Configuration Spaces on n-gon Linkages

John J.Millson Millson和他的合作者将从判定这些模空间何时非空(寻找广义三角不等式)的问题入手，探索李代数、对称空间、紧李群和欧几里得建筑中n-方形连杆的配置空间的几何性质。此外，他们还将研究这些模空间的更精细的结构。特别是，他们将研究这些空间上的交换和非交换的哈密顿微分方程组，并利用这些系统的量子化来获得关于紧李群和李型Artin群的表示理论的结果。Millson和B.Leeb得到了李代数中n元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发现了欧几里德建筑和对称空间中n-角连杆的模空间非空的充要条件。在某些情况下，Millson和H.Flaschka在这些空间上构造了(可交换)可积的哈密顿系统，但流不是周期的。对于紧李群表示理论的应用，关键是找到相关的“作用变量”，即找到具有周期流的哈密顿量，它是原始泊松交换哈密顿量的函数。Millson和Toledano-Laredo通过量子化某些非对易的哈密顿系统，构造了李型Artin群的表示。他们希望构建他们量子系统的三角类比。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始于高中几何中最早的定理之一--如果两个三角形的边长相同，则它们是全等的。四边形的类比显然是错误的：人们可以把正方形变成菱形，而不改变边长。因此，人们试图将所有具有相同边长的平面n边形的集合参数化。从那里可以引出19世纪数学中最受欢迎的一个主题，平面连杆机构(柔性杆和铰链系统)的研究。在十九世纪，这样的研究具有巨大的实际意义--问题是通过连杆将(活塞杆的)直线运动转换为圆周运动(转动车轮)。这个问题是由法国海军军官皮切利耶解决的。事实证明，从现代的观点来看，19世纪的工作是不够精确的。Millson和Kapovich已经纠正了错误，并证明了一个结果(通常归因于瑟斯顿)，即给定任何光滑流形M，都存在一个平面连杆机构，其配置空间与M的若干副本的不相交并并不同胚。这一结果将发表在《拓扑学》杂志上。上述关于平面连杆机构的工作导致了对空间中n边形连杆机构的研究。这一理论非常丰富，与辛几何、可积哈密顿系统和表示理论联系在一起。球面和双曲三维空间及其推广(紧李群及其复化的对称空间)中的类似理论似乎与几何和代数中的一些最新对象联系在一起，例如泊松李群和量子群。

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.