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.
