Semester one

Lecture 1: Overlap.

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.

Lecture 2: Expander graphs.

U. First, October 22, 2013.

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

Lecture 3: Mixing lemma.

A. Lubotzky, October 29, 2013.

Mixing lemma for bipartite biregular graphs.

Lecture 4: Introduction to simplicial complexes.

A. Lubotzky, November 5, 2013.

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

Lecture 5: Universal cover.

A. Lubotzky, November 12, 2013.

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

Lecture 6: Discrepancy.

A. Lubotzky, November 19, 2013.

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

Lecture 7: The affine building of PGL.

U. First, November 26, 2013.

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

Lecture 8: More on the affine building of PGL.

A. Lubotzky, December 3, 2013.

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

Lecture 9: Preparations for the Ramanujan graphs.

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.

Lecture 10: More on Ramanujan graphs.

A. Lubotzky, December 24, 2013.

More on Ramanujan graphs: constructing a lattice.

Lecture 11: Arithmetic groups. Girth of Ramanujan graphs.

A. Lubotzky, December 31, 2013.

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

Lecture 12: Cheeger Inequality for simplicial complexes.

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.

Lecture 13: The chromatic number of Ramanujan graphs. Spectra of the Ramanujan complexes.

A. Lubotzky, January 14, 2014.

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

Semester two

Lecture 14: Overlap theorems.

G. Kalai, February 18, 2014.

There is no video recording of this lecture.

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

Lecture 15: Colored overlap theorems.

G. Kalai, February 25, 2014.

Barany's theorem aka Colored Caratheodory's theorem. Colored Helly's theorem. Tverberg's theorem.

Lecture 16: Pach's theorem, part 1.

G. Kalai, March 4, 2014.

Around Pach's theorem.

Lecture 17: Homology of simplicial complexes and related topology.

G. Kalai, March 11, 2014.

Homology of a simplicial complex. Contractable complexes.

Lecture 18: Collapses. Wagner's theorem.

G. Kalai, March 18, 2014.

A definition of a collapse. Wagner's theorem. Fractional Helly's theorem.

Lecture 19: Pach's theorem, part 2.

G. Kalai, March 25, 2014.

Pach's theorem, cohomology and Laplacians.

Lecture 20: Counting trees.

G. Kalai, May 1, 2014.

Counting trees and Cauchy-Bint theorem.

Lecture 21: An expansion of simplicial complexes.

A. Lubotzky, May 13, 2014.

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

Lecture 22: property testing.
(notes: written, typed)

A. Lubotzky, May 20, 2014.

Linearity is testable. Cocycle tester. Tensor power question.

Lecture 23: Pach's theorem, part 3.

G. Kalai, May 27, 2014.

Colorful Tverberg's theorem. Fractional Helly's theorem. Ham Sandwich theorem.

Lecture 24: To come.

Video is to come.