Finite element approximations for the eigenvalue problem of the Laplace operator is discussed. A gradient recovery scheme is proposed to enhance the accuray of the numerical eigenvalues. By reconstructing the numerical solution and its gradient, it is possible to produce more accurate numerical eigenvalues. Furthermore, the recovered gradient can be used to form an a posteriori error estimator to guide an adaptive mesh refinement. Therefore, this method works not only for structured meshes, but also for unstructured and adaptive meshes.

Additional computational cost for this post-processing technique is only $O(N)$ ($N$ is the total degrees of freedom), comparing with $O(N^2)$ cost for the original problem.

In this talk we will present numerical methods for the simulation of several
two-phase flow problems requiring high accuracy at the interface. Throughout,
we will motivate and illustrate our approach using examples such as falling
liquid films, partial coalescence, bouncing droplets on a soap film, and
walking droplets on a periodically excited bath.
We identify the different domains present in such systems using a level set
function. A common challenge is how to correctly enforce boundary conditions
across the different domains. Another challenge is how to represent and track
small structures (possibly with sub-grid features) without loss of mass over
long computations.
Toward achieving these goals, we will introduce a new method to solve the
advection equation for the level set. Our approach relies on carrying both the
values and gradients of the level set function as coupled quantities via a
projection step. Using a local stencil, our method achieves third order global
accuracy and removes the need to solve a reinitialization equation. We will
then show the connection of the new method with enforcing sub-grid interfacial
jump conditions in the two-phase Navier-Stokes equations.
Finally, we will discuss some advantages of the proposed approach e.g.
locality
and higher order, and also extensions beyond two-phase liquid/gas systems.

We consider the spectrum of the Fibonacci Hamiltonian for small
values of the coupling constant. It is known that this set is a Cantor set
of zero Lebesgue measure. We show that as the value of the coupling constant
approaches zero, the thickness of this Cantor set tends to infinity, and,
consequently, its Hausdorff dimension tends to one. Moreover, the length of
every gap tends to zero linearly. Finally, for sufficiently small coupling,
the sum of the spectrum with itself is an interval. The last result provides
a rigorous explanation of a phenomenon for the Fibonacci square lattice
discovered numerically by Even-Dar Mandel and Lifshitz. The proof is based
on a detailed study of the dynamics of the so called trace map. This is a
joint work with David Damanik.

The local Langlands correspondence for GSp(4)
gives a classification of irreducible complex representations of GSp(4,k),
where k is a p-adic field in terms of 4-dimensional symplectic
Galois representations (plus some additional data). I will describe the
precise statement and give an idea of its proof. I will also mention
some further questions in this direction. This is joint work with
Shuichiro Takeda.