Geometrization is the process of converting algebra, with its powerful
computational techniques, into a geometric framework, which is often 
more intuitive and amenable to generalizations and connections with
other fields.
My interests concern the geometrization of
the complicated and fascinating algebraic structures, that have been
introduced to mathematics from physics, especially string theory. In
my dissertation, I clarify and expand the role of geometry in
the study of the soliton equations of mechanics. The first part of my
proposal develops this new understanding into a method for approaching
a range of questions in 
integrable mechanical systems and the associated symmetry groups.

In the second part of my proposal, I develop a detailed program to
geometrize some of the more elaborate intractable algebras coming from
string theory, the $\W$--algebras. This involves describing a new class
of geometries on surfaces, of which these algebras are the
symmetries. I expect these ``jet geometries'' to display a rich interplay
of different fields, generalizing many of the extensive facets of the
classical theory of surfaces. Moreover, they provide new
insights into old questions and conjectures.







