Hitchin representations over real closed fields

The space of representations of the fundamental group of a closed surface to PSL(n,R) contains a distinguished connected component consisting entirely of discrete and faithful representations: the Hitchin component. For n = 2, this component is identified with the Teichmüller space of the surface and the study of its compactifications has led to many influential results. The aim of this talk is to present the Hitchin component, its real spectrum compactification and how we can interpret its boundary points geometrically as representations to PSL(n,F), where F is a closed real field, which admit positive limit maps in flag varieties over F.