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.

See the abstract here.

We consider holomorphic Poisson structures as a special kind of
generalized geometry in the sense of Hitchin and Gaultieri.
A consideration on local deformation leads us to compute their associated
Lie algebroid cohomology spaces. As this cohomology is represented by the
limit of a bi-complex, we consider various situations early degeneration of
the associated spectral sequence of the bi-complex occurs. Cases for
discussion include Kahlerian manifolds and nilmanifolds with abelian complex
structures.

Two permutations are conjugate if and only if they have the same cycle structure, and two complex unitary matrices are conjugate if and only if they have the same set of eigenvalues. Motivated by the large literature on cycles of random permutations and eigenvalues of random unitary matrices, we study  conjugacy classes of random elements of finite classical groups. For the case of GL(n,q), this amounts to studying rational canonical forms. This leads naturally to a probability measure on the set of all partitions of all natural numbers. We connect this measure to symmetric function theory, and give algorithms for generating partitions distributed according to this measure. We describe analogous results for the other finite classical groups (unitary, symplectic, orthogonal). We were excited to learn that (at least for GL(n,q)), exactly the same random partitions arise in the “Cohen-Lenstra heuristics” of number theory.