Monday, June 20, 2016
Freie Universität Berlin
Takustr. 9
14195 Berlin
room 005
Lecture - 14:15
Abstract:
Joint work with Boris Aronov, Xavier Goaoc, and Michael Dobbins
We show that the union of n translates of a convex body in~$\R^3$
can have~$\Theta(n^3)$ holes in the worst case, where a \emph{hole}
in a set~$X$ is a connected component of $\R^3 \setminus X$. This
refutes a 20-year-old conjecture. As a consequence, we also obtain
improved lower bounds on the complexity of motion planning problems
and of Voronoi diagrams with convex distance functions.
Colloquium - 16:00
Abstract:
A shadow of a geometric object A in a given direction v is
the orthogonal projection of A on the hyperplane orthogonal to v.
In particular, we define the i-th coordinate shadow of A as the image
of A by the orthogonal projection to the i-th coordinate hyperplane.
In this talk, I will present intuitive proofs of the following
statements:
(i) for any d>=1, there exists a d-sphere in (d+2)-space, all of
whose coordinate shadows are contractible,
(ii) for any d>=3, a simple closed polygonal curve in d-space has at
most two coordinate shadows that are simple paths.
The case d=3 of (ii) is joint work with
Prosenjit K. Bose, Jean-Lou De Carufel, Michael Gene Dobbins, and Giovanni
Viglietta. The case d>3 of (ii) is joint work with
Michael Gene Dobbins, Luis Montejano, and Edgardo Roldan-Pensado.