Representations of the Fundamental Group and Connections with Deformation Theory, Geometry and Integrable Systems

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9803520 Millson Millson will continue to explore connections among spaces of representations of finitely-generated groups into Lie groups, linkages in model spaces of constant curvature, arrangements of subspaces, and integrable Hamiltonian systems and their quantizations. Much of this work will be done with Michael Kapovich of the University of Utah. Millson and Kapovich used the above relations to construct Artin groups that were not the fundamental groups of smooth complex algebraic varieties. This work has just been accepted by the Publ. Math. IHES. Millson is also working with Hermann Flaschka of the University of Arizona on constructing integrable systems on symplectic quotients (by the adjoint representation) of products of orbits in simple Lie algebras. They have recently found such systems for minimal orbits. A critical role is played by the Aronszajn-Weinstein formula of perturbation theory. Their end goal is to give new insight into the work of Lusztig and others on decomposing tensor products of irreducible representations. 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 (turning of 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. 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 appears to connect up with some of the newest objects in geometry and algebra, Poisson Lie groups and quantum groups. ***