本项目开展拟共形映射、Teichmüller空间、渐进Teichmüller空间几何性质研究。我们将推进拟共形映射极值性研究，并在此基础上对Teichmüller空间几何进行更深入的探讨，特别地，我们将在有限维Teichmüller空间几何里引入角度的概念，并利用它来研究Teichmüller空间的凸性和曲率性质以及测地盘的属性，这些研究将有助于更好地理解有限维Teichmüller空间。同时我们拟将Teichmüller空间的一些理论成果推广到渐进Teichmüller空间上，对两个空间上的各种联系进行考察，并力图完全解决渐进Teichmüller空间中的测地线不唯一性猜测。我们还将把Teichmüller空间理论与调和映射理论结合起来，以图更深入的了解两者间紧密的联系。我们希望通过对这些重要问题的研究来丰富与发展Teichmüller空间理论，并推动国内在这一方面研究的发展。

This project aims at studying quasiconformal mappings and the geometry in Teichmüller spaces and asymptotic Teichmüller spaces. We will promote the study on the extremality of quasiconformal mappings, on the basis of which we can explore the geometry in Teichmüller spaces in deeper direction. In particular, we will introduce angle in a finite-dimensional Teichmüller space and use it to characterize the convexity, the curvature property and geodesic disks in a new sight. Such new consideration will help us understand a finite-dimensional Teichmüller space to a new extent. Meanwhile, we will try to extend some classical results in Teichmüller spaces to the asymptotic Teichmüller spaces. We will consider the relations between these spaces and try to solve the non-uniqueness conjecture on geodesics in the asymptotic Teichmüller spaces. We also study the relations between Teichmüller spaces and harmonic mappings. We believe that our project will enrich and develop the theory of Teichmüller spaces and will strengthen the study in the field in China.