# Tutorial on Computable Analysis

A compact course on computable analysis will be offered at Hagen, Germany, during

February 21-25, 2000,

by the theoretical computer science group:

Prof. Dr. Klaus Weihrauch
Theoretische Informatik I
FernUniversität Hagen
D-58084 Hagen
Germany

All over the world numerous computers are used for real number computation. They evaluate real functions, find zeroes of functions, determine eigenvalues and integrals and solve differential equations, and so they perform or at least are expected to perform computations on sets like the set of real numbers, the set of open subsets of real numbers or the set of differentiable real functions. The increasing demand for reliable as well as fast software in scientific computation and engineering requires a sound and broad foundation. Computable analysis is the mathematical theory of those functions on the real numbers and other sets from analysis, which can be computed by machines. It connects the two classical disciplines analysis/numerical analysis and computability/complexity theory combining in particular the central concepts of limit and approximation on the one hand and of machine models and computation on the other hand. Computable analysis may serve as an additional framework for numerical analysis and all other disciplines which need an exact concept of computation for real functions.

Though computable analysis started in the early years of computability theory, the field is still in its infancy. It has a great potential for further development, since there are numerous challenging open problems, many basic questions have not yet been studied systematically and only occasionally its concepts have been applied to advanced problems.

At present computable analysis appears as a juxtaposition of several approaches producing a number of very interesting results but developing more or less independently. Therefore, for the interested newcomer the situation is bewildering and learning the state of the art is a laborious undertaking.

This tutorial will give a self-contained introduction which will focus on the representation based approach to computable analysis. In this approach computability is defined explicitly on infinite strings and transferred to other sets like the real numbers by means of representations, where infinite strings are used as names of abstract objects. The following topics will be treated, among others: computability on infinite strings, representations of objects via infinite strings and relative computability via representations, computability over the real numbers, computability on closed or open sets, computability on spaces of continuous functions, computational complexity. In order to make the workshop as efficient as possible, lectures will alternate with exercises and discussions. It is aimed at graduate students and researchers in computer science or mathematics who are interested in an introduction to computable analysis. Only basic knowledge in analysis, computability theory and complexity theory will be assumed.

## Time table

 Morning Session 8:30 to 12:00 Afternoon Session 14:00 to 17:30 Monday February 21 Introduction Computability on Cantor Space Tuesday February 22 Naming Systems Computability on the Real Numbers Wednesday February 23 Computability on Closed, Open and Compact Sets Excursion Thursday February 24 Spaces of Continuous Functions Analytic Functions and Power Series Friday February 25 Computational Complexity Extensions and other Approaches

All lectures take place in room number F 09, Informatikzentrum, Feithstraße 142, FernUniversität Hagen, D-58097 Hagen

## Participants

This picture shows some participants of the tutorial

## Contact

by letter:

Tutorial
Prof. Dr. Klaus Weihrauch
Theoretische Informatik I
FernUniversität Hagen
D-58084 Hagen
Germany

by email:

[Email remove]

or one of the lecturers:

Klaus Weihrauch
Vasco Brattka
Peter Hertling

## Registration

There will be only a small fee of DM 30,-. For registration please send an email or a letter to the contact address given above.

## Accommodation

Here you can find a

List of hotels in Hagen

The hotels in "Hagen-Nord" are quite close to the university; e.g. Birkenhof (20 min. by feet) and Zum Bauernhaus (3 min. by feet). Please ask us in time if you need help with finding accommodation. (For students a limited number of reasonably priced accommodations is available.)

## Further Information

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