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From Ramanujan graphs to Ramanujan complexes
on Wednesday, January 28, 2015, at 3:00 PM in Science Center 507
Ramanujan graphs are optimal expanders (from spectral point of view). Explicit constructions of such graphs were given in the 80's as quotients of the Bruhat-Tits tree associated with GL(2) over alocal field F, by suitable congruence subgroups. The spectral bounds were proved using works of Hecke, Deligne and Drinfeld on the "Ramanujan conjecture" in the theory of automorphic forms. The work of Lafforgue, extending Drinfeld from GL(2) to GL(n), opened the door for the construction of Ramanujan complexes as quotients of Bruhat-Tits buildings. This gives finite simplical complexes which on one hand are "random like" and at the same time have strong symmetries. Recently various applications have been found in combinatorics, coding theory and in relation to Gromov's overlapping properties. We will describe these developments and some recent applications. In particular, we will present a joint work with Tali Kaufman and David Kazhdan in which these complexes are used to (partially) answer a question of Gromov.