Sensory systems in individual living cells, as well as in multi-cellular organisms, employ a variety of adaptation mechanisms in order to produce behaviors that are invariant to certain characteristics of environmental inputs, such as symmetries or background signal levels, while at the same time allowing the extraction of relevant features of these inputs. These mechanisms and behaviors are responsible for phenomena ranging from chemotaxis in bacteria to the logarithmic sensitivities to forces, sounds, and vision in humans revealed through psychophysical measurements.

Much of our recent research has been devoted to the understanding of feedforward and feedback circuits that produce adaptation behavior. While closely related to standard concepts in control theory such as disturbance rejection, completely new questions arise. For example, while the internal model principle (IMP) would predict that feedback systems must be present in order to guarantee robust adaptation, the lack of separation between plant and controller components makes the significance of the IMP questionable. More so, the need to perform coordinate changes to exhibit the internal model (transformations which, if at all possible, require strong nonsingularity and global properties on vector fields) typically leads to uninterpretable variables. Moreover, questions such as the invariance of transient behaviors to symmetries appear not to have been systematically studied in this context. We will discuss one such behavior (fold-invariance) and mention new experimental results that confirm theoretical predictions.

Heteroepitaxial growth is a process where crystals are grown
one layer at a time using a molecular beam in a vacuum.
When strain is present these systems can form three dimensional
islands often called quantum dots. This is a nanoscale process
and continuum models struggle to capture many of the phenomena
that occur. Kinetic Monte Carlo (KMC) is an alternative approach which
is quite promising but has had limited use in strained systems
due to a variety computational bottle necks. In this talk, I will
outline KMC models for strained epitaxial growth and
how one can go about performing simulations in an efficient manner.

Model assimilation of data strives to determine optimally the state of an evolving physical system from a limited number of observations. The first attempt of applying the extended Kalman filter (EKF) method of data assimilation to shock-wave dynamics induced by high-speed impact will be presented. Additionally, we will mention current work on estimating hydrocode model parameters using EKF.

Conformal mappings have been widely applied in medical imaging and computer graphics, such as in brain registration and texture mapping, where the mappings are constructed to be as conformal as possible to reduce geometric distortions. A direct generalization of conformal mappings is quasiconformal mappings, where the mappings are allowed to have bounded conformality distortions. In this talk, we explore how the theories of quasiconformal mappings and their computations can be applied to areas where conformal mappings are used traditionally. These includes registration of biological surfaces, shape analysis, medical morphometry, compression and refinement of texture mappings, and the inpainting of surface diffeomorphisms.