The Cartan-Kahler theory for exterior differential systems (EDS) is a powerful tool in the study of over-determined systems and geometric systems with symmetry. I will present the machinery for linear Pfaffian systems, the setting in which the theory is most commonly applied. (The prolongation of any EDS is linear Pfaffian.) We will discuss tableau, torsion, involutivity, prolongation and Cartan's Test. I will illustrate the method with many examples, and hope to convince the audience of the power and elegance of the machinery.

If time permits, I will briefly discuss the Cartan-Kalher theory for general EDS: this is the machinery R. Bryant used to prove the existence (and construct examples) of metrics of exceptional holonomy in 1987.

The tree property at \kappa^+ states that every tree with height \kappa^+ and levels of size at most \kappa has an unbounded branch. There is a tension between the tree property and the Singular Cardinal Hypothesis (SCH). Woodin asked if the failure of SCH at a singular cardinal \kappa implies the tree property at \kappa^+. Recently Neeman answered this question in the negative. Here we show that this result can be obtained at small cardinals. In particular we will show that given \omega many supercompact cardinals there is a generic extension in which the tree property holds at \aleph_{\omega^2+1} and SCH fails at \aleph_{\omega^2}.

We study the two-phase Stefan problem that models heat propagation and phase transitions in a material with two distinct phases, such as
water and ice. For this problem, we introduce a notion of viscosity
solutions that allows for an appearance of the so-called mushy region. We prove a comparison principle and use this result to establish well-posedness of the viscosity solutions. As a corollary, we show that the viscosity solutions and the weak solutions defined in the sense of distributions coincide.

We will be looking at the trace map of the discrete Schrodinger operator with potential given by the period doubling sequence. It is known that for any positive coupling constant the spectrum of the corresponding operator is a Cantor set of Lebesgue measure zero. We are interested in the structure of the spectrum for small coupling constant, specifically the Hausdorff dimension and thickness.