By degenerating a smooth curve to an curve with one node (irreducible or reducible), we establish two recurrence relations for the dimensions of spaces of generalized theta functions on moduli spaces of semi-stable parabolic bundles on smooth curves of genus $g$, which imply an explicit formula of dimension (Verlinde formula). There are two steps to establish such recurrence relations: (1) factorizations of generalized theta functions over nodal curves;(2) invariance of dimensions during degeneration, which are implied by vanishing theorem of cohomolgy on moduli spaces. The step (1) and step (2) for $g>2$ were done by myself around 2000. However vanishing theorem for $g<3 $ remains open.

Recently, we solve this characteristic zero problem by a mothed of characteristic p>0. Namely, we prove that moduli spaces of semi-stable parabolic bundles and generalized parabolic sheaves with fixed determinants are of globally Frobenius regular type, which imply the vanishing theorem for any genus.

孙笑涛，现任天津大学数学学院院长、教授。他主要从事代数几何的研究,2000年获国家杰出青年基金; 2012年获国家自然科学二等奖（唯一完成人），2013年获第十四届陈省身数学奖。
 
Xiaotao Sun is the professor and dean of School of Mathematics, Tianjin University. His research is mainly focusing on Algebraic geometry. He won the National Science Fund for Distinguished Young Scholars in 2000, the Second Prize of National Natural Science Award in 2012 and the Shiing-Shen Chern Mathematics Award in 2013.