Relative Noether inequality and geography of irregular varieties

In this talk, I will introduce a relative version of the classical Noether inequality on surfaces of general type. Some applications of this inequality will be discussed, such as a new proof of Cornalba-Harris-Xiao inequality and some results for surfaces in positive characteristic. I will also talk about the generalization of this inequality for higher dimensional varieties, which leads to some new geographical inequalities for irregular varieties. Part of the work is joint with Xinyi Yuan.