The Schwarzchild, respectively the Kerr space-times are solutions for the vacuum Einstein equation which model a spherically symmetric, respectively a rotating black hole. In this talk I will discuss the decay properties of solutions to the linear wave equation on
such backgrounds.

Suppose g is a square integrable function on the real line. The principal shift invariant space, , generated by g is the closure of the span of the system
B ={g(.-k): k an integer}. These spaces are most important in many areas of Analysis. This is particulrly true in the theory of Wavelets. We begin by describing a very simple method for obtaining the basic properties of and the systems B.
The systems obtained by applying, in addition to the integral translations, also the integral modulations (these are the multiplication of a function by exp(-2pinx)) are known as the Gabor systems. By using the Zak transform we show how the same methods can be used to study the basic properties of the Gabor systems and their span.
We will define the Zak transform and explain all this
in a very simple way that will be easily understood by all who know only a "smidgeon" of mathematics. A bit more challenging will be the explanation how all this can be extended to general locally compact abelian groups and their duals.
This is joint work with E. Hernandez, H. Sikic and E. N.
Wilson.

In this talk, we present a characterization of the Christoeffel pairs of timelike isothermic surfaces in the four-dimensional split-quaternions. When restricting the receiving space to the three-dimensional imaginary split-quaternions, we establish an equivalent condition for a timelike surface in $R^3_2$ to be real or complex isothermic in terms of the existence of integrating factors. This is joint work with M. Magid (Wellesley College).

As the complex version of Ricci flow, K\"ahler-Ricci flow enjoys the special feature, i.e., cohomology information for the evolving K\"ahler metric. The flow can thus be reduced to scalar level as first used by H. D. Cao in the alternative proof of Calabi's Conjecture. People have mostly been focusing on the situation when the K\"ahler class is fixed. As first considered by H. Tsuji, by allowing the class to evolve, the flow can be applied in the study of degenerate class, for example, class on the boundary of K\"ahler cone. We discuss some results in this drection. This is the geometric analysis aspect of Tian's program, which aims at applying K\"ahler-Ricci flow in the study of algebraic geometry objects with great interests.

By trace map we mean the following polynomial map of R^3:
T(x,y,z)= (2xy-z, x, y).

Despite of its simple form, it is related to complicated mathematical objects such as character varieties of some surfaces, Painlev\'e sixth equation, and discrete Schr\"odinger operator with Fibonacci potential. We will present some very recent results on dynamics of the trace map and discuss their applications. These is a joint project with D.Damanik.