We will discuss a strong law of large numbers, an annealed CLT, and
the limit law of the ``environment viewed from the particle" for transient
random walks on a strip (product of Z with a finite set). The model was
introduced by Bolthausen and Goldsheid and includes in particular RWRE
with bounded jumps on the line as well as some one-dimensional RWRE with a
memory.

In this talk we will survey the development on the Green's functions of the Boltzmann equations. The talk will include the motivation from the field of hyperbolic conservation laws, the connection between the Boltzmann equation
and the hyperbolic conservation laws, and the particle-like and the wave-like duality in the Boltzmann equation. With all these components one can realize a clear layout of the Green's function of the Boltzmann equation. Finally we will present the application of the Green's function the an initial-boundary value problem in the half space domain.

In this talk, we shall design and analyze additive and multiplicative multilevel methods on adapted grids obtained by newest vertex bisection. The analysis relies on a novel decomposition of newest vertex bisection which give a bridge to transfer results on multilevel methods from uniform grids to adaptive grids. Based on this space decomposition, we will present a unified approach to the multilevel methods for $H^1$, $H(\rm curl)$, and $H(\rm div)$ systems.

numbers greater than or equal to 1 such that 1/p+1/q is less than or equal to 1. The Return Times Theorem proved by Bourgain asserts the following: For each function f in L^{p}(X) there is a universal subset X_0 of X with measure 1, such that for each second dynamical system (Y,Sigma_2,m_2,S), each g in L^{q}(Y) and each x in X_0, the averages 1/N\sum_{n=1}^{N}f(T^nx)g(S^ny) converge for almost every y in Y.
We show how to break the duality in this theorem. More precisely, we prove that the result remains true if p is greater than 1 and q is greater than or equal to 2. We emphasize the strong connections between this result and the Carleson-Hunt theorem on the convergence of the Fourier series. We also prove similar results for the analog of Bourgain's theorem for signed averages, where no positive results were previously known. This is joint work with Michael Lacey, Terence Tao and Christoph Thiele.