A linear time-invariant dynamical system is
controllable if its trajectory can be adjusted
to pass through any pair of points by the proper selection of an
input.
Controllability can
be equivalently characterized as a rank
problem and therefore cannot be verified
reliably numerically in finite precision.
To measure the degree of controllability of a system
the <em>distance to uncontrollability</em> is introduced as the
spectral or
Frobenius norm of the
smallest perturbation yielding an uncontrollable system.
For a first order system we present a polynomial time
algorithm to find the nearest uncontrollable system
that improves the computational costs of the previous techniques.
The algorithm locates the global
minimum of a singular value optimization problem
equivalent to the distance to uncontrollability.
In the second part for higher-order systems we derive a singular-value
characterization and exploit
this characterization for the computation of the higher-order distance
to
+uncontrollability to low
precision.

We generalize the approach of Adams by obtaining a character fornula relating the irreducible characters of $PGL(n)$ with those of certain covers of $SL(n)$. We then study the lifting of functions between covers of $SL(n)$ and $PGL(n)$. We use orbital integrals to obtain a formula for the lifting of characters as a dual to the formula relating the characters. This is based on the approach of Kazhdan and Flicker.

In work in the mid-90's, John Stembridge noted that several combinatorial statistic generating functions carried enumerative information when the variable was set to $-1$. He dubbed this the "$q=-1$ phenomenon. In joint work with Vic Reiner and Dennis Stanton, we have noted that Stembridge's phenomenon generalizes to when $q$ is an appropriate root of unity. We called this the "cyclic sieving phenomenon." I will describe several instances of this phenomenon and some open problems.