Representations of the Fundamental Group and Connections with Deformation Theory, Geometry and Integrable Systems

9803520米尔森·米尔森将继续探索有限生成群到李群的表示空间、常曲率模型空间中的联系、子空间的排列以及可积哈密尔顿系统及其量子化之间的联系。这项工作的大部分将由犹他大学的迈克尔·卡波维奇完成。Millson和Kapovich利用上述关系构造了不是光滑复代数簇的基本群的Artin群。这项工作刚刚被公共部门接受。数学课。他就是。米尔森还与亚利桑那大学的赫尔曼·弗拉施卡合作，在单李代数中轨道乘积的辛商(通过伴随表示)上构造可积系统。他们最近发现了这样的最小轨道系统。微扰理论中的Aronszajn-Weinstein公式起着至关重要的作用。他们的最终目标是为Lusztig和其他人在分解不可约表示的张量积方面的工作提供新的见解。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：人们可以在不改变边长的情况下将正方形变成菱形。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(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(车轮的转动)。法国海军军官皮切利耶解决了这个问题。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纠正了错误，并给出了一个结果的证明(通常归因于瑟斯顿)：给定任意光滑流形M，存在一个平面连杆机构，它的构形空间与M的若干个副本的不相交并是微分同胚的。上述关于平面连杆机构的工作引出了对空间中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中的类似理论似乎与几何和代数中的一些最新对象、泊松李群和量子群联系在一起。***

9803520 Millson Millson will continue to explore connections among spaces of representations of finitely-generated groups into Lie groups, linkages in model spaces of constant curvature, arrangements of subspaces, and integrable Hamiltonian systems and their quantizations. Much of this work will be done with Michael Kapovich of the University of Utah. Millson and Kapovich used the above relations to construct Artin groups that were not the fundamental groups of smooth complex algebraic varieties. This work has just been accepted by the Publ. Math. IHES. Millson is also working with Hermann Flaschka of the University of Arizona on constructing integrable systems on symplectic quotients (by the adjoint representation) of products of orbits in simple Lie algebras. They have recently found such systems for minimal orbits. A critical role is played by the Aronszajn-Weinstein formula of perturbation theory. Their end goal is to give new insight into the work of Lusztig and others on decomposing tensor products of irreducible representations. 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 side lengths. 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 of rods 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 (turning of a wheel) by a linkage. The problem was solved 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. 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 appears to connect up with some of the newest objects in geometry and algebra, Poisson Lie groups and quantum groups. ***