In this talk, I will discuss the existence of a unique global weak solution
to the general Ericksen-Leslie system in $R^2$, which is smooth away from possiblyfinite many singular times, for any initial data. This is a joint work with Jinrui Huang and Fanghua Lin.

We show that the set of limit-periodic Schroedinger operators with
purely absolutely continuous spectrum is dense in the space of
limit-periodic
Schroedinger operators in arbitrary dimensions. This result was previously
known only in dimension one.
The proof proceeds through the non-perturbative construction of
limit-periodic
extended states. The proof relies on a new estimate of the probability (in
quasi-momentum) that the Floquet Bloch operators have only simple
eigenvalues.

In the first part, we discuss penalty decomposition (PD) methods for solving
a more general class of $l_0$ minimization in which a sequence of penalty
subproblems are solved by a block coordinate descent (BCD) method. Under
some suitable assumptions, we establish that any accumulation point of the
sequence generated by the PD methods satisfies the first-order optimality conditions
of the problems. Furthermore, for the problems in which the $l_0$ part is the only
nonconvex part, we show that such an accumulation point is a local minimizer of the
problems. Finally, we test the performance of the PD methods by applying them to sparse
logistic regression, sparse inverse covariance selection, and compressed sensing
problems. The computational results demonstrate that our methods generally
outperform the existing methods in terms of solution quality and/or speed.  

In the second part, we consider $l_0$ regularized convex cone programming problems.
In particular, we first propose an iterative hard thresholding (IHT) method and
its variant for solving $l_0$ regularized box constrained convex programming. We
show that the sequence generated by these methods converges to a local minimizer.
Also, we establish the iteration complexity of the IHT method for finding an
$\epsilon$-local-optimal solution. We then propose a method for solving $l_0$
regularized convex cone programming by applying the IHT method to its quadratic
penalty relaxation and establish its iteration complexity for finding an
$\epsilon$-approximate local minimizer. Finally, we propose a variant of this
method in which the associated penalty parameter is dynamically updated, and
show that every accumulation point is a local minimizer of the problem.
 

T. Y. Lam proposed abstracting algebraic properties of the Peirce
corners eRe associated with an idempotent e in a ring R and introduced the
notion of general corners. We consider this notion principally in the
context of C*-algebras and some operator spaces in place of a ring R. Our
aim is to characterise such corners as fully as we can, ideally by
establishing that they are related to the ranges of the more well-known
completely positive conditional expectations.