Cohomology of Shimura varieties
Shimura varieties are a certain class of algebraic varieties over number fields with lots of symmetries, introduced by Shimura and Deligne nearly half a century ago. They have been playing a central role in number theory and other areas. Langlands proposed a program to compute the L-functions and cohomology of Shimura varieites in 1970s; this was refined by Langlands-Rapoport and Kottwitz in 1980s. I will review some old and recent results in this direction.