High Dimensional Expanders

Explore high-dimensional expanders in this lecture from the Hausdorff Trimester Program on Applied and Computational Algebraic Topology. Delve into two key notions of expansion: coboundary expansion and spectral expansion. Examine their differences in higher dimensions, despite equivalence in the graphical case. Investigate the existence and construction of high-dimensional expanders, estimate expansion in common complexes, and uncover combinatorial and geometric implications. Cover topics such as simplicial cohomology, random complexes, homological connectivity, weighted expansion, topological overlap property, expander graphs and complexes, Latin square complexes, harmonic cochains, and the relationship between eigenvalues and cohomology. Gain insights into this growing field of research with applications in mathematics and theoretical computer science.

Intro
Graphical Spectral Gap
High Dimensional Expansion
Simplicial Cohomology
Expansion of a Complex
A Model of Random Complexes
Homological Connectivity of Random Complexes
Weighted Expansion
The Topological Overlap Property
Topological Overlap and Expansion
Expander Graphs
Expander Complexes
The Complete 3-Partite Complex
Latin Square Complexes
Random Latin Squares Complexes
Large Deviations for Latin Squares
Harmonic Cochains
Eigenvalues and Cohomology
Flag Complexes