Computer-assisted proofs - tools, methods and applications

Summary

Our seminars on computer-assisted proofs are intended to assemble a diverse group of scientists working on differing aspects of computer-assisted proofs and verification methods. The current one is the fifth initiated by Rump.

Computer-assisted proofs in general are characterized by the fact that part of a mathematical proof is assisted in an algorithmic way. This includes numerical calculations, taking account of all numerical errors, as well as symbolic computations.

This concept of computer-assisted proofs can be regarded as a special approach to constructive mathematics. In recent years, various mathematical problems have been solved by computer-assisted proofs, among then the Kepler conjecture (a 3 dimensional sphere packing problem), the existence of chaos, the existence of the Lorenz attractor, and more.

This concept of computer assisted proofs can be regarded as a special approach to constructive mathematics. In recent years, various mathematical problems have been solved by computer-assisted proofs, among then the Kepler conjecture (a 3 dimensional sphere packing problem), the existence of chaos, the existence of the Lorenz attractor, and more.

A major representative of computer-assisted proofs are so-called verification methods. These are algorithms verifying the correctness of the assumptions of mathematical theorems with rigor. These methods use solely floating-point arithmetic estimating all numerical errors. Therefore these methods are particularly fast. Besides the conference, a 163-page review article on verification methods by Rump was discussed which appeared in 2010 in Acta Numerica.

The organizers refrained from presenting talks to give more space to the participants. As always, they and the 46 participants from 10 different countries of the seminar enjoyed the pleasant and stimulating atmosphere in Dagstuhl. Our own assessment is that computer-assisted proofs have several new exciting directions pursued by a number of established and young researchers, and we are already looking forward to the next seminar.