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.

Since 2000, there has been a growing interest in the study of negative
index metamaterials across many disciplinaries. In this talk, I'll first derive
the Maxwell's equations resulting from negative index metamaterials. Then I'll discuss some time-domain mixed finite elements developed for solving these equations, followed by succinct error estimate. Finally, some numerical
results will be presented.

The study of singular solutions of the NLS goes back to the 1960s, with applications in nonlinear optics and, more recently, in BEC. Asymptotic and numerical studies conducted in the 80s showed that singular solutions of the critical NLS collapse with the Townes (R) profile at a blowup rate known as the loglog law. Recently (2003) Merle and Raphael proved this result rigorously for a large class of initial conditions. Concurrently, it was demonstrated experimentally that the profile of collapsing laser beams is given by the Townes profile. Thus, all the research that was carried out from the eighties until these days leads to the belief that the Townes profile is the only attractor of blowup solutions of the critical NLS.
In this talk I will present new families of singular solutions of the critical and supercritical NLS that collapse with a self-similar ring profile, and whose blowup rate is different from the one of the "old" singular solutions. I will show, experimentally and theoretically, that these new blowup profiles are attractors for large super-Gaussian initial conditions.
In addition, I will present in the talk a semi-static adaptive grid method we have developed for the numerical simulations involved in this study for solutions which focus over 15 orders of magnitude.