Title: The complexity of counting homomorphisms

Abstract:
The complexity of counting homomorphisms from an input graph G to
a fixed graph H was established by Dyer and Greenhill (2000), and
generalised to partition functions by Bulatov and Grohe (2005).
These results give a dichotomy between problems which are in polynomial
time and problems which are hard for the counting class #P.
These problems are closely related to the study of the complexity 
of constraint satisfaction problems. This has revealed fundamental 
connections with universal algebra. We will survey recent work
and describe a new dichotomy theorem for uniform hypergraphs 
with edge size r at least 3. This gives an algebraic characterisation 
which is not obvious in the graph case, r=2.