Abstract: I will start with some introduction to Shimura varieties and their completed cohomology, and report on my joint work in progress with Lue Pan which shows that, in the rational p-adic completed cohomology of a general Shimura variety, "sufficiently regular" infinitesimal weights (whose meaning will be explained) can only show up in the middle degree.  I will give some examples and explain the main ingredients in our work, if time permits.

Bio-Sketch: Kai-Wen Lan received his Ph.D. in 2008 from Harvard University, under the supervision of Richard Taylor.  He spent four years as a Veblen Research Instructor at Princeton University and the Institute for Advanced Study, and then moved to the University of Minnesota, Twin Cities, in 2012, where he became a professor in 2020. His research interests lie in the relations among the algebraic, analytic, and geometric aspects of number theory.  He has made major contributions to the geometry of Shimura varieties, the construction of Galois representations for automorphic representations without polarization conditions, and p-adic Riemann--Hilbert correspondences. He received the Alfred P. Sloan Research Fellowship in 2014.