Monday, November 12, 2012, 11:30am, Phys

John Steenbergen

- Given a data set, one often assumes the data comes from an underlying space. By imposing some discrete structure on the data, namely, a simplicial complex, one can attempt to study the geometry of the space the data is coming from by studying the geometry of the simplicial complex. Classically, this has been accomplished via graphs (which are 1-dimensional simplicial complexes) through the graph Laplacian, which approximates the Laplace-Beltrami operator on the underlying space. However, higher-dimensional Laplacians exist which could give more information. Recent applications of higher-dimensional Laplacians include statistical ranking and parametrizing data via angular coordinates. In this talk, some recent research on higher-dimensional Laplacians will be discussed. Historically, graph Laplacians were fruitfully studied via Cheeger numbers through the Cheeger inequality. A long-standing open problem has been to extend those results to higher dimensions, a problem which is partly addressed by our work. Finally, as time permits, the behavior of higher-dimensional Laplacians (and Cheeger numbers) will be illustrated via some examples, along with a possible future application of higher-dimensional Laplacians.

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