Positivity is meant as a generalisation of the cyclic order on the circle. Associated to that is the notion of monotone maps from a cyclically ordered set in the circle.
Generalisations of the idea of positivity appeared to be crucial in understanding some connected components of the space of representations of a surface group in a Lie group G, although the common phenomenon was not figured out until recently.
In this talk, based on a preprint with Olivier Guichard and Anna Wienhard, I will start by examples generalizing this notion of cyclic order on the circle: convex curves or configurations in the plane, time like curve in Minkowski space. Then I will move to the general geometry of parabolic spaces and explain why the notion of positivity relates to special configurations of pairwise transverse triples and quadruples of points. This notion of positivity, which abides simple combinatorial properties, allows to define positive — or monotone — curves, then positive representations of surface groups.
I will then sketch the proof of our main result: positive representations are Anosov and fill up connected components of the space of representations.
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and where the social distancing is not possible;
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