We present numerical schemes for the incompressible Navier-Stokes
equation with open or traction boundaries. We use pressure Poisson
equation formulation and propose new boundary conditions for
pressure on the open or traction boundaries. For Stokes equation
with open boundary condition, we prove unconditional stability of
a first order semi-discrete scheme with explicit treatment of the
pressure. Using either boundary condition, the schemes for full
Navier-Stokes equations that treat both convection and pressure
terms explicitly work well with various spatial discretization
including spectral collocation and $C^0$ finite elements. Besides
standard stability and accuracy check, various numerical results
including backward facing step, flow past a cylinder and a
bifurcation tube (or h-shaped tube) are reported. In all the
numerics, we do not have to require the inf-sup compatibility
condition between finite element spaces for velocity and pressure.
Even though we treat pressure and convection terms explicitly, time
step size of $O(1)$ is allowed in benchmark computations with
either boundary condition when Reynolds number is of $O(1)$ and
when first order time stepping is used. Our results extend that of
H. Johnston and J.-G. Liu (J. Comp. Phys. 199 (1) 2004, 221--259)
which deals with Dirichlet boundary condition.

"Distinguished varieties" are a special class of algebraic
curves in C^2 that exit the bidisk through the distinguished boundary
(aka the torus). We shall discuss connections with the polynomials
that define these curves and polynomials with no zeros on the bidisk,
and use a powerful "sums of squares" formula (actually a two variable
version of the Christoffel-Darboux formula for orthogonal polynomials)
to a prove a determinantal representation of distinguished varieties.
As an application of our approach, we will prove a certain bounded
analytic "extension" theorem.

I will first give a brief introduction of the connection between
a Hamilton-Jacobi equation and the Aubrey-Mather theory. This is the so
called weak KAM theory. An extremely interesting project in weak KAM
theory is to understand what kind of dynamical information is encoded in
the
effective Hamiltonian. I will present a result about the connection
between linear pieces on level curves of the effective Hamiltonian and the
structure of correspondent Aubry sets.