The total variation based image denoising model of Rudin, Osher,
and Fatemi
has been generalized and modified in many ways in the literature; one of
these modifications is to use the L1 norm as the fidelity term. We study the
interesting consequences of this modification, especially from the point of
view of geometric properties of its solutions. It turns out to have
interesting
new implications for data driven scale selection and multiscale image
decomposition.

(joint work with Selim Esedgolu).

We show that certain abelian surfaces come from p-adic Siegel cusp forms (in the sense that they have the same p-adic Galois representation). This relates to the question: unless it is isogenous to a product of elliptic curves, does an abelian surface come from a Siegel cusp form?