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Singular reduction and quantization
13.6.2019, 16:15 - 17:15
Speaker:Prof. Dr. Pablo Ramacher, Philipps-Universit├Ąt Marburg
Location:Mathematisches Institut, Bunsenstr 3-5SitzungszimmerGras Geo Map
Organizer:Mathematisches Institut
In this talk, we study singular situations arising in equivariant spectral geometry, quantum ergodicity, and cohomology by means of Fourier analysis and resolution of singularities. In particular, we derive a local Weyl law for the reduced spectral function of an invariant elliptic differential operator on a compact manifold carrying the action of a compact Lie group, and characterize its caustic behaviour near singular orbits. From this we deduce statements about the asymptotic distribution of eigenvalues, together with pointwise and $L^p$-bounds for eigenfunctions, showing that the orbit structure is reflected in the shape of eigenfunctions. Based on these results, we prove subconvex bounds for Hecke--Maass forms on compact arithmetic quotients in the eigenvalue and isotypic aspect, as well as an equivariant quantum ergodicity theorem in case that the reduced Hamiltonian flow is ergodic. To conclude, we derive certain residue formulae in equivariant cohomology for $S^1$-actions. As we shall explain, our work relies on the description of the asymptotic behaviour of certain oscillatory integrals of Witten type.
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