Poisson geometry, riemannian geometry, and applications

最近，由于数学物理(弦和膜理论)和形变理论的原因，非结合代数结构以及不满足雅可比恒等式的括号运算正受到越来越多的关注。Weinstein应用他以前发展的Courant代数体和Dirac结构理论来研究其中一些非结合结构的几何模型，推广了Poisson结构。第二部分研究了哈密顿对称群动量映射的凸性定理以及与真李群的线性化有关的问题。在Poisson几何中寻找“最优”凸性定理，涉及到用群作用来描述真群胚的局部结构的问题。对真群胚的分析又引出了黎曼几何中子流形的平均族问题。以前关于这个问题的结果正在被推广和改进，一个目的是理解非参数化子流形的无限维空间的度量几何。最后，我们研究了它在力学中的应用：应用于离散化拉格朗日系统的群胚和应用于非完整系统的广义泊松结构。温斯坦的研究涉及几何中的对称性，以及在物理学中的应用。传统上，对称是用群的运算和相应的李代数的无穷小运算来描述的。最近，由于数学物理(弦和膜理论)以及数学中的形变理论的原因，人们越来越关注更一般的对称概念，包括缺乏群运算的结合性的代数结构，以及不满足李代数概念所必需的Jacobi恒等式的括号运算。对这些(更传统的)对称性的研究也涉及到“群胚”的使用，即不同的群在空间的不同部分具有对称性。温斯坦对这些对称性的研究涉及到代数和微分几何的结合，以及一些非常具体的几何问题，其中一个简单的形式是“空间中相邻两条非参数曲线的平均值是多少？”这个问题很容易提出，可能在计算机图形学中有应用，但它的解决却出人意料地困难。最后，温斯坦正在使用群胚分析离散滞后系统，用于创建数值算法。

DMS-0204100Alan WeinsteinRecently, for reasons arising from mathematical physics (string andmembrane theory) as well as from deformation theory, increasedattention is being paid to nonassociative algebraic structures and,along with them, bracket operations which do not satisfy the Jacobiidentity. Weinstein applies his previously developed theory ofCourant algebroids and Dirac structures to investigate geometricmodels for some of these nonassociative structures, generalizingPoisson structures. A second part of the research concernsconvexity theorems for momentum mappings of hamiltonian symmetrygroups and related questions about the linearization of proper Liegroupoids. The search for the ``optimal'' convexity theorem inPoisson geometry is related to the problem of describing the localstructure of proper groupoids in terms of group actions. The analysisof proper groupoids leads in turn to the problem of averaging familiesof submanifolds in riemannian geometry. Previous results on thisproblem are being extended and improved, with one goal being anunderstanding of the metric geometry of infinite dimensional spaces ofunparametrized submanifolds. Finally, applications to mechanics arebe investigated: groupoids applied to discretized lagrangian systems,and generalized Poisson structures applied to nonholonomic systems.Weinstein's research concerns symmetry in geometry, with applicationsto physics. Traditionally, symmetry has been described mathematicallyin terms of the operation of groups, and the correspondinginfinitesimal operation of Lie algebras. Recently, for reasonsarising from mathematical physics (string and membrane theory) as wellas from deformation theory in mathematics, increased attention isbeing paid to more general notions of symmetry, involving algebraicstructures lacking the associativity of a group operation and, alongwith them, bracket operations which do not satisfy the Jacobi identityessential to the notion of Lie algebra. The study of these (and moretraditional) symmetries also involves the use of ``groupoids,'' wheredifferent groups are operating on different parts of a space withsymmetry. Weinstein's study of these symmetries involves acombination of algebra and differential geometry, as well as some veryconcrete geometric questions, a simple form of which is ``what is theaverage of two nearby unparametrized curves in space?'' This questionis easy to pose and may have applications in computergraphics, but it is surprisingly difficult to solve. Finally, Weinstein is using groupoids in the analysis of discrete lagrangiansystems used to create numerical algorithms.