Consider the generalized Anderson Model
$H^\omega=\Delta+\sum_{n\in\mathcal{N}}\omega_n P_n$, where $\mathcal{N}$ is a countable set, $\{\omega_n\}_{n\in\mathcal{N}}$ are iid randomvariables and $P_n$ are rank $N<\infty$ projections. For these models one can prove theorems analogous to that of Jak\v{s}i\'{c}-Last on the
equivalence of measures.

We show that if the projection $Q_m^\omega P_n$ (where $Q^\omega_m$ is
cannonical projection on the subspace generated by $H^\omega$ and range of
$P_m$) has same rank as $P_n$, then the trace measure
$\sigma_i(\cdot)=tr(P_iE_{H^\omega}(\cdot)P_i)$ and absolute continuous
part of the measure $P_iE_{H^\omega}(\cdot)P_i$ are equivalent for $i=n,m$.
 

We synthesize the double Fourier sphere method and
low rank function techniques to develop a collection of
fast numerical algorithms for computing with functions based
on the fast Fourier transform.  Furthermore, by imposing certain
partial regularity conditions on the solutions of PDEs we derive
optimal complexity and stable spectral methods for long-time
simulations of the Navier-Stokes equations on the disk, spiral waves
on the surface of the sphere, and geophysical flows in the solid
sphere. 

Preserving Academinc Honesty 

This is the second meeting of our sequence of teaching seminars. We will continue to illustrate (and practice) best strategies to bring active learning into our classrooms (especially in the calculus discussion sessions). In addition, we will focus on understanding and preventing academic dishonesty with the help of a special guest, Don Williams, Director of PS Student Affairs. 

Finite element exterior calculus (FEEC) is an framework to design and understand
finite element discretizations for a wide variety of systems of partial
differential equations. The applications are already made to the Hodge Laplacian,
Maxwell's equations, the equations of elasticity, and elliptic eigenvalue problems.
We propose fast solvers for several numerical schemes based on the discretization
of this approach and present theoretical analysis. Specifically, in the first part,
we propose an efficient block diagonal preconditioner for solving the discretized
linear system of the vector Laplacian by mixed finite element methods. A variable
V-cycle multigrid method with the standard point-wise Gauss-Seidel smoother is
proved to be a good preconditioner for the Schur complement $A$. The major benefit
of our approach is that the point-wise Gauss-Seidel smoother is more algebraic and
can be easily implemented as a `black-box' smoother. The multigrid solver for the
Schur complemEnt will be further used to build preconditioners for the original
saddle point systems. In the second part, we propose a discretization method for
the Darcy-Stokes equations under the framework of FEEC. The discretization is shown
to be uniform with respect to the perturbation parameter. A preconditioner for the
discrete system is also proposed and shown to be efficient . In the last part, we
focus on the stochastic Stokes equations. The stochastic saddle-point linear systems
are obtained by using finite element discretization under the framework of FEEC in
physical space and generalized polynomial chaos expansion in random space. We prove
the existence and uniqueness of the solutions to the continuous problem and its
corresponding stochastic Galerkin discretization. Optimal error estimates are also
derived. We construct block-diagonal/triangular preconditioners for use with the
generalized minimum residual method and the bi-conjugate gradient stabilized method.
An optimal multigrid solver is applied to efficiently solve the diagonal blocks
that correspond to deterministic discrete Stokes systems. To demonstrate the
efficiency and robustness of the discretization methods and proposed block
preconditioners, various numerical examples also are provided.

Abstract:

I will discuss various geometric gluing constructions. First I will discuss constructions for Constant Mean Curvature hypersurfaces in Euclidean spaces including my earlier work for two-surfaces in three-space which settled the Hopf conjecture for surfaces of genus two and higher, and recent generalizations in collaboration with Christine Breiner in all dimensions. I will then briefly mention gluing constructions in collaboration with Mark Haskins for special Lagrangian cones in Cn. A large part of my talk will concentrate on doubling and desingularization constructions for minimal surfaces and on applications on closed minimal surfaces in the round spheres, free boundary minimal surfaces in the unit ball, and self-shrinkers for the Mean Curvature flow. Finally I will discuss my collaboration with Simon Brendle on constructions for Einstein metrics on four-manifolds and related geometric objects.