Alexander Lubotzky: From Ramanujan graphs to Ramanujan complexes (NTWS 056)
Channel: Number Theory Web Seminar
Date: 2023-08-06

Abstract: Ramanujan graphs  are k-regular graphs with all non trivial eigenvalues  bounded (in absolute value) by 2SR(k-1). They 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 a local field F, by the action of suitable congruence subgroups of arithmetic groups. The spectral bound was 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 the Bruhat-Tits buildings associated with GL(n) over F.  This way one gets  finite simplicial complexes which on one hand are "random like" and at the same time have strong symmetries. These seemingly contradicting properties make them very useful for constructions of various  external  objects. 
Recently various applications have been found in combinatorics, coding theory and in relation to Gromov's overlapping properties.  We will survey some of these applications. 

Link to slides: https://drive.google.com/file/d/1hW6ysJRfYj9T11toT5bAOrwGuT0SSkk-

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Thursday, December 3, 2020 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, December 4, 2020 (12am China Standard Time, 3am AEDT, 5am NZDT)