High Dimensional Expanders and Ramanujan Complexes

High Dimensional Expanders and Ramanujan Complexes - Alex Lubotzky

Alex Lubotzky
Hebrew University
February 12, 2013

Expander graphs, in general, and Ramanujan graphs, in particular, have been objects of intensive research in the last four decades. Many application came out, initially to computer science and combinatorics and more recently also to pure mathematics (number theory, geometry, group theory ). In recent years, there has been an interest in generalizing this theory to higher dimensional simplical complexes. We plan to survey first the classical theory and then describe the more recent developments. Some directions of current research will be presented as well as suggestions for future research.