Woong Kook, Combinatorial Laplacians and high dimensional tree numbers

Combinatorial Laplacians provide important enumeration methods in topological combinatorics. For a finite chain complex , combinatorial Laplacians on are defined by </p>

We will review applications of in computing the tree numbers for graphs and in solving discrete Laplace equations for networks. In general, the boundary operators are used to define high-dimensional trees as a generalization of spanning trees for graphs. We will demonstrate an intriguing relation between high-dimensional tree numbers and for acyclic complexes, based on combinatorial Hodge theory. As an application, a formula for the top-dimensional tree-number of matroid complexes will be derived. If time permits, an important role of combinatorial Laplacians in topological data analysis (TDA) will be briefly discussed.