Millson and his collaborators will explore the geometric properties of configuration spaces of n-gon linkages in Lie algebras, symmetric spaces, compact Lie groups and Euclidean buildings beginning with the question of deciding when these moduli spaces are nonempty (finding the generalized triangle inequalities). In addition they will study the finer structure of these moduli spaces. In particular, they will study commutative and noncommutative Hamiltonian systems of differential equations on these spaces and use the quantizations of these systems to obtain results about the representation theory of compact Lie groups and of the Artin groups of Lie type. Millson and B.Leeb have obtained necessary and sufficient conditions for the moduli spaces of n-gon linkages in Lie algebras to be nonempty generalizing well-known results of Klyachko for sl(n,C). Millson, M.Kapovich and B.Leeb have found necessary and sufficient conditions for the moduli spaces of n-gon linkages in Euclidean buildings and symmetric spaces to be nonempty. In some cases, Millson and H.Flaschka have constructed (commutative) integrable Hamiltonian systems on these spaces but the flows are not periodic. For applications to the representation theory of compact Lie groups it is critical to find the associated "action variables," i.e. find Hamiltonians with periodic flows which are functions of the original Poisson-commuting Hamiltonians. Millson and Toledano-Laredo have constructed representations of Artin groups of Lie type by quantizing certain noncommutative Hamiltonian systems. They hope to construct the trigonometric analogues of their quantum systems. The results just mentioned may be found at www.math.umd.edu/~jjm.
Millson's work begins with one of the first theorems of high-school geometry - the theorem that if two triangles have the same set of side lengths then they are congruent. The analogue for quadrilaterals is clearly false: one can change a square into a rhombus without changing the side lengths. So one is led to try to parametrize the set of all planar n-gons with the same side lengths. From there one is led to a favorite theme of nineteenth century mathematics, the study of planar linkages (systems of rods and hinges). In the nineteenth century such a study was of immense practical significance - the problem was to convert linear motion (of a piston rod) to circular motion (turn a wheel) by a linkage. The problem was solved by a French naval officer, Peaucellier. It turns out that from the modern point of view the nineteenth century work is insufficiently precise. Millson and Kapovich have corrected the errors and written up a proof of a result (often attributed to Thurston) that, given any smooth manifold M, there is a planar linkage whose configuration space is diffeomorphic to a disjoint union of a number of copies of M. This result will appear in the journal "Topology". The above work on planar linkages led to a study of n-gon linkages in space. This theory is enormously richer, connecting with symplectic geometry, integrable Hamiltonian systems and representation theory. The analogous theory in spherical and hyperbolic three-space and their generalizations (compact Lie groups and the symmetric spaces of their complexifications) appears to connect up with some of the newest objects in geometry and algebra, for example Poisson Lie groups and quantum groups.
