Optical mapping is a new technique to generate restriction maps of DNA easily and quickly. DNA restriction maps can be aligned by comparing corresponding restriction fragment lengths. To relate, organize, and analyse these maps it is necessary to rapidly compare maps. The issue of the statistical significance of approximately matching maps then becomes central, as in BLAST with sequence scoring. In this paper, we study the approximation to the distribution of counts of matched regions of specified length when comparing two DNA restriction maps. Distributional results are given to enable us to compute p-values and hence to determine whether or not the two restriction maps are related. The key tool used is the Chen-Stein method of Poisson approximation. Certain open problems are described.

We obtain a necessary condition for a cohomology class on a compact locally symmetric space S(Γ)=Γ\X (a quotient of a symmetric space X of the non-compact type by a cocompact arithmetic subgroup Γ of isometries of X) to restrict non-trivially to a compact locally symmetric subspace SH(Γ)=Δ\Y of Γ\X. The restriction is in a ‘virtual’ sense, i.e. it is the restriction of possibly a translate of the cohomology class under a Hecke correspondence. As a consequence we deduce that when X and Y are the unit balls in $\Bbb C$n and $\Bbb C$m, then low degree cohomology classes on the variety S(Γ) restrict non-trivially to the subvariety SH(Γ); this proves a conjecture of M. Harris and J-S. Li. We also deduce the non-vanishing of cup-products of cohomology classes for the variety S(Γ).