平移曲面是当前非常活跃的研究方向，吸引着 C.McMullen、M.Mirzakhani、M.Kontsevich、A.Avila、A.Okounkov、J.-C.Yoccoz 等菲尔兹奖获得者以及 A.Eskin、A.Zorich、W.Veech 等数学家的研究兴趣。它与 Teichmuller 理论、动力系统、代数几何等研究方向密切相关。近年来，平移曲面的组合结构备受关注，如马鞍线复形、圆柱复形、以及周期方向复形。本项目将主要研究圆柱复形，具体包括： (1) 平移曲面上圆柱复形的连通性问题, 包括连通分支的数目与直径，以及连通的充分必要条件；(2) 圆柱复形的刚性问题，也即圆柱复形间的同构与平移曲面间的仿射同胚这两者之间的关系。这些研究将有助于我们更加深入地认识和运用平移曲面。

Translation surface is today one of the most active research areas in mathematics, which is closely related to several other fields, including Teichmuller theory, dynamical systems, algebraic geometry. Recently, people in this filed are getting interested in the combinatorial structures related to translation surfaces, such as the saddle connection complex, the cylinder complex, and the periodic direction complex. This program will focus on the cylinder complex. More precisely, we will consider: (1) the connectivity problem, including the number and diameter of connected components, and the sufficient and necessary condition for the cylinder complex being connected, (2) the rigidity problem, i.e. the relationship between simplicial isomorphisms among cylinder complexes and affine homeomorphisms among translation surfaces. This program will improve the developments and applications of translation surfaces.