**UBC Mathematics Department**

*http://www.math.ubc.ca*

## Colloquium Abstract: Professor Niky Kamran,
Department of Mathematics & Statistics, McGill University

*An abstract structure for the infinite pseudogroups of Lie and Cartan*

A pseudogroup of local diffeomorphisms of an analytic manifold is said
to be a *Lie pseudogroup* if it is the set of solutions of an
involutive exterior differential system. E. Cartan showed that any Lie
pseudogroup can be characterized geometrically as the set of local automorphisms
of a reduction of the frame bundle of a manifold associated to the
given differential system. The Lie pseudogroups defined by exterior
differential systems which are completely integrable in the sense
of the Frobenius theorem are said to be of *finite type*, since
their elements are parametrized locally by arbitrary constants. Classical
examples are given by the local isometries of a Riemannian manifold,
or the symmetries of a system of ordinary differential equations of
order n\geq 2. Any Lie pseudogroup of finite type can be regarded as
the representation by local diffeomorphisms of a finite-dimensional
(local) Lie group. In contrast, we have the Lie pseudogroups of
*infinite type*, whose elements are parametrized by arbitrary
functions. In this case, the defining exterior differential system
is involutive in the sense of the Cartan-Kahler Theorem. Simple examples
are given by the local automorphisms of a symplectic or a contact
manifold. There are basic questions about Lie pseudogroups of infinite
type which are not yet fully understood. One would like for example
to define an infinite-dimensional Lie group structure for these
pseudogroups. This is of course not possible in general, because of
the absence of a Frobenius theorem in most of the relevant
infinite-dimensional spaces. We will review a number of classical
results from the theory of Lie pseudogroups and sketch some recent
developments concerning the latter problem.

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