Title: Introduction to Taylor-Wiles-Kisin method and some applications Speaker: Zijian Yao (Harvard Univ.) Time: 2017-8-17, 8:30-12:00 Place: S817 Abstract: This will be mostly a survey talk on modularity patching arguments, and is divided into roughly three parts. In the first part I will survey motivations and background for (potential) modularity and modularity lifting theorems, and outline the more recent techniques (i.e.,refinements of Taylor-Wiles patching method due to Diamond, Kisin, etc.). In the second talk, I will introduce background on integral p-adic Hodge theory to control deformation rings at $p$, which is a crucial step in the patching method. In the last talk, I will give some applications of the patching method, including my work on an analog of Breuil-Mezard conjecture in the function field setup. If time permits, I will also mention the result of Buzzard-Taylor on classicality of weight 1 forms, and some improvements. The last part is work in progress joint with Chi-Yun Hsu. Attachment: