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，人们越来越关注非结合代数结构，沿着它们的是不满足雅可比恒等式的括号运算。 温斯坦应用他以前开发的理论的柯朗代数体和狄拉克结构调查几何模型的一些这些nonassociative结构，generalizingPoisson结构。 第二部分研究了Hamilton群的动量映射的凸性定理和真李群胚线性化的相关问题。 在Poisson几何中寻找“最优”凸性定理与用群作用描述真群胚的局部结构问题有关。 对真群胚的分析又引出了黎曼几何中子流形族的平均问题。 以前关于这个问题的结果正在被推广和改进，一个目标是理解非参数化子流形的无限维空间的度量几何。 最后，研究了在力学中的应用：群胚应用于离散拉格朗日系统，广义泊松结构应用于非完整系统。 传统上，对称性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）以及数学中的变形理论的原因，人们越来越关注更一般的对称性概念，包括缺乏群运算结合性的代数结构，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（以及更传统的）对称性的研究还涉及到“群胚”的使用，即不同的群在具有对称性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，其中一个简单的形式是“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来，而且可能在计算机图形学中有应用，但解决起来却出奇地困难。 最后，温斯坦是使用groupoid在离散lagrangian系统的分析用于创建数值算法。

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.