Linkages and bending flows

The key mathematical notion in mechanics and robotics is the configuration space describing all possible positions of a given mechanical system. A typical example of such a system is the {\it linkage}, which is a mathematical idealisation of the systems like the robot arm. The complexity of such systems can be measured using the Betti numbers of the corresponding configuration space and studied within topological robotics. Even for linkages such complexity depends very substantially on the lengths of parts, so the general situation is still to be understood.

The aim of the project is to study the geometry of the configuration space Sigma_n of a linkage in R^3 with n joints using different idea going back to Kapovich and Millson, who have shown that Sigma_n admits integrable systems generated by bending flows. So Sigma_n can be represented (in many ways, depending on triangulation of n-gon) as a singular Lagrangian fibration, which can be described using combinatorial structure of the corresponding polytopes determined by the moment map. The study of the singularities of the corresponding moment maps should give an important information about global properties of Sigma_n.

The bending flows are the degenerate cases (corresponding to the vertices of Stasheff polytope) of the Gaudin integrable systems, for which the corresponding moduli space is Deligne-Mumford moduli space M_{0,n} of stable rational curves with n marked points (Aguirre et al). The quantum version of the bending flows is interesting by itself and closely connected to the representation theory of symmetric groups and double affine Hecke algebras.