New development of the research of Lefschetz fibrations by Teichmuller theory

滋贺得到了全纯Riemann曲面族的有效界，并证明了Teichmuller曲线的刚性和有限性.滋贺在与Miyachi的合作中研究了Lefschetz纤维化的完整性和斜率不等式的关系，Miyachi阐明了Teichmuller距离的边界行为。Mark和J.货车Horn-Morries，Endo发现了合理排污的单值排列，并发展了一些用有限图研究Lefschetz纤维化的方法。

Shiga obtained effective bounds of holomorphic families of Riemann surfaces, and also showed the rigidity and the finiteness of Teichmuller curves. In a joint work with Miyachi, Shiga studied relations of the holonomy and the slope inequality of Lefschetz fibrations.Miyachi clarified the boundary behavior of the Teichmuller distance.In a joint work with T. Mark and J. Van Horn-Morries, Endo found monodromy permutations for rational blow-downs and developed some method to study Lefschetz fibrations by using finite graphs.