The course will give an introduction to high dimensional expanders. This is a topic at the intersection of several areas, both math and computer science. We will explore topological, combinatorial, and group theoretic aspects of this topic; as well as applications to computer science.
Lecture 1-2: HDX spectral expansion; link expansion, up-down operators, double sampler. Spectral analysis of expansion in graphs and bipartite graphs ?
Lecture 3: Coboundary expansion definition and the complete complex
Lecture 4: Spherical building
Lecture 5: coboundary expansion of Spherical Building
Lecture 6: Cosystolic expansion in Ramanujan complexes
Lecture 7: Group theoretic construction of HDX (Kaufman-Oppenheim)
Lecture 8: Local testing and HDX (A computer science concept and how it relates to high dimensional expansion)
Lecture 9: Agreement testing (A concept that comes from PCPs yet relates to high dim expansion and to local testing)
Lecture 10: Application of double samplers / HDX : list decoding
Lecture 11: Covering spaces and strong agreement; Bogdanov's example
Lecture 12: Fourier analysis on HDX
more topics
* complement random walk and expander mixing lemma
* isoperimetric inequality for LSV complex; and application to 3LIN lower bound
* Grassmannian and cube vs cube test and graph