报告题目：On Thurston's "geometric triangulation" conjecture
摘要：After the Hyperbolization Theorem, the existence of hyperbolic structure is clear. However, for a hyperbolic 3-manifold, the existence of a geometric triangulation is still open. My talk is along the direction of solving this conjecture. Using combinatorial Ricci flow methods, we show: Let M be a compact 3-manifold with boundary consisting of surfaces of genus at least 2. If M admits an ideal triangulation with valence at least 10 at all edges, then there exists a unique hyperbolic metric on M with totally geodesic boundary under which the ideal triangulation is geometric. This provides the first existence result of a geometric triangulation on such 3-manifolds, and shows a deep connection between the topology and the geometry of 3-manifolds. Moreover, the combinatorial Ricci flow provides an effective tool of finding geometric structures and geometric triangulations of 3-manifolds. The talk is based on joint work with Ke Feng and Bobo Hua.
报告人简介：葛化彬，中国人民大学数学学院教授，博士生导师，数学系系主任。主要研究方向为几何拓扑，推广了柯西刚性定理和Thurston圆堆积理论，部分解决Thurston的“几何理想剖分”猜想、完全解决Cheeger-Tian、林芳华的正则性猜想，相关论文分别发表在Geom. Topol., Geom. Funct. Anal., Amer. J.Math., Adv. Math.等著名数学期刊。