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.
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.
U. First, October 22, 2013.
Definition of expander graphs, Laplacian and spectral gap. Cheeger's inequality.
A. Lubotzky, October 29, 2013.
Mixing lemma for bipartite biregular graphs.
A. Lubotzky, November 5, 2013.
Simplicial complexes: a definition and basic notions. A coloring of a simplicial complex, spherical buildings over finite fields.
A. Lubotzky, November 12, 2013.
Spherical buildings. View on the Ramanujan graphs from their universal covers.
A. Lubotzky, November 19, 2013.
The discrepancy of a simplicial complexes and an upper bound for it.
U. First, November 26, 2013.
Local fields. The affine building associated to PGL_d over a local field.
A. Lubotzky, December 3, 2013.
The type function of a building. An action of (P)GL on it. The link of a vertex.
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.
A. Lubotzky, December 24, 2013.
More on Ramanujan graphs: constructing a lattice.
A. Lubotzky, December 31, 2013.
Arithmetic groups. Strong approximation theorem. The girth of the Ramanujan graphs.
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.
A. Lubotzky, January 14, 2014.
A lower bound on the chromatic number of non-bipartite Ramanujan graphs. Spectra of the Ramanujan complexes.
G. Kalai, February 18, 2014.
There is no video recording of this lecture.
Radon's theorem. Helly's theorem. Caratheodory's theorem.
G. Kalai, February 25, 2014.
Barany's theorem aka Colored Caratheodory's theorem. Colored Helly's theorem. Tverberg's theorem.
G. Kalai, March 4, 2014.
Around Pach's theorem.
G. Kalai, March 11, 2014.
Homology of a simplicial complex. Contractable complexes.
G. Kalai, March 18, 2014.
A definition of a collapse. Wagner's theorem. Fractional Helly's theorem.
G. Kalai, March 25, 2014.
Pach's theorem, cohomology and Laplacians.
G. Kalai, May 1, 2014.
Counting trees and Cauchy-Bint theorem.
A. Lubotzky, May 13, 2014.
An expansion of simplicial complexes following Gromov and Linial-Meshulam. A proof of the complete complex being an expander.
A. Lubotzky, May 20, 2014.
Linearity is testable. Cocycle tester. Tensor power question.
G. Kalai, May 27, 2014.
Colorful Tverberg's theorem. Fractional Helly's theorem. Ham Sandwich theorem.
Video is to come.