M.Sc Thesis
M.Sc Student Lew Alan Spectral Gaps of Generalized Flag Complexes Department of Mathematics Professor Roy Meshulam

Abstract

Let X be a simplicial complex on vertex set V of size n. The missing faces of X are the subsets of V not contained in X that are minimal with respect to inclusion. Assume that all the missing faces of X are of dimension at most d. Let Lj denote the j-Laplacian acting on real j-cochains of X. We call the minimal eigenvalue of Lj the j-th spectral gap of  X.  A classical result of Garland relates the spectral gaps of a complex with the spectral gaps of the links of its faces. Our main result is a global counterpart of Garland’s result , connecting between the k-th spectral gaps (for k ≥ d) and the (d-1)-th spectral gap of X. These results extend theorems of Aharoni , Berger and Meshulam for flag complexes (the case d = 1) . As an application we prove a fractional extension of a Hall-type theorem of Holmsen , Martınez-Sandoval and Montejano for general position sets in matroids .

We also prove a different lower bound on the k-th spectral gaps, in terms of the number of vertices n and the minimal degree of a k-dimensional face. This bound follows by an application of Gersgorin’s circle theorem to the k-Laplacian .

The last part of the thesis is dedicated to the study of some families of simplicial complexes arising from finite geometries, which have interesting spectral and homological properties .