CAREER: Higher Teichmuller Theory

AbstractAward：DMS-0846408首席研究员：安娜温哈德拟议项目的目标是进一步发展新兴领域的高等泰希穆勒理论。高阶Teichmueller空间是曲面的基本群到半单李群的各种表示的连通分支，它们与Teichmueller空间具有相同的基本性质。经典Teichmueller空间由于其丰富的结构和它是Riemann曲面模空间的光滑复盖这一事实，在数学的各个领域中起着重要的作用. 对于更高的Teichmueller空间，许多潜在的有趣的结构尚未被发现，Riemann曲面与模空间的关系仍需澄清.建议的研究重点是识别几何对象（类似于双曲曲面的情况下，经典Teichmueller空间），这是参数化的higherTeichmueller空间。这些对象将被用来定义高阶Teichmueller空间上的代数、几何和动力学结构，我们期望它们将有助于构造一个从高阶Teichmueller空间到经典Teichmueller空间的自然映射。因为它参数化所有的表面，Teichmueller空间是一个基本的数学对象，它也在理论物理中起着重要的作用。Teichmueller空间中的每一点都对应于欧氏平面、二维球面或在大多数情况下双曲平面被一个多边形及其在随后的侧面反射下的图像的平铺。(Nice受到这种瓷砖启发的图片出现在M.C.的工作）。曲面的组合类型由多边形的顶点数和胶合数据决定，而曲面的几何形状则取决于多边形的边长和顶点处的夹角。Teichmueller空间是一个模空间的例子，它具有丰富的对称性（多边形的反射）。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。高阶Teichmueller空间是在高维空间中具有更复杂模式的模空间。然而，这些模式的对称群与经典Teichmueller空间相关的平铺群相同。因此，人们期望更高的Teichmueller空间来参数化几何对象，这些对象本身不是曲面，但与曲面密切相关。该项目提出了各种活动，让研究生参与研究和教学，例如一年一次的数学务虚会，让研究生在一个小组中学习、探索和提出一个数学主题，以及小型研讨会和迷你课程，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.