This talk will describe the simulation, design and optimization of a qubit
for use in quantum communication or quantum computation. The qubit is
realized as the spin of a single trapped electron in a semi-conductor
quantum dot. The quantum dot and a quantum wire are formed by the
combination of quantum wells and gates. The design goal for this system is a
"double pinchoff", in which there is a single trapped electron in the dot
and a single (or small number of) conduction states in the wire. Because of
considerable experimental uncertainty in the system parameters, the optimal
design should be "robust", in the sense that it is far away from
unsuccessful designs. We use a Poisson-Schrodinger model for the
electrostatic potential and electron wave function and a semi-analytic
solution of this model. Through a Monte Carlo search, aided by an analysis
of singular points on the design boundary, we find successful designs and
optimize them to achieve maximal robustness.

I will present a rigorous study of the perfect Bose-gas in the
presence of a homogeneous ergodic random potential. It is
demonstrated that the Lifshitz tail behaviour of the one-particle
spectrum reduces the critical dimensionality of the (generalized)
Bose-Einstein Condensation (BEC) to $d=1$. To tackle the
Off-Diagonal Long-Range Order (ODLRO) I will introduce the
space averaged one-body reduced density matrix. For a one
dimensional Poisson-type random potential we proved that
randomness enhances the exponential decay of this matrix in domain
free of the BEC.
These general results will then be applied to the Luttinger-Sy model in
which I can explicitely compute any of the physical quantities
(pressure, density, type of condensation, ODLRO...).

In this talk, I will use the Korteweg-de
Vries equation, non-linear Schrodinger equation, and
the sine-Gordon equation as models to explain some
remarkable properties of a certain class of non-linear
wave equations, the so called "soliton equations".
Some relations to differential geometry will be
discussed. I will also use Richard Palais'
3D-XplorMath Visualization Computer Program to help us
"see" some of these properties.