Spaces of real places play an important role in real algebra and real algebraic
geometry. They appear in all settings where non-archimedean orderings have to be
studied. We will show with an easy example that such settings come up in a very
natural way.
The first systematic study of real places was done by Serge Lang in 1953. Since
then, spaces of real places have been studied by Murray Marshall, Eberhard
Becker, Danielle Gondard, Ron Brown and others. But many questions have
remained open. It is not known which topological spaces can be represented as
spaces of real places, but partial results were obtained in the past few
years. In the talk, I will present these results. I will also show an example
of a space of real places with an amazingly rich topological and algebraic
structure. There are plenty of self-similarities in this space, which has led
us to the question whether some sort of fractality can be defined.
The spaces we study are not necessarily metric, and therefore their natural
topology is not easy to handle.
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