Abstract: In this mini-series, I am reporting on the recent joint work with Xinwen Zhu. I hope to cover the following content:
A quick introduction to the geometric Satake theory
Local Shtukas and spectral operators (which extends the geometric Satake theory in some sense)
Application to the geometry of Affine Deligne-Lusztig varieties (describing some of the irreducible components)
Cycles on the special fiber of Shimura varieties and relation to Tate conjecture