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Of current research interest in the area concerning the disorder effects on the dynamics under random Schroedinger operators is the occurrence of energy regimes (phases) in which extended states are formed from resonating local quasi-modes.
The corresponding eigenstates are ``non-ergodic'', in the sense that they violate a heuristic version of the equidistribution principle, yet they do not exhibit Anderson localization. Such phases were proven to occur in the the random Schroedinger operator on tree graphs (in a joint work with Simone Warzel), and are also expected to show up in many-particle systems which are the subject of ongoing work (with SW and Mira Shamis).

Echolocating bats exhibit an extraordinary array of solutions to the challenges of maneuvering in cluttered environments, pursuing evasive prey, taking food from water surfaces, and landing on the ceiling or walls of confined spaces. Moreover, they are equipped with a biological sonar system that permits spatial navigation and target tracking in complete darkness. By actively controlling the directional aim, timing, frequency content, and duration of echolocation signals to “illuminate” the environment, the bat directly influences the acoustic input available to its sonar imaging system. Detailed analyses of the bat’s sonar behavior suggests that the animal’s actions play into a rich 3-D representation of the environment, which then guides motor commands for subsequent call production, head aim and flight control in an adaptive feedback system. Somatosensory signaling of airflow along the wing membrane also contributes to the exquisite flight control of bats. Recent research reveals that microscopically small hairs embedded in the bat wing play a functional role in sensing air flow, which is important to it to carry out rapid and agile aerial maneuvers. Neurons in bat primary somatosensory cortex (S1) respond to directional stimulation of the wing hairs with low-speed air flow, and this response is diminished after removal of the hairs. The directional preference of cortical S1 neurons indicates that the hairs respond strongest to reverse airflow, and might therefore act as stall detectors. Further, depilation of different functional regions of the wing membrane alters flight behavior in obstacle avoidance tasks by reducing aerial maneuverability, as indicated by decreased turning angles. Collectively, these findings suggest that bat aerial navigation engages multisensory processes that guide a suite of adaptive motor behaviors.