Poisson geometry, riemannian geometry, and applications

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

DMS-0204100 Alan Weinstein Recently, for reasons arising from mathematical physics (string and membrane theory) as well as from deformation theory, increased attention is being paid to nonassociative algebraic structures and, along with them, bracket operations which do not satisfy the Jacobi identity. Weinstein applies his previously developed theory of Courant algebroids and Dirac structures to investigate geometric models for some of these nonassociative structures, generalizing Poisson structures. A second part of the research concerns convexity theorems for momentum mappings of hamiltonian symmetry groups and related questions about the linearization of proper Lie groupoids. The search for the ``optimal'' convexity theorem in Poisson geometry is related to the problem of describing the local structure of proper groupoids in terms of group actions. The analysis of proper groupoids leads in turn to the problem of averaging families of submanifolds in riemannian geometry. Previous results on this problem are being extended and improved, with one goal being an understanding of the metric geometry of infinite dimensional spaces of unparametrized submanifolds. Finally, applications to mechanics are be investigated: groupoids applied to discretized lagrangian systems, and generalized Poisson structures applied to nonholonomic systems. Weinstein's research concerns symmetry in geometry, with applications to physics. Traditionally, symmetry has been described mathematically in terms of the operation of groups, and the corresponding infinitesimal operation of Lie algebras. Recently, for reasons arising from mathematical physics (string and membrane theory) as well as from deformation theory in mathematics, increased attention is being paid to more general notions of symmetry, involving algebraic structures lacking the associativity of a group operation and, along with them, bracket operations which do not satisfy the Jacobi identity essential to the notion of Lie algebra. The study of these (and more traditional) symmetries also involves the use of ``groupoids,'' where different groups are operating on different parts of a space with symmetry. Weinstein's study of these symmetries involves a combination of algebra and differential geometry, as well as some very concrete geometric questions, a simple form of which is ``what is the average of two nearby unparametrized curves in space?'' This question is easy to pose and may have applications in computer graphics, but it is surprisingly difficult to solve. Finally, Weinstein is using groupoids in the analysis of discrete lagrangian systems used to create numerical algorithms.