Take a piece of wire of length L and move the endpoints
together. What is the shape of the resulting curve. Experimentally
it looks somewhat like the silhouette of an icecone. The calculation
should be done by minimizing the integral of of the square of the curvature
along the wire. There are various parameters in the problem,
such as the length L of the curve, and the relative angle at which
the endpoints meet.

Formulate the laws of motion of rod in a 2-dimensional
strip. Iterate numerically. Generalize to the equations
of motion of a rod between two horizontal plates.
This problem has important applications in the study
of granular flow.

This is a model for avalanches. As the particle falls down
the staircase, it may eventually stop, accellerate, or exhibit
a motion with bounded, non-zero velocity. What is the nature
of the boundary between these types of motion?

Certain models predict that it is possible to transport
granular material by depositing it on a vibrating staircase.
The idea to write the equations of motion for one
particle on a vibrating staircase, and numerically verifying the
predictions of these models.

Let M be a smooth n-manifold with two marked points a and b.
The mediatrix L is the set of of points x such that the
distance from x to a equals the distance from x to b.
The conjecture is that L is locally an "almost" smooth hypersurface
in M. In particular if M is 2-dimensional, then at every x_0 in L,
L consists of finitely many spokes, each of which is radially differentiable
at x_0.

The following model simulates the dynamics of an earth quake.
Consider a block placed on a flat surface with friction which is
dependent on the velocity. A spring is attached to the block.
To move the block the loose end of the spring is pulled with
a constant velocity. The force needed to get the block in motion
is greater than the frictional force, the block will move in
erratic fashion, similar to the stick-slip motion that occurs between
two active tectonic plates. The main features of this model are
well-known but the details of the dynamics of this system
remain a challenging problem.

This arises from a problem in statistical physics.
Is it possible or reasonable for a dynmical system to have
a measure whose mean appears to move on a strange attractor.
As we iterate the measure (using the push-forward), does
the measure itself lie on some attractor?