An approach to the analysis of the Laplace operator
via Lie algebras of vector fields
ABSTRACT:
"Several problems in pure and
applied mathematics lead to singular
differential operators. An example is
the Dirichlet boundary value problem for the
Laplace operator on a polygon or on a polyhedron.
The resulting differential operators are
Fuchs type differential operators (in
the case of a polygon) or generalizations
of these operators (in the case of a polyhedron).
A unified approach to the analysis of singular
differential operators has been suggested
by Melrose. His approach is based on Lie
algebras of vector fields. My talk will
be devoted to, first, explaining how the
Fuchs type operators and their generalizations
appear naturally if one considers
the Dirichlet boundary value problem for the
Laplace operator. Then I will explain, following
Melrose, how Lie algebras of vector
fields provide a unified language for treating
singular differential operators. In the end, I
will present some progress in Melrose's program
and some applications."