#
Monday Lecture and Colloquium

**Monday, April 26, 2010 **

(FU) Freie Universität Berlin

Institut für Informatik

Takustr. 9,

14195 Berlin

room 005

** Lecture - 14:15**

###
Oleg Pikhurko, Carnegie Mellon University, Pittsburgh

### Hypergraph Turan Problem

*Abstract:*

The Turan function of a given k-graph (that is, a k-uniform set system) F
is the maximum number of edges in an F-free k-graph on n vertices. This
problem goes back to the fundamental paper of Turan from 1941 who solved
it for complete graphs. Unfortunately, when we move to k-graphs with k>2,
then very few non-trivial instances of the problem have been solved. In
particular, the problem of Turan from 1941 (also a $1000 question of
Erdos) to determine the Turan function of the tetrahedron is still open in
spite of decades of active attempts.
In this talk we will survey some results and methods on the hypergraph
Turan problem. In particular, we describe the recent progress on a
tetrahedrom-related problem obtained by Razborov using his flag algebra
technique as well as the corresponding exact result. Time permitting, we
discuss the stability approach of Simonovits that greatly helps in
converting asymptotic computations, such as those obtained via flag
algebras or (hyper)graph limits, into exact results.

**Colloquium - 16:00**

###
Benjamin Lorenz - Freie Universität Berlin

### Classification of Smooth Lattice Polytopes with few Lattice Points

*Abstract:*

Lattice polytopes appear in many different areas, for example in toric geometry
where an interesting object of study are projective toric varieties and
especially the smooth ones. This smoothness can again be found as an intrinsic
property of the polytope defining the variety.
Christian Haase et al have shown that there are only finitely many smooth
lattice polytopes for fixed dimension and an upper bound on the number of
lattice points in the polytope. Based on this proof I have developed an
algorithm that can generate the list of all these smooth lattice polytopes given
certain minimal smooth fans and an upper bound on the number of lattice points.
The talk will give an outline of the proof of the finiteness and then discuss
the classification algorithm.

Letzte Aktualisierung:
15.04.2010