Graduiertenkolleg: Methods for Discrete Structures

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Monday Lecture and Colloquium


Monday, June 19, 2017

Freie Universität Berlin
Takustr. 9
14195 Berlin
room 005



Lecture - 14:15

Rainer Sinn - Georgia Institute of Technology, Atlanta

Positive Semidefinite Matrix Completion

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

Hannah Sjöberg - Freie Universität Berlin

Semi-algebraic sets of f-vectors

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)



Letzte Aktualisierung: 13.06.2017