Conceptual questions of quantum gravity and mathematical structures in three-dimensional gravity

引力是四种基本相互作用中唯一一种至今无法构建量子理论的相互作用。这是由于现有的量化方法无法解决的概念问题的存在。这包括在不依赖于固定背景时空的情况下构建量子理论-在量子引力中，时间和空间本身被量子化-时间在量子理论中的作用，时间和空间是连续还是离散的问题以及低能量引力的出现。虽然四维引力的完整量子理论目前还遥不可及，但三维引力（一个空间，两个时间维度）可以作为一个玩具模型，它允许人们在完全量子化的理论中解决这些概念问题，并获得与更高维度相关的见解。此外，3D重力对数学非常重要。该理论将当前研究的许多领域联系在一起，导致该主题的重大进展。该研究项目旨在解决3d量子引力的几个重要概念问题。这些问题除了对量子引力的研究很重要之外，也与数学密切相关。特别是，他们涉及和相关的概念和方法，从理论的量子群，模空间的平坦连接，双曲几何和拓扑量子场论。

Gravity is the only one of the four fundamental interactions for which quantum theory could not be constructed so far. This is due to the presence of conceptual problems which cannot be addressed within existing quantisation approaches. This includes the construction of a quantum theory without relying on a fixed background spacetime - in quantum gravity it is time and space themselves that are to be quantised - the role of time in the quantum theory, the question if time and space are continuous or discrete and the emergence of gravity for low energies. While a full quantum theory of four-dimensional gravity is currently not within reach, 3d gravity (one space, two time dimensions) serves as a toy model which allows one to address these conceptual questions in a fully quantised theory and to gain insights relevant to higher dimensions. Moreover, 3d gravity is of high importance for mathematics. The theory ties together many areas of current research, which lead to major progress in the subject. The research project aims to address several important conceptual questions of 3d quantum gravity. Besides their importance to the research subject of quantum gravity, these questions are also highly relevant to mathematics. In particular, they involve and relate concepts and methods from the theory of quantum groups, moduli spaces of flat connections, hyperbolic geometry and topological quantum field theory.