Title: New approaches to Garland’s method
Abstract:
The k-dimensional Laplacian of a simplicial complex X, introduced by Eckmann in 1944, is a linear operator acting on the space of k-chains of X, which is closely related to the topology of the complex. In 1973, Garland developed a method for estimating the spectral gaps of high-dimensional Laplacian operators of a complex X in terms of the spectral gaps of lower-dimensional Laplacians of its links (certain “local” subcomplexes of X). Since its introduction, numerous extensions and variants of Garland’s method have been developed, with applications in areas such as group theory, matching theory, the study of random simplicial complexes, and the analysis of random walks. In this talk, I will give an accessible introduction to Laplacian operators on simplicial complexes and to Garland’s method, and then present some new results, focusing on new techniques for proving Garland-type theorems.