Imaging of obscured targets in random media is a difficult and important problem. One of the central questions is that of stability which is particularly relevant to imaging in stochastic media. The main goal of the research is to develop a general criterion for multiple-frequency array imaging of multiple targets in stochastic media. An important feature of the cluttered media we considered here is that the coherent or mean signals do not vanish. It is called Rician fading medium in communication. Foldy-Lax formulas are used to simulate the exact wave propagations in the random media. In the talk, I will propose two models: passive array model and active array model. Then the stabilities of the imaging function with multiple point targets are given in both cases, followed by the numerical simulations to show the consistency with the analysis. If time permits, I will give some preliminary results about the stability conditions of the multiple extended targets, as well as the simulations.

We consider a directed polymer pinned by one-dimensional quenched randomness, modeled by the space-time trajectories of an underlying Markov chain which encounters a random potential of form u + V_i when it visits a particular site, denoted 0, at time i. The polymer depins from the potential when u goes below a critical value. We consider in particular the case in which the excursion length (from 0) of the underlying Markov chain has power law tails. We show that for certain tail exponents, for small inverse temperature \beta there is a constant D(\beta), approaching 0 with \beta, such that if the increment of u above the annealed critical point is a large multiple of D(\beta) then the quenched and annealed systems have very similar free energies, and are both pinned, but if the increment is a small multiple of D(\beta), the annealed system is pinned while the quenched is not. In other words, the breakdown of the ability of the quenched system to mimic the annealed occurs entirely at order D(\beta) above the annealed critical point.

I will discuss the behavior of N bosons trapped in a potential well subject
to a pairwise interaction. In the mean-field limit -- i.e., when N tends to
infinity while keeping the interaction per particle bounded -- the evolution
of a product state remains, asymptotically, a product state. The single
particle wave-function then evolves according to a non-linear Hartree
equation. Versions of this result have been proved before, e.g., by
Hepp in 1977, Spohn in 1980 or, recently, by Rodnianski and Schlein, but
the proofs are often quite technically involved. I will describe a very simple,
and ideologically correct, proof (for bounded interaction potentials) based
on exchengeability and Stormer's (aka quantum deFinetti) theorem.
Based on recent discussions with Nick Crawford.