I will discuss the properties of discrete random Schrödinger operators in which the random part of the potential is supported on a sublattice. For the standard Anderson model, no results concerning localization/delocalization transition are rigorously established. For trimmed Anderson model described above, one can trace out the onset of the localization breakup, in the strong disorder regime (for some examples). This is a joint work with Sasha Sodin.

We present a joint work with Richard Bamler.
We consider Ricci flows that satisfy certain scalar curvature bounds. It is found that the time derivative for the solution of the heat equation and the curvature tensor have better than expected bounds. Based on these, we derive a number results. They are: bounds on distance distortion at different times and Gaussian bounds for the heat kernel, backward pseudolocality, L^2-curvature bounds in
dimension 4.

We show that the total mass, i.e. the sum over all points in the d-dimensional integer lattice of the solution to the parabolic Anderson model with initial function the point mass at the origin goes to zero in the high disorder regime. This talk is basedon joint work with L. Chen, D. Khoshnevisan, and K. Kim.