Monday, June 8, 2015
Technische Universität Berlin
Institut für Mathematik
Straße des 17. Juni 136
10623 Berlin
room MA 041
Lecture - 14:15
Abstract:
We consider geometric Ramsey statements involving subsets of
points or curves in general position, for various definitions of the
latter. This line of results stems from an old and still unsolved
problem of Erdős about the largest number α(n) such that
every set of n points in the plane with no four on a line contains
α(n) points with no three on a line. We show how to use
geometric incidence bounds to obtain new results in this vein,
involving plane curves, hyperplanes, and lines in 3-space.
Part of this talk is joint work with
Michael Payne, Noam Solomon, Csaba D. Tóth, and David R. Wood.
Colloquium - 16:00
Abstract:
The covariogram of a finite subset K ⊆ ℝd is the
function gK associating to each u ∈ ℝd the
cardinality of K ∩ (K+u). Two sets in ℝd are
called homometric if they have the same covariogram; they are
called nontrivially homometric if they are homometric but do not
coincide up to translations and point reflections.
We study nontrivially homometric pairs of lattice-convex sets,
where a set K ⊆ ℝd is called lattice-convex
if K is the intersection of ℤd and a convex subset
of ℝd . This line of research was initiated in 2005 by
Daurat, Gérard and Nivat and, independently, by Gardner,
Gronchi and Zong, who raised the problem of the reconstruction of
lattice-convex sets K from gK. We report on recent positive
and negative results to this problem: Under mild extra
assumptions, gK determines a planar lattice-convex set K up
to translations and reflections. The latter is not true in full
generality as we can give an infinite family of planar
nontrivially homometric pairs. Still, all known nontrivially
homometric pairs share a specific structure: They can be written
as direct sums S ⊕ T and S ⊕ (-T), where S and
T have some further properties. We present a necessary and a
sufficient condition for the existence of (nontrivially
homometric) lattice-convex sets having this particular form.
This allows us to explicitly describe all nontrivially homometric
pairs in dimension two, under the above `direct sum assumption',
and to construct examples of nontrivially homometric pairs of
lattice-convex sets for each dimension d ≥ 3.
This is joint work with Gennadiy Averkov.