Configuration Spaces on n-gon Linkages

n 边形链接上的配置空间

• 批准号: 0104006
• 负责人: John Millson
• 金额: $ 17.42万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104006&HistoricalAwards=false
• 关键词: Configuration; Spaces; gon; Linkages

DMS-0104006John J. MillsonMillson and his collaborators will explore the geometric properties of configuration spaces of n-gon linkages in Lie algebras, symmetric spaces, compact Lie groups and Euclidean buildings beginning with the question of deciding when these moduli spaces are nonempty (finding the generalizedtriangle inequalities). In addition they will study the finer structure of these moduli spaces. In particular, they will study commutative and noncommutative Hamiltonian systems of differential equations on these spaces and use the quantizations of these systems to obtain results about the representation theory of compact Lie groups and of the Artin groups of Lie type. Millson and B.Leeb have obtained necessary and sufficient conditions for the moduli spaces of n-gon linkages in Lie algebras to be nonempty generalizing well-known results of Klyachko for sl(n,C). Millson, M.Kapovich and B.Leeb have found necessary and sufficient conditions for the moduli spaces of n-gon linkages in Euclidean buildings and symmetric spaces to be nonempty. In some cases, Millson and H.Flaschka have constructed (commutative) integrable Hamiltonian systems on these spaces but the flows are not periodic. For applications to the representation theory of compact Lie groups it is critical to find the associated "action variables," i.e. find Hamiltonians with periodic flows which are functions of the original Poisson-commuting Hamiltonians. Millson and Toledano-Laredo have constructed representations of Artin groups of Lie type by quantizing certain noncommutative Hamiltonian systems. They hope to construct the trigonometric analogues of their quantum systems. The results just mentioned may be found at http://www.math.umd.edu/~jjm. 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 sidelengths. 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 ofrods 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 (turn a wheel) by a linkage. The problem wassolved 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. This result will appear inthe journal "Topology". 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 and their generalizations (compact Lie groups and the symmetric spaces of their complexifications) appears to connect up with some of the newest objects in geometry and algebra, for example Poisson Lie groups and quantum groups.

DMS-0104006 John J. Millson Millson和他的合作者将探索李代数、对称空间、紧李群和欧几里得建筑中n边形连杆的构型空间的几何性质，从确定这些模空间何时非空的问题开始（找到广义三角形不等式）。此外，他们将研究这些模空间的更精细结构。特别是，他们将研究交换和非交换的哈密顿系统的微分方程在这些空间和使用这些系统的量子化，以获得有关的代表性理论的紧凑李群和阿廷集团的李型。Millson和B.Leeb得到了李代数中n边形连杆的模空间为非空的充要条件，推广了Klyachko关于sl（n，C）的著名结果. Millson，M.Kapovich和B.Leeb已经找到了欧氏建筑物和对称空间中n边形连杆的模空间非空的充要条件。在某些情况下，Millson和H.Flaschka已经在这些空间上构造了（交换的）可积Hamilton系统，但流不是周期的。对于紧致李群表示论的应用，关键是找到相关的“作用变量”，即找到具有周期流的哈密顿算子，这些哈密顿算子是原始泊松交换哈密顿算子的函数。Millson和Toledano-Laredo通过量子化某些非对易Hamilton系统，构造了李型Artin群的表示。他们希望构造出量子系统的三角类似物。刚才提到的结果可以在http://www.math.umd.edu/~jjm上找到。 米尔森的工作开始于高中几何的第一个定理之一-定理，如果两个三角形有相同的一套边长，然后他们是全等的。 四边形的类比显然是错误的：一个人可以把一个正方形变成一个菱形而不改变边长。因此，人们试图参数化所有具有相同边长的平面n维的集合。从那里人们被引导到十九世纪数学的一个最喜欢的主题，平面连杆机构（杆和铰链系统）的研究。 在19世纪这样的研究是巨大的实际意义-问题是转换直线运动（活塞杆）的圆周运动（转一个轮子）的联系。法国海军军官波塞利耶解决了这个问题。 事实证明，从现代的观点来看，世纪的作品不够精确。 米尔森和卡波维奇已经纠正了错误，并写了一个结果的证明（通常归因于瑟斯顿），即给定任何光滑流形M，存在一个平面连杆，其位形空间是M的若干副本的不交并集的同构。 这一结果将发表在“拓扑学”杂志上。 平面连杆机构的上述工作导致了空间n边形连杆机构的研究。 这个理论非常丰富，与辛几何，可积哈密顿系统和表示论有关。球面和双曲三空间中的类似理论及其推广（紧李群及其复化的对称空间）似乎与几何和代数中的一些最新对象有关，例如泊松李群和量子群。