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.

As one of the deepest and most beautiful theorems in geometry, the
Hodge theorem builds a bridge between Riemannian metric and
topological invariants. It gives an isomorphism between the space of
harmonic p forms on a Riemannian manifold and the pde Rham cohomology group of a smooth structure. By the de Rham theorem, we see the
isomorphism between the space of harmonic p forms and p real singular cohomology group.

The Hodge theorem is a good example of how PDEs help us understand geometric structure and even topological structure. In this talk, we
will give an introduction to this theorem, explain the idea behind it, and give some applications in Riemannian geometry.

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}.