Monday, January 25, 2016
Technische Universität Berlin
Institut für Mathematik
Straße des 17. Juni 136
10623 Berlin
room MA 041
Lecture - 14:15
Abstract:
Although the roots of this work
can be traced back more than 40 years, there
has been a real surge of research in the
last five years exploring connections between
the dimension of a poset and the structural/topological
properties of its order diagram and the associated
cover graph. After reviewing essential preliminary
material, we will explore the origins and
motivation for the following three theorems,
and we will outline the proof techniques used to
prove them:
These results represent joint work with Dave Howard, Noah Streib, Bartosz Walczak, Ruidong Wang and Stephen Young.
Colloquium - 16:00
Abstract: In 2012, Streib and Trotter uncovered a connection between the dimension of posets and the topology of their order diagram by showing that posets with planar cover graphs have dimension bounded from above by a function of their height. That statement begged to be generalized---what about other classes of sparse graphs for instance?---and indeed several extensions of the Streib-Trotter theorem have been found recently, cf. Tom Trotter's talk.
In this talk, I will go back to the roots of this area and
address the question of how good a bound on the dimension one can get for
posets with planar cover graphs in terms of the height. Nowadays, we know of
four genuinely different ways of proving the Streib-Trotter theorem but they
all give exponential bounds, while Streib and Trotter suggested that a linear
bound might exist. I will report on ongoing work on this problem, and in
particular sketch a proof that there is a linear bound if the diagram of the
poset can be drawn in a planar way. This is joint work with Piotr Micek and
Veit Wiechert.