Title: Eigenvalues of high dimensional Laplacian operators
| Speaker: | Alan Lew |
| Affiliation: | Carnegie Melon University |
| Location: | Please contact Sabrina Lato for Zoom link. |
Abstract: A simplicial complex is a topological space built by gluing together simple building blocks (such as vertices, edges, triangles and their higher dimensional counterparts). Alternatively, we can define a simplicial complex combinatorially, as a family of finite sets that is closed under inclusion. In 1944, Eckmann introduced a class of high dimensional Laplacian operators acting on a simplicial complex. These operators generalize the Laplacian matrix of a graph (which can be seen as a 0-dimensional Laplacian), and are strongly related to the topology of the complex (and in particular, to its homology groups).
In this talk, I will show different ways in which one can obtain information on the spectrum of a high dimensional Laplacian operator on a simplicial complex from the spectra of its lower dimensional Laplacians, and present several applications to the study of the topology of the complex. In addition, I will discuss one of our main technical tools, the use of "additive compound matrices" for studying sums of eigenvalues of a linear operator, which may be of independent interest.