In this talk, a multidimensional, multiphysics, two-phase transport model of proton exchange membrane fuel cell (PEMFC), which is based on the multiphase mixture formulation and encompasses all components in a PEMFC using a single computational domain, is specifically presented and simulated by a combined finite element-upwind finite volume method together with Newton's linearization, where flow, species, charge-transport and energy equations are simultaneously solved. To investigate the essential fuel cell model, I begin with a 2D simplified single-component two-phase PEFC model. Numerical simulations in 3D are carried out as well to explore and design efficient and robust numerical algorithms for the sake of fast and convergent nonlinear iteration. A more reasonable source term for water transport equation is given, and a series of efficient numerical algorithms and discretizations are designed and analyzed to achieve this goal. Our numerical simulations show that the convergent physical solutions can be reached within one hundred more steps, against the standard finite-volume based commercial CFD solvers which always produce oscillating iterations and never reach convergent solutions. Attained reasonable and comparable numerical solutions illustrate that our numerical methods and iterative algorithms are efficient and robust.

Colding and Minicozzi have shown that if an embedded minimal disk in $B_R\subset\Real^3$ has large curvature then in a smaller ball, on a scale still proportional to $R$, the disk looks roughly like a piece of a helicoid. In this talk, we will see that near points whose curvature is relatively large the description can be made more precise. That is, in a neighborhood of such a point (on a scale $s$ proportional to the inverse of the curvature of the point) the surface is bi-Lipschitz to a piece of a helicoid. Moreover, the Lipschitz constant goes to 1 as $Rs$ goes to $\infty$ . This follows from Meeks and Rosenberg's result on the uniqueness of the helicoid of which, time permitting, we will discuss a new proof. Joint work with C. Breiner.

A few years ago -- following a suggestion by I. M. Gelfand-- I discovered an intriguing connection between the topological degree of a map from the circle into itself and its Fourier coefficients. This relation is easily
justified when the map is smooth. However, the situation turns out to be much more delicate if one assumes only continuity, or even Holder continuity.
I will present recent developments and open problems.
I will also discuss new estimates for the degree of maps from S^n into S^n, leading to unusual characterizations of Sobolev spaces.
The initial motivation for this direction of research came from the analysis of the Ginzburg-Landau model.

In this lecture, we present a method which has broad applicability to studies of nonlinear stability of periodic traveling-wave solutions for equations of KdV-type. In particular we obtain the existence and stability of a family of periodic traveling-wave solutions for the Benjamin-Ono equation via the classical Poisson summation theorem and positivity properties of the Fourier transform.