Topological robotics: motion planning in projective spaces
  We study an elementary problem of topological robotics: rotation of a line, which is fixed by a revolving joint at a base point: one wants to bring the line from its initial position to a final position by a continuous motion in the space. The final goal is to construct an algorithm which will perform this task once the initial and final positions are given.   Any such motion planning algorithm will have instabilities, which are caused by topological reasons. A general approach to study instabilities of robot motion was suggested recently by the first named author. With any path-connected topological space X one associates a number TC(X), called the topological complexity of X. This number is of fundamental importance for the motion planning problem: TC(X) determines character of instabilities which have all motion planning algorithms in X.   In the present paper we study the topological complexity of real projective spaces. In particular we compute TC(RP^n) for all n<24. Our main result is that (for n distinct from 1, 3, 7) the problem of calculating of TC(RP^n) is equivalent to finding the smallest k such that RP^n can be immersed into the Euclidean space R^{k-1}. 
Michael Farber, Serge Tabachnikov and Sergey Yuzvinsky