfunctions represent the quantum                  
equivalent of free particle motion we expect to see some echoes of                 
classical dynamics in their concentration properties. In particular we 
expect that classical systems displaying chaotic behaviour should 
correspond with quantum systems displaying reduced eigenfunction  
concentration. In this talk I will discuss the history of such 
estimates for general manifolds and more recent work where the                     
geometry of the underlying manifold causes reduction in concentration.