Tutorial on
Computable Analysis

About computable analysis

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.

About the tutorial

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:

by email:

or one of the lecturers:

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

Further Information

• Hagen
• Map of Hagen
• Railway time table
• Bus time table (Ort="Hagen", Haltestelle="FernUni")
• Weather forecast (PLZ="58000")
• Currency converter
• Köln