Monday, June 19, 2017
Freie Universität Berlin
Takustr. 9
14195 Berlin
room 005
Lecture - 14:15
Abstract:
We focus on geometric (convex and algebraic) aspects of the positive semidefinite matrix completion problem. Of particular interest will be the geometry of Hankel spectrahedra.
Colloquium - 16:00
Abstract:
The set of all f-vectors of 3-dimensional polytopes is the set of integer
points of a 2-dimensional rational cone. This is due to Steinitz (1906).
Grünbaum and others characterized the projection of the set of f-vectors
of 4-polytopes to any two of its coordinates. These sets can be described
as all the integer points between some upper and lower bounds, minus a
finite number of points and curves. The set of f-vectors of simplicial
polytopes is given by the g-theorem, its structure is in general not so
simple. We define "simple descriptions" in the following way: We call a
set of integer points a "semi-algebraic set of integer points" if it is
the set of all integer points of some semi-algebraic set. It turns out
that the 2-dimensional coordinate projections of the set of f-vectors of
4-polytopes are all semi-algebraic, with the exception of the pair
(f_1,f_2). Additionally, the set of f-vectors of simplicial d-polytopes is
semi-algebraic for d<6, but not for d=6. This also implies that the set of
all f-vectors of 6-polytopes is not semi-algebraic.
(Work in progress, joint work with Günter M. Ziegler)