Let k be a local field, and let K be a separable quadratic field extension of k. It is known that an irreducible complex representation π_1 of the unitary group G_1 = U_n(k) has a multiplicity free restriction to the subgroup G_2 = U{n−1}(k) fixing a non-isotropic line in the corresponding Hermitian space over K. More precisely, if π_2 is an irreducible representation of G_2 , then π = π_1 ⊗ π_2 is an irreducible representation of the product G = G_1 G_2 which we can restrict to the subgroup H = G_2 , diagonally embedded in G. The space of H-invariant linear forms on π has dimension ≤ 1.

In this talk, I will use the local Langlands correspondence and some number theoretic invariants of the Langlands parameter of π to predict when the dimension of H-invariant forms is equal to 1, i.e. when the dual of π_2 occurs in the restriction of π_1 . I will also illustrate this prediction with several examples, including the classical branching formula for representations of compact unitary groups. This is joint work with Wee Teck Gan and Dipendra Prasad.

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.