Thierry Coquand (Göteborg University, Sweden)

Title: A logical approach to abstract algebra

Abstract: We survey recent works in constructive mathematics, mainly
in algebra. These works show that Hilbert's program works
for a large part of abstract algebra. Using in an essential way the ideas
contained in the classical arguments, we can transform most of highly
abstract proofs of ``concrete'' statements into elementary proofs, that can
be seen directly as idealised algorithms. Surprisingly the arguments we
get are not only elementary but also mathematically clearer and not
necessarily longer. In some cases the simplification
is significant enough to suggest improved versions of classical theorems.
One (still distant?) goal is to connect research in constructive mathematics
with computer algebra.