Invited speaker : Marcel ERNE

Talk :

Galois connections provide the passage between two worlds of our thinking, under preservation or inversion of certain ordered, implicational or hierarchical structures. The celebrated classical example of a Galois connection is the dual isomorphism between suitable closure systems of field extensions and the corresponding automorphism groups, in the spirit of Artin's modern Galois theory. But do such connections actually occur in Galois' aphoristic original work? Do they perhaps already show up in the earlier research by Lagrange? Or was Dedekind the first mathematician dealing consciously with Galois connections? What about Hilbert's contribution to that theme? We shall pursue some of these and related questions and show that the basic ideas of Galois connections - hence the mathematical foundations of Formal Concept Analysis - are inherent in many famous theorems of classical algebra, logic or topology. Of course, a prominent role is played in that scenery by the pioneers of the 19th century: Boole, Schröder and Dedekind, but also by the ``inventors'' of order-theoretical Galois connections in the 20th century, Birkhoff and Ore. We shall touch upon some fundamental discoveries in that area, like polarities and axialities, extents and intents, adjoint situations, pseudo-negation and residuation - and we shall discuss some controversies about these topics. Certainly, it cannot be the intention of the talk to give a presentation of the whole historical development; but it should demonstrate how the elementary ideas of contexts, concept lattices and adjunctions facilitate many mathematical considerations and make the involved relationships more transparent than classical algebraic or topological methods.

Short biography

Since 1981, Marcel Erné is a professor of Mathematics at the University of Hannover. He received his Diploma in Mathematics at the University of München in 1970 and his Ph.D. at the University of Münster in 1972. In the years 1988 and 1989 he was a guest at the University of Toledo, Ohio, and at the TH Darmstadt, where he became acquainted with and interested in Formal Concept Analysis. His principal research areas are: order theory with applications in topology, algebra and computer science, set theory, categorical topology and universal algebra. In these fields, he has published around 100 papers. Current research topics are: dualities between categories of ordered sets, concept lattices and topological structures; distance functions and quasi-uniform spaces; combinatorics of finite ordered structures; ideal and radical theory in general algebras; domain theory, pointfree topology and choice principles. He is a coeditor of the journal ``Applied Categorical Structures''. In 2004, together with K. Denecke and S.L. Wismath, he edited a volume about ``Galois Connections and Applications'', initiated during a conference on the subject in Potsdam 2001.