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

基本群的表示以及与变形理论、几何和可积系统的联系

• 批准号: 9803520
• 负责人: John Millson
• 金额: $ 8万
• 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: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2002-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803520&HistoricalAwards=false
• 关键词: Representations; Fundamental; Group; Connections; Deformation

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. ***

小行星9803520 米尔森将继续探索空间之间的连接有限生成群的陈述成李群，链接模型空间的常曲率，安排子空间，和可积的哈密顿系统及其量化。 这项工作的大部分将由犹他州大学的迈克尔·卡波维奇完成。 Millson和Kapovich利用上述关系构造了非光滑复代数簇的基本群的Artin群。 这项工作刚刚被接受的公共数学。IHES。 米尔森还与赫尔曼Flaschka亚利桑那大学的建设可积系统的辛concurents（由伴随表示）产品的轨道在简单的李代数。 他们最近发现了这种系统的最小轨道。 一个关键的作用是发挥阿龙szajn-Weinstein公式的微扰理论。 他们的最终目标是给新的洞察工作Lusztig和其他人分解张量积的不可约表示。 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边长，那么他们是全等的。 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一个最喜欢的主题，十九世纪数学，研究平面连杆机构（系统的杆和铰链）。 在19世纪，这样的研究具有巨大的实际意义--问题是通过连杆将直线运动（活塞杆）转换为圆周运动（车轮的转动）。 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，有一个平面连杆，其位形空间与M的若干副本的不交并同构。 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。 这个理论是非常丰富的，与辛几何，可积哈密顿系统，和表示论。 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，泊松李群和量子群有关。 ***