Abstract: Can we make objects invisible? This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc. including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter. In the last decade or so there have been several scientific proposals to achieve invisibility. We will introduce some of these in a non-technical fashion concentrating on the so-called "transformation optics" that has received the most attention in the scientific literature.

The almost Matthieu operator arises as a model for Bloch electrons in a magnetic field. Aubry and Andre famously made a conjecture about the spectral properties of this operator more than thirty years ago. In the process of its study, variations of the conjecture arose naturally. Although the original conjecture was recently settled, new problems remain unsettled. We discuss some of these open problems and possible methods (and their shortcomings) to their solution.

The discrete directed polymer model is a well studied example of a Gibbsian disordered system and a random walk in a random environment. The usual goal is to understand how the random environment affects the behavior of the underlying walk and how this behavior varies with a temperature parameter that determines the strength of the environment. At infinite temperature the environment has no effect and the walk is the simple random walk, while at zero temperature the environment dominates and the walk follows a single path along which the environment is largest. For temperatures in between there is a competition between the walk wanting to behave diffusively (like simple random walk) and following a path of highest energy (like last passage percolation).

In this talk I will describe recent joint work with Kostya Khanin and Jeremy Quastel for taking a scaling limit of the directed polymer model to construct a continuous path in a continuum environment. We end up with a one-parameter family of random probability measures (indexed by the temperature parameter) that we call the continuum directed random polymer. As the temperature parameter varies the paths cross over from Brownian motion to what is conjectured to be a continuum limit of last passage percolation. This cross over is an inherent feature of the KPZ universality class, which I will also briefly describe.

Certain organisms undergo phototaxis, that is they migrate toward light. In this talk we will discuss our recent results on modeling phototaxis in order to understand the functionality of the cell and how the motion of individual cells is translated into emerging patterns on macroscopic scales. This is a joint work with Amanda Galante, Susanne Wisen, Tiago Requeijo, and Devaki Bhaya.

In this talk, I will present two deblurring methods, one exploits the spatial interactions in images, i.e. the self-similarity; and the other explicitly takes into account the sparse characteristics of natural images and does not entail solving a numerically ill-conditioned backward-diffusion.

In particular, the self-similarity is defined by a weight function, which induces two types of regularization in a nonlocal fashion. Furthermore, we get superior results using preprocessed data as input for the weighted functionals.

The second part of the talk is based on the observation that the sparse coefficients that encode a given image with respect to an over-complete basis are the same that encode a blurred version of the image with respect to a modified basis. An explicit generative model is used to compute a sparse representation of the blurred image, and the coefficients of which are used to combine elements of the original basis to yield a restored image.