Michael Dobbins (Binghamton)
Realization Spaces of Arrangements of Convex Bodies
In this talk I will define combinatorial types of arrangements of
convex bodies, extending order types from point sets to arrangements
of convex bodies, and study their realization spaces. Our main results
witness a trade-off between the combinatorial complexity of the bodies
and the topological complexity of their realization space. On one hand,
we show that every combinatorial type can be realized by an arrangement
of convex bodies and (under mild assumptions) its realization space is
contractible. On the other hand, we prove a universality theorem that
says that the restriction of the realization space to arrangements of
convex polygons with a bounded number of vertices can have the homotopy
type of any primary semialgebraic set. This is joint work with Andreas
Holmsen and Alfredo Hubard.
To the Combinatorics Seminar Web page.