The differential equation $\dot{x}(t) = Ax(t) + Bu(t)$ coupled with the
algebraic equation $y(t) = Cx(t) + Du(t)$ where $A\in\mathbb{C}^{n\times n}$,
$B\in\mathbb{C}^{n\times m}$, $C\in\mathbb{C}^{p\times n}$ is
called a state space system and commonly employed to represent
a linear operator from an input space to an output space in control
theory. One major challenge with such a representation is that
typically $n$, the dimension of the intermediate state function $x(t)$,
is much larger than $m$ and $p$, the dimensions of the input
function $u(t)$ and the output function $y(t)$. To reduce the order of
such a system (dimension of the state space) the traditional
approaches are based on minimizing the $H_{\infty}$ norm of the
difference between the transfer functions of the original system and
the reduced-order system. We pose a backward error minimization
problem for model reduction in terms of the norms of the
perturbations to the coefficients $A$, $B$ and $C$ such that the
perturbed systems are equivalent to systems of order $r<n$. It follows
from the fact that singular values are insensitive to perturbations that
a system with a small backward error has a small forward error, that
is the difference between the transfer functions is small in $H_{\infty}$
norm. We derive a singular value characterization for a simplified
version of the backward error minimization problem. The singular
value characterization is a generalization of a formula recently
derived for the Wilkinson distance problem, the norm of the smallest
perturbation to a matrix so that the perturbed matrix has a multiple
eigenvalue. We suggest methods to estimate the Wilkinson distance
and minimize the backward error for model reduction.

The Riemannian Penrose inequality in dimensions less than 8
Abstract: The Positive Mass Theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost minimal surface, and equality is achieved only for Schwarzschild metrics. The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole. In 1999, H. Bray extended this result to the general case of multiple black holes using a different technique. In this talk we extend Bray's technique to dimensions less that 8. This is joint work with H. Bray.

The Positive Mass Theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost minimal surface, and equality is achieved only for Schwarzschild metrics. The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole. In 1999, H. Bray extended this result to the general case of multiple black holes using a different technique. In this talk we extend Bray's technique to dimensions less that 8. This is joint work with H. Bray.

I will discuss Feynman's path integral interpretation of quantum mechanics over curved configurations spaces, i.e. Riemannian manifolds. We will see how curvature of the configuration space enters in the interpretation (and the ambiguity) of Feynman's path integral prescription.

I will begin with some elementary remarks about Hermitian
symmetric functions on complex manifolds. I will introduce
various positivity condtions for such functions and discuss
the relationships among them. Examples include globalizable metrics on Hermitian
line bundles. A version of the Cauchy-Schwarz inequality for
Hermitian symmetric functions and how it relates to complex geometry will
be a major part of the talk. To conclude I will pose several accessible problems

An optimization problem is ill-posed if its solution is not unique or is acutely sensitive to data perturbations. A common approach to such problems is to construct a related problem with a well-behaved solution that deviates only slightly from the original solution set. The strategy is often used in data fitting applications, and also within optimization algorithms as a means for stabilizing the solution process.

This approach is known as regularization, and deviations from solutions of the original problem are generally accepted as a trade-off for obtaining solutions with other desirable properties.

In fact, however, there exist necessary and sufficient conditions such that solutions of the regularized problem continue to be exact solutions of the original problem. We present these conditions for general convex programs, and give some applications of exact regularization.

(Joint work with Paul Tseng.)