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.

We consider the question: ``How bad can the deformation space of an
object be?'' (Alternatively: ``What singularities can appear on a
moduli space?'') The answer seems to be: ``Unless there is some a
priori reason otherwise, the deformation space can be arbitrarily
bad.'' We show this for a number of important moduli spaces.
More precisely, up to smooth parameters, every singularity that can be
described by equations with integer coefficients appears on moduli
spaces parameterizing: smooth projective surfaces (or
higher-dimensional manifolds); smooth curves in projective space (the
space of stable maps, or the Hilbert scheme); plane curves with nodes
and cusps; stable sheaves; isolated threefold singularities; and more.
The objects themselves are not pathological, and are in fact as nice
as can be. This justifies Mumford's philosophy that even moduli
spaces of well-behaved objects should be arbitrarily bad unless there
is an a priori reason otherwise.
I will begin by telling you what ``moduli spaces'' and ``deformation
spaces'' are. The complex-minded listener can work in the holomorphic
category; the arithmetic listener can think in mixed or positive
characteristic. This talk is intended to be (mostly) comprehensible
to a broad audience.