Anders Björner
Department of Mathematics
Royal Institute of Technology
S-10044 Stockholm, Sweden
e-mail: bjorner@math.kth.se
Consider the family 
of all graphs with nodes
having a certain property P. If the
property P is monotone (meaning stable under removal of
edges) then can be thought of as an
abstract simplicial complex. Namely, the vertex set of
consists of all 
pairs
of nodes, and a collection of such pairs form a
simplex if the corresponding edges determine a graph with
property P.
It turns out that several graph properties P exhibit
interesting topological structure via the complex
, and that questions about such complexes arise
in several mathematical contexts. In the lecture I will
give a survey of results of this type from the last few years.
The areas where
questions about the topology of such graph properties arise
include complexity theory, reliability theory, knot
theory, commutative algebra and combinatorial geometry.
Also, monotone properties of directed graphs and of hypergraphs will be discussed. No previous knowledge of the area will be assumed.