In this colloquium style talk, I will first give a quick introduction to etale cohomology and Tate conjecture. Then I will explain my joint work with Yichao Tian and Xinwen Zhu on constructing cycles on the special fibers of Shimura varieties. The talk should be accessible to students who are reasonably familiar with the language of algebraic geometry.

Xiao Liang is currently a professor at BICMR at PKU. He graduated from MIT in 2009 and held positions at University of Chicago (postdoc), University of California at Irvine (tenure-track) and University of Connecticut (tenured). Liang Xiao's has broad interest in number theory and arithmetic geometry, including geometry of Shimura varieties and p-adic aspects of Langlands program.