"Computation on topological algebras""

**PLEASE NOTE SPECIAL DAY AND PLACE**

Wednesday, March 1, 4:30pm, Baker Hall 231B

ABSTRACT: Computations on topological algebras cannot be performed in
general since the spaces may be uncountable.  By considering
computations on approximations rather than on the actual elements it
is possible to establish a theory for computations on topological
algebras.  One way to do this (if the approximations constitute the
compact elements of a Scott-Ersbov domain) we call a domain
rcpresentation of the space.  The natural computability theory of
domains may be then imported into the topological spaces.
Introduction of computability into topological algebras via domain
representations can be shown to work for a large class of metric
spaces, the effective metric spaces, including several Lp-spaces.  In
fact, domain representations can be given for any To topological
space.  However, the domains constructed for general topological
spaces are not always countably based and are therefore unsuitable for
computations.

The general computability introduced by a domain representation of the
reals corresponds to the classical theory of Computable Analysis.
However, there exists many sources for uncountable topological
algebras, e.g., streams.  We view streams as arbitrary functions from
time to data.  Streams may model both continuous and discrete
phenomena with respect to both time and data.  Hence, discontinuous
streams (functions) appear naturally, for example, digital signals
seen over continuous time constitute streams that are discontinuous
unless they are constant.  By introducing a notion of streams being
approximatively represented, we may actually consider computations
containing discontinuous streams (functions).  Another way of viewing
approximative representations of functions is that they represent the
partial but continuous function obtained by restricting the functions
to the points where they are continuous.