CAREER: Higher Teichmuller Theory

项目编号：dms -0846408项目负责人：Anna wienhard项目目标是进一步发展高等特希穆勒理论这一新兴领域。更高的teichmueller空间是曲面的基群的各种表示的连接组成的半单李群，它们与teichmueller空间共享基本性质。经典的Teichmueller空间以其丰富的结构和对黎曼曲面的模空间的光滑覆盖在数学的各个领域中发挥着重要的作用。对于更高的Teichmueller空间，许多潜在的有趣结构尚未被发现，并且与黎曼曲面的模空间的关系仍然需要澄清。提出的研究重点是识别由更高的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，这些对象将用于定义更高的Teichmueller空间上的代数、几何和动态结构，我们期望它们将有助于构建从更高的Teichmueller空间到经典的Teichmueller空间的自然映射。Teichmueller空间是所有具有固定组合型的几何曲面的空间。由于Teichmueller空间参数化了所有曲面，因此它是一个基本的数学对象，在理论物理中也起着重要的作用。Teichmueller空间中的每个点都对应于欧几里得平面，二维球面或大多数情况下的双曲面的一个多边形及其在随后的侧面反射下的图像的平铺。(《m.c.》中出现了受这种瓷砖启发的漂亮图片。埃舍尔的作品)。表面的组合类型由多边形顶点的数量和粘合数据决定，但表面的几何形状取决于边的长度和多边形顶点的角度。Teichmueller空间是具有丰富对称性（多边形侧面反射）的模空间（平铺）的例子。这种模空间可以通过模式的对称性所产生的群来有效地研究。高等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Teichmuellerspace相关的瓷砖的对称群是相同的。因此，人们期望更高的Teichmueller空间来参数化几何对象，这些对象本身不是表面，但与表面密切相关。提出的研究重点是识别这些几何物体并研究它们的精细结构。该项目提出了各种各样的活动，让研究生参与研究和教学，例如，每年一次的数学静修，让研究生在一个小组中学习、探索和展示一个数学主题，以及小型讲习班，让研究生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.