Mixing, coloring and expansion of Ramanujan complexes
Ramanujan graphs, constructed by Lubotzky, Phillips and Sarnak and known also as the LPS graphs, are certain quotients of the Bruhat-Tits building of PGL_2(Q_p). These graphs form a family of expander graphs, and serve as an explicit construction of graphs of high girth and large chromatic number. High dimensional counterparts of the LPS graphs are the Ramanujan Complexes, constructed by Lubotzky, Samuels and Vishne, as quotients of the Bruhat-Tits building of PGL_d over a non-archimedean field of finite characteristic. I'll talk about the mixing of these complexes, which implies that they have good expansion and large chromatic number.
If time permits, I'll talk about analogous results for general hypergraphs satisfying certain regularity condition.
Joint work with S.Evra, A.Lubotzky.