Character varieties of surfaces: classical and quantum aspects

该项目建议调查的字符品种的表面，从一个表面的基本群的李群同态组成。字符变体出现在数学和数学物理的许多领域。该项目结合了字符品种的几个方面，特别涉及几何，量子拓扑和动力系统。 其总体思路是利用多年来为某些低维群开发的各种技术和见解，并将其应用于更广泛的李群和问题。该项目沿着两个主题展开。第一个主题涉及经典几何，重点是李群是特殊线性群的情况。它建议分析相应的字符品种的所谓希钦组件。第二个主题是围绕着量化的性格品种。特别是，该项目的主要目标之一是分类考夫曼绞代数的所有表示。该项目的最后一部分是致力于3维的影响，这种分析表示的考夫曼绞代数，特别是应用体积猜想。数学和数学物理中的许多现象都涉及曲面上平坦丛的数学概念，或者等价地，从曲面的基本群到李群的同态。特别地，李群由2 × 2矩阵组成的情况现在相对较好地理解，因为它与（非欧几里德）双曲几何的关系。该项目的主旨是建立在这方面的专业知识，攻击更复杂的李群，并调查更复杂的现象。它借用了动力系统理论（以“混沌理论”的名义推广）和物理学中的量子场论的思想。

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.