Speaker : Professor Sheng Rao (Wuhan University) Power series proofs for local stabilities of $K\ddot{a}hler$ and balanced structures with mild $\partial \bar{\partial }$-lemma 2017-01-20 (Fri) 10:30 - Seminar Room 617, Institute of Mathematics (NTU Campus) By use of a natural map introduced recently by Quanting Zhao and the speaker from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira-Spencer's local stability theorem of $K\ddot{a}hler$ structures. We also obtain two new local stability theorems, one of balanced structures on an $n$-dimensional balanced manifold with the $\left ( n-1,n \right )$-th mild $\partial \bar{\partial }$-lemma by power series method and the other one on $p$-$K\ddot{a}hler$ structures with the deformation invariance of $\left ( p, p \right )$-Bott-Chern numbers. This talk is based on a recent work arXiv: 1609.05637v1 joint with Xueyuan Wan and Quanting Zhao.