Woong Kook, Combinatorial Laplacians and high dimensional tree numbers

Combinatorial Laplacians and high dimensional tree numbers
Woong Kook
Seoul National University
2014/05/08 Thursday 4PM-5PM
Room 1409
Combinatorial Laplacians provide important enumeration methods in topological combinatorics. For a finite chain complex \(\{C_{i},\partial_{i}\}\), combinatorial Laplacians \(\Delta_{i}\) on \(C_{i}\) are defined by

\(\Delta_{i}=\partial_{i+1}\partial_{i+1}^{t}+\partial_{i}^{t}\partial_{i}\, .\)

We will review applications of \(\Delta_{0}\) in computing the tree numbers for graphs and in solving discrete Laplace equations for networks. In general, the boundary operators \(\partial_{i}\) 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 \(\det\Delta_{i}\) 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.

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