#
Monday Lecture and Colloquium

**Monday, May 10, 2010**

Technische Universität Berlin

Fakultät II, Institut für Mathematik

Str. des 17. Juni 136

room MA 041

10623 Berlin

** Lecture - 14:15**

###
Mihyun Kang - Berlin

### Enumeration methods for planar graphs and beyond

*Abstract:*

In this talk we present recent results on the number of labelled planar graphs and corresponding enumeration methods. We
also discuss the power and limitations of each method. In addition we discuss typical properties of planar graphs.

Gimenez and Noy recently settled the long-standing open problem of determining the asymptotic number of labelled planar
graphs with $n$ vertices and $M$ edges, using the singularity analysis of generating functions. The number of edges
considered in the work of Gimenez and Noy is restricted to be of the form $an$ for $1 < a <3$. In this range of $M$, a
random planar graph $P(n,M)$ contains a large component of size $n-O(1)$, all planar graphs of finite size appear as
its subgraphs. Thus, it corresponds to late stages of the evolution of the Erdos-Renyi random graph $G(n,M)$. Kang and
Luczak studied the number of planar graphs and its typical properties in a more interesting range, when $M\le n$,
combining counting arguments with analytic and probabilistic techniques. Apart from these methods, the matrix integral
method from theoretical physics was successfully applied to planar graphs. Kang and Loebl showed that the enumeration of
graphs embeddable on a surface can be formulated as the Gaussian matrix integral of an ice-type partition function,
where the number of planar graphs appears as a leading term.

**Colloquium - 16:00**

###
Tom Banchoff - Brown University, Providence RI

###
Self-Linking, Inflections, and the Normal Euler Class for Smooth and Polyhedral Surfaces in Four-Space

*Abstract:*

The normal Euler class for a simplicial surface in four-space can be defined by analogy with the geometric description of this class for
a smooth surface. We present a combinatorial formula for the normal Euler class in terms of the self-linking numbers of spherical polygons and
inflection faces of polyhedra, related to a construction of Gromov, Lawson, and Thurston. The talk will feature computer graphics illustrations.

Letzte Aktualisierung:
30.04.2010