Yaroslav Kurylev: Laplacian on collapsing manifolds and stability of inverse problems

We consider an inverse problems of the reconstruction of a Riemannian manifold from its local heat data, namely, the heat kernel H(x,y,t) when t>0 and x and y vary on an open subset of (an unknown) M. Our main interest is in the stability of this inverse problem. However, since inverse problems are highly unstable, we should put some a priori conditions on the class of considered manifolds. We deal with the classical Gromov class M(K, D) of manifolds with bounded sectional curvature and bounded diameter which allows for the collapse to lower dimensions. We analyse the spectral behaviour due to collapse and obtain stability of the inverse Problem. (with M. Lassas and T. Yamaguchi)