This is a web-page of the course on High Dimensional Expanders organized by Prof. Gil Kalai and Prof. Alex Lubotzky at The Hebrew University of Jerusalem in the 2013/14 academic year.

This course is the second in a series. The first one was carried out under the same title in the 2011/12 academic year. Both courses followed one goal -- studying high dimensional expanders -- and have a non-empty intersection in the material. Nevertheless these courses differ from each other. Typed notes for the 2011/12 may be found here: Semester One Semester Two

Videos are filmed by Adam Nishri. Hand-written notes are by Konstantin Golubev.

(notes)

*A. Lubotzky, October 15, 2013.*

Boros-Furedi theorem. Barany's theorem. Gromov's theorem on the topological overlap property of the full d-dimensional complex on n vertices. Linial-Meshulam model of random complexes.

(notes)

*U. First, October 22, 2013.*

Definition of expander graphs, Laplacian and spectral gap. Cheeger's inequality.

(notes)

*A. Lubotzky, October 29, 2013.*

Mixing lemma for bipartite biregular graphs.

(notes)

*A. Lubotzky, November 5, 2013.*

Simplicial complexes: a definition and basic notions. A coloring of a simplicial complex, spherical buildings over finite fields.

(notes)

*A. Lubotzky, November 12, 2013.*

Spherical buildings. View on the Ramanujan graphs from their universal covers.

(notes)

*A. Lubotzky, November 19, 2013.*

The discrepancy of a simplicial complexes and an upper bound for it.

(notes)

*U. First, November 26, 2013.*

Local fields. The affine building associated to PGL_d over a local field.

(notes)

*A. Lubotzky, December 3, 2013.*

The type function of a building. An action of (P)GL on it. The link of a vertex.

(notes)

*A. Lubotzky, December 10, 2013.*

The link of a vertex of a building. 1-dimensional building is the infinite regular tree. Preparations for the Ramanujan graphs.

(notes)

*A. Lubotzky, December 24, 2013.*

More on Ramanujan graphs: constructing a lattice.

(notes)

*A. Lubotzky, December 31, 2013.*

Arithmetic groups. Strong approximation theorem. The girth of the Ramanujan graphs.

(notes)

*R. Rosenthal, January 7, 2014.*

Cheeger's inequality for simplicial complexes, following works of O. Parzanchevski, R. Rosenthal, R. Tassler and of A. Gundert, M. Szedlak.

(notes)

*A. Lubotzky, January 14, 2014.*

A lower bound on the chromatic number of non-bipartite Ramanujan graphs. Spectra of the Ramanujan complexes.

(notes)

*G. Kalai, February 18, 2014.*

There is no video recording of this lecture.

Radon's theorem. Helly's theorem. Caratheodory's theorem.

(notes)

*G. Kalai, March 11, 2014.*

Homology of a simplicial complex. Contractable complexes.

(notes)

*A. Lubotzky, May 13, 2014.*

An expansion of simplicial complexes following Gromov and Linial-Meshulam. A proof of the complete complex being an expander.

(notes: written, typed)

*A. Lubotzky, May 20, 2014.*

Linearity is testable. Cocycle tester. Tensor power question.