Professor Peter Schneider
of University of Illinois, Chicago & University of
Munster
will speak on
"Is there a p-adic local Langlands correspondence?"
Among other things, algebraic number theory is about the study of Galois groups of algebraic number fields. By some local-to-global philosophy this very often is undertaken first for local number fields. This explores the fact that number fields possess a whole series of very interesting absolute values other than the standard archimedean absolute value. The corresponding completions are called local number fields. For example, the rational numbers $\mathfrak{Q}$ give rise to the p-adic number fields $\mathfrak{Q}_p$ for each prime number p. Unlike with $\mathfrak{C}$ over $\mathfrak{R}$ the algebraic closure $\overline{\mathfrak{Q}_p$ of $\mathfrak{Q}_p$ is of infinite degree. The task at hand therefore is to understand the automorphism group $Aut(\overline{\mathfrak{Q}_p}/\mathfrak{Q}_p)$. I will describe in this lecture the history of this problem. It starts with local class field theory during the first half of the last century which identifies the maximal abelian quotient of $Aut(\overline{\mathfrak{Q}_p}/\mathfrak{Q}_p)$. It proceeds to the local Langlands correspondence which approaches the full group by representation theoretic means and which was proved a few years ago. I will finish with explaining why this apparent climax in fact is, in the light of new developments, far from being the end of the story.