CAREER: Higher Teichmuller Theory

摘要奖：DMS-0846408首席研究员：安娜·维恩德提议的项目的目标是进一步发展新兴的高等泰希穆勒理论领域。高阶Teichmueller空间是曲面的基本群表示为半单李群的连通分支，其本质性质与Teichmueller空间相同。经典Teichmueller空间由于其丰富的结构和它是黎曼曲面的模空间的光滑覆盖而在数学的各个领域都扮演着重要的角色。对于高阶Teichmueller空间，许多潜在有趣的结构尚待发现，与Riemann曲面的模空间的关系也有待进一步研究。所提出的研究集中于识别被高阶TeichMueller空间参数化的几何对象(类似于经典TeichMueller空间中的双曲曲面)。然后，这些对象将被用来定义高阶TeichMueller空间上的几何、几何和动力结构，我们期望它们将有助于构造从高阶TeichMueller空间到经典TeichMueller空间的自然映射。TeichMueller空间是具有固定组合类型的所有几何曲面的空间。TeichMueller空间是一个基本的数学对象，因为它将所有的曲面都参数化，在理论物理中也扮演着重要的角色。每个Pointin Teichmueller空间对应于欧几里德平面、二维球面或在大多数情况下双曲平面由一个多边形及其在随后的侧面反射下的图像的平铺。(受到这种瓷砖启发的漂亮图片出现在M.C.埃舍尔的作品中。)曲面的组合类型由多边形顶点数和粘合数据决定，而曲面的几何形状取决于多边形边的长度和顶点的角度。Teichmueller空间是一个具有丰富对称性(在多边形边的反射)的模空间的例子。这种模空间可以通过由模式的对称性生成的群来有效地研究。高维Teichmueller空间是高维空间中具有更复杂模式的模空间。然而，这些图案的对称群与经典Teichmueller空间中的镶嵌的对称群相同。因此，人们期望更高的Teichmueller空间将不是曲面本身，但与曲面密切相关的几何对象参数化。这项拟议的研究集中于识别这些几何物体并研究它们的精细结构。该项目提出了各种活动，让毕业生参与到研究和教学中来，例如，研究生可以参加一年的数学静修，让他们以小组形式学习、探索和展示一个数学主题，以及带有迷你课程的小研讨会，让研究生深入了解现代研究合作的工作。

AbstractAward: DMS-0846408Principal Investigator: Anna WienhardThe goal of the proposed project is to further develop theemerging field of higher Teichmueller theory. Higher Teichmuellerspaces are connected components of the variety of representationsof the fundamental group of a surface into semisimple Lie groups,which share essential properties with Teichmuellerspace. Classical Teichmueller space plays an important role invarious fields of mathematics due to its rich structure and thefact that it is a smooth cover of the moduli space of Riemannsurfaces. For higher Teichmueller spaces many potentiallyinteresting structures are yet to be discovered and the relationto the moduli space of Riemann surfaces still needs to beclarified. The proposed research focusses on identifying thegeometric objects (similar to hyperbolic surfaces in the case ofclassical Teichmueller space) which are parametrized by higherTeichmueller spaces. These objects will then be used to definealgebraic, geometric and dynamical structures on higherTeichmueller spaces, and we expect that they will help toconstruct a natural map from higher Teichmueller spaces toclassical Teichmueller space.Teichmueller space is the space of all geometric surfaces with afixed combinatorial type. Because it parametrizes all surfaces,Teichmueller space is a fundamental mathematical object whichalso plays an important role in theoretical physics. Every pointin Teichmueller space corresponds to a tiling of the Euclideanplane, the two-dimensional sphere or in most cases the hyperbolicplane by one polygon and its images under subsequent reflectionsin the sides. (Nice pictures inspired by such tilings appear inM.C. Escher's work.) The combinatorial type of the surface isdetermined by the number of vertices of the polygon and thegluing data, but the geometry of the surface depends on thelength of the sides and the angles at the vertices of thepolygon. Teichmueller space is the example of a moduli space of apattern (the tiling) with rich symmetries (the reflections in theside of the polygon). Such moduli spaces can be efficientlystudied via the group generated by the symmetries of thepattern. Higher Teichmueller spaces are moduli spaces of morecomplicated patterns in higher dimensional spaces. However, thesymmetry group of these patterns is the same as the symmetrygroup of the tilings associated to classical Teichmuellerspace. Therefore one expects higher Teichmueller spaces toparametrize geometric objects which are not surfaces themselves,but closely related to surfaces. The proposed research focusseson identifying these geometric objects and studying their finerstructure.The project proposes various activities to involve graduatestudents in research and teaching, as for instance a yearlymathematical retreat for graduate students to study, explore andpresent a mathematical topic in a group, and small workshops withmini-courses which allow graduate students to get insight intothe working of modern research collaborations.