I'll discuss some recent results on small cardinals with the tree property (joint work with my student Spencer Unger).

We study the stochastic heat equation ∂tu = u+σ(u)w in (1+1) dimensions, where w is space-time white noise, σ:R→R is Lipschitz continuous, and is the generator of a Lvy process. We assume that the underlying Lvy process has finite exponential moments in a neighborhood of the origin and u0 has exponential decay at ∞. Then we prove that under natural conditions on σ: (i) The νth absolute moment of the solution to our stochastic heat equation grows exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation. Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.
This is joint work with Daniel Conus (University of Utah)

A new characterization for Carleson measures in terms of integration on a non-tangential cone is established. Applications on the bounedness of area operator and Volterra operator on Hardy spaces are discussed.

This is a joint work with Lu Xu. We establish a geometric lower bound for the principal curvature of the level surfaces of solutions to $F(D^2u, Du, u, x)=0$ in convex ring domains, under a refined structural condition introduced by Bianchini-Longinetti-Salani.