Character varieties of surfaces: classical and quantum aspects

该项目建议研究曲面的特征变量，包括从曲面的基本群到李群的同态。字符变化出现在数学和数学物理的许多领域。该项目结合了特征变化的几个方面，涉及到特别是几何、量子拓扑和动力学系统。它的主线是利用多年来为某些低维群开发的各种技术和见解，并将这些应用于更广泛的Lie群和问题。该项目围绕两个主要主题进行阐述。第一个主题涉及经典几何，集中在李群是特殊线性群的情况下。它建议分析所谓的希钦成分对应的人物变体。第二个主题是人物变体的量化。特别是，该项目的主要目标之一是对考夫曼斜线代数的所有表示进行分类。该项目的最后部分致力于考夫曼斜线代数表示的这种分析的三维含义，特别是对体积猜想的应用。数学和数学物理中的许多现象都涉及到曲面上的平丛的数学概念，或者等价地，从曲面的基本群到李群的同态。特别是，李群由2乘2矩阵组成的情况现在被相对较好地理解，因为它与(非欧几里得)双曲几何的关系。该项目的主旨是建立在这些专业知识的基础上，以攻击更复杂的谎言团体，并调查更复杂的现象。它借鉴了动力系统理论(以“混沌理论”的名义推广)和物理学中的量子场论。

The Project proposes to investigate the character varieties of surfaces, consisting of homomorphisms from the fundamental group of a surface to a Lie group. Character varieties occur in many areas of mathematics and mathematical physics. The Project combines several aspects of character varieties, involving in particular geometry, quantum topology and dynamical systems. Its overarching thread is to capitalize on various techniques and insights that, over the years, have been developed for certain low-dimensional groups and to apply these to a wider range of Lie groups and problems. The Project is articulated along two main themes. The first theme involves classical geometry and is focused on the case where the Lie group is the special linear group. It proposes to analyze the so-called Hitchin component of the corresponding character variety. The second theme is centered on the quantization of character varieties. In particular, one of the main goals of the Project is to classify all representations of the Kauffman skein algebra. A final part of the Project is devoted to the 3-dimensional implications of this analysis of representations of the Kauffman skein algebra, and in particular to applications to the Volume Conjecture. Many phenomena in mathematics and mathematical physics involve the mathematical notion of flat bundle over a surface or, equivalently, of homomorphisms from the fundamental group of a surface to a Lie group. In particular, the case where the Lie group consist of 2-by-2 matrices is now relatively well understood, because of its relationship to (non-euclidean) hyperbolic geometry. The thrust of the Project is to build on this expertise to attack more complicated Lie groups and to investigate more complex phenomena. It borrows ideas from the theory of dynamical systems (popularized under the name of "chaos theory") and from quantum field theory in physics.