A sketch of the proof of Daubechies, Landau and Landau (1995) using elementary von Neumann algebra theory of Rieffel's (1981) incompleteness theorem: If ab>1, then there does not exist a square integrable function g whose Gabor lattice G(g,a,b) is dense in L^2.

In this talk I will introduce the manifest interconductance rank (MIR) form and contrast it to another long-known canonical form used in the
data-driven identification of ion channel gating kinetics: the uncoupled model (UCM). (The UCM has every open state connected to every closed state and vice versa). MIR form has significantly fewer parameters and provides more insight into gating kinetics than the uncoupled model. Beyond the new canonical form the principle results to be presented are
(1)All topologies with interconductance rank=1 and with the same number of open and closed states result in identical steady-state statistics
(2)detailed balance is preserved under transformation to either UCM or MIR forms and
(3) a general detailed balance preserving transformation. These results should facilitate maximum likelihood methods for finding models that best fit a given data set.

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?