#### Event Title

The Spectrum of Sparse Random Complex

#### Session Number

O04

#### Advisor(s)

Dominic Dotterrer, University of Chicago

#### Location

B-116

#### Start Date

28-4-2016 9:15 AM

#### End Date

28-4-2016 9:40 AM

#### Abstract

The combinatorial Laplacian is an operator that has numerous applications in physics, finance, randomized algorithms, and graph theory. It contains information that describes certain properties of the graph. One important set of information is its set of eigenvalues (also known as its spectrum). For example, one can determine the relative connectivity of a graph with this. Furthermore, one can determine global geometric properties through the spectrum. The graph Laplacian can be extended to simplicial complexes (higher dimension versions of graphs). The importance of the graph Laplacian therefore leads us to study the Laplacian for simplicial complexes. Through this operator, one can quantitatively measure the vanishing of cohomology – in other words a measure of higher dimensional connectivity. This investigation will present the ideas that have been researched

The Spectrum of Sparse Random Complex

B-116

The combinatorial Laplacian is an operator that has numerous applications in physics, finance, randomized algorithms, and graph theory. It contains information that describes certain properties of the graph. One important set of information is its set of eigenvalues (also known as its spectrum). For example, one can determine the relative connectivity of a graph with this. Furthermore, one can determine global geometric properties through the spectrum. The graph Laplacian can be extended to simplicial complexes (higher dimension versions of graphs). The importance of the graph Laplacian therefore leads us to study the Laplacian for simplicial complexes. Through this operator, one can quantitatively measure the vanishing of cohomology – in other words a measure of higher dimensional connectivity. This investigation will present the ideas that have been researched