In 1966, Almgren showed that any immersed minimal surface in
S^3 of genus 0 is totally geodesic, hence congruent to the equator. In
1970, Blaine Lawson constructed many examples of minimal surfaces in S^3
of higher genus; he also constructed numerous examples of immersed minimal
tori. Motivated by these results, Lawson conjectured that any embedded
minimal surface in S^3 of genus 1 must be congruent to the Clifford
torus.

In this talk, I will describe a proof of Lawson's conjecture. The proof
involves an application of the maximum principle to a function that depends
on a pair of points on the surface.

The Stokes-Darcy model arises in many interesting real world applications, including groundwater flows in karst aquifers, interaction between surface and subsurface flows, industrial filtrations, oil reservoir in vuggy porous medium, and so on. This model describes the free flow of a liquid by the Stokes or Navier-Stokes equation and the confined flow in a porous media by the Darcy equation; the two flows are coupled through interface conditions. For the problems mentioned, the resulting coupled Stokes-Darcy model has higher fidelity than either the Darcy or Stokes systems on their own. However, coupling the two constituent models leads to a very complex system.

This presentation discusses multi-physics domain decomposition methods for solving the coupled Stokes-Darcy system. Robin boundary conditions based on the physical interface conditions are utilized to decouple the Stokes and Darcy parts of the system. A parallel iterative domain decomposition method is first constructed for the steady state Stokes-Darcy model with the Beavers-Joseph interface condition. Then two parallel non-iterative domain decomposition methods are proposed for the time-dependent Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition. Numerical examples are presented to illustrate the features of these methods and verify the theoretical results.

We will introduce the rudiments of a new theory of non-smooth
solutions which applies to fully nonlinear PDE systems and extends
Viscosity Solutions of Crandall-Ishii-Lions to the general vector case.
Key ingredient is the discovery of a notion of Extremum for maps which
extends min-max uniquely and allows for ``nonlinear passage of
derivatives" to test maps. The notions supports uniqueness, existence
and stability results, preserving most features of the scalar viscosity
counterpart. We will also discuss applications in vector-valued Calculus
of Variations in $L^\infty$ and Hamilton-Jacobi PDE with vector
solution.

In this talk, we focus on the study of mathematical theory of
the complex fluids. In the first part, we discuss the global existence for
weak solutons to multidimensional compressible flow of nematic liquid
crystals and the incompressible limits. In the second part, we establish
global existence and uniqueness results for weak solutions to
multidimensional Navier-Stokes-Vlasov equations.