We propose to continue our study of the small time asympotics of the trace of the heat operator for the Laplacian on singular analytic varieties. Actually we aim at the clarification of the asymptotic expansion proved so far in order to obtain index theorems on singular spaces as well as the extension to more general singularities. So far we have treated very particular singularities, different than the well studied conical case, that however suggest certain model cases that would be expected in the general case. However we would like to suggest two other directions since interestingly enough the techniques that have been developed for the preceding problem allowed the derivation of estimates of nodal sets of the laplacian on smooth compact manifolds. Moreover these techniques combined with classical methods in differential geometry gave some preliminary results on model cases of singularities of minimal hypersurfaces, specifically perturbations of cylindrical cones.
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