Let X be a proper scheme over a field K. There are two ways of organizing the points of X into equivalence classes using rational curves. One is the notion of R-equivalence introduced by [Manin72]. Two points x1, x2 ∈ X(K) are called directly R-equivalent if there is a morphism p : P → X with p(0:1) = x1 and p(1:0) = x2. This generates an equivalence relation called R-equivalence. The set of R-...

We describe and prove uniqueness of a natural homomorphism (called the Orchard morphism) between some groups associated to finite sets.

208 NOTICES OF THE AMS VOLUME 51, NUMBER 2 As far as I know, the notion of a quasi-morphism does not have much to do with category theory. This very natural idea underlies several interesting recent developments at the crossroads of algebra, topology, geometry, and dynamics. Other, perhaps better, names currently in use for the same concept are quasi-homomorphism and pseudo-character. Let G be ...