We will give a mathematically-oriented review about the
geometry of the internal six-dimensional space M_6 in string theory
(with particular attention to the "type II" variety). In particular
we will be interested in vacua which have a property called
"supersymmetry." We will show what kind of constraints this
physical requirement puts on M_6. One reason this is interesting
mathematically is that the conditions we will get are a natural
generalization of the concept of Calabi-Yau manifold.

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.