Poisson geometry, riemannian geometry, and applications

最近，由于数学物理（弦和膜理论）以及变形理论的原因，人们越来越多地关注非结合代数结构，以及与它们一起不满足雅可比恒等式的括号运算。温斯坦运用他以前发展的courant代数和狄拉克结构理论来研究这些非结合结构的几何模型，推广了泊松结构。第二部分研究了哈密顿对称群动量映射的非凸性定理和适当李氏群线性化的相关问题。在泊松几何中寻找“最优”凸性定理涉及到用群作用来描述固有群拟的局域结构问题。适当群的分析引出黎曼几何中子流形的平均族问题。先前关于这个问题的结果正在得到扩展和改进，其目标之一是理解无参数子流形的无限维空间的度量几何。最后，讨论了在力学中的应用：群类在离散拉格朗日系统中的应用，广义泊松结构在非完整系统中的应用。温斯坦的研究涉及几何中的对称性，并将其应用于物理学。传统上，对称性在数学上被描述为群的运算，以及相应的李代数的无穷小运算。最近，由于数学物理（弦和膜理论）以及数学变形理论的原因，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，以及与之相关的不满足李代数概念所必需的雅可比恒等式的括号运算。对这些（以及更传统的）对称性的研究还涉及到“群形”的使用，即不同的群在对称的空间的不同部分上运行。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，其简单形式是“空间中两条邻近的非参数化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也有应用，但令人惊讶的是，它很难解决。最后，Weinstein在离散拉格朗日系统的分析中使用群类群来创建数值算法。

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.