• Professor,
    • Jan van Mill
    • Vrije Universiteit Amsterdam
    • Topology

Brouwer’s Contributions to Topology

Brouwer brought about several revolutions. One of them was in topology. His most well-known result in that area is probably his widely used fixed-point theorem. The easiest proof of it is due to Sperner and is of a combinatorial nature. The fixed-point theorem is also a consequence of Brouwer’s result on the topological invariance of degree. Brouwer was also the founder of dimension theory. The dimension of a topological space measures its topological complexity. In this lecture we will briefly discuss these fundamental results. In the early days of dimension theory there was a lot of confusion. We will try to elucidate the causes of the confusion, which still persist today.


Within the field of mathematics, Jan van Mill carries out research into topological groups, the topology of infinite dimensional manifolds, dimension theory and set-theoretic topology. He also has an interest in the history of topology. Knowledge of the symmetry groups of geometric objects is essential when conducting research into the structure of the latter. Due to the nature of geometric objects, their symmetry groups are complex and consist of more than simply rigid motions. For example, these groups in particular are of an infinite size. This means that not only is the algebraic structure of these groups important, but also their topological structure. Topology formalises the concept of proximity of transformations. Van Mill's work concentrates on the topological classification of such objects. Van Mill is Emeritus Professor of Mathematics at the Vrije Universiteit, to which he has been affiliated since 1974. From 2006 to 2010, Van Mill served as Dean of the Faculty of Sciences at the Vrije Universiteit. He was affiliated with the University of Amsterdam for a number of years on a part-time basis in the 1980s. In 2014 he was appointed Professor of Topology at the University of Amsterdam, he retired in 2017. Van Mill is a member of several foreign academic committees and is affiliated with various academic journals, for instance as editor-in-chief of the journal 'Topology and its Applications'.

J. van Mill's Homepage