Laver functions for supercompact cardinals appear in many forcing constructions, including all known constructions of models of strong forcing axioms. Viale proved that the Proper Forcing Axiom implies the existence of a "generic" Laver function from $\omega_2 \to H_{\omega_2}$. I will discuss his result and some recent work of mine on generic Laver functions.

I will discuss a sharp lower bound for the first positive eigenvalue of the sublaplacian on a closed, strictly pseudoconvex pseudo-hermitian manifold of dimension $2m+1\geq 5$. We prove that the equality holds iff the manifold is equivalent to the CR sphere up to a scaling. The essential step is a characterization of the CR sphere when there is a nonzero function satisfying a certain overdetermined system.
This is joint work with Song-Ying Li.

Abstract: For the site percolation model on the triangular lattice and
certain generalizations for which Cardy’s Formula has been established
we acquire a power law estimate for the rate of convergence of the
crossing probabilities to Cardy’s Formula.

Using as examples the Schroedinger equation and the wave equation we show that homogenization of many linear operators with oscillating coefficients could be done in a frame of the adiabatic approximation based on pseudodifferential operators (functions of noncommuting operators) and the Maslov methods. This approach allows one to reproduce well known homogenization results in the other way, but also take into account so-called dispersion effects leading to a change of structure of original equation. We discuss as example the asymptotic of the solution to the Cauchy problem with localized initial data and rapidly oscillating velocity.
This work was done together with J.Bruening, V.Grushin and S.Sergeev.