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dc.contributor.advisor Forman, Robin
dc.creatorCrowley, Katherine Dutton
dc.date.accessioned 2009-06-04T06:57:55Z
dc.date.available 2009-06-04T06:57:55Z
dc.date.issued 2001
dc.identifier.urihttp://hdl.handle.net/1911/17951
dc.description.abstract Understanding the conditions under which a simplicial complex collapses is a central issue in many problems in topology and combinatorics. Let K be a simplicial complex endowed with the piecewise Euclidean geometry given by declaring edges to have unit length, and satisfying the property that every 2-simplex is a face of at most two 3-simplices in K. Our main theorem is that if |K| is nonpositively curved (in the sense of CAT(0)) then K simplicially collapses to a point. The main tool used in the proof is Forman's discrete Morse theory (see section 2.2), a combinatorial version of the classical smooth theory. A key ingredient in our proof is a combinatorial analog of the fact that a minimal surface in R3 has nonpositive Gauss curvature (see theorem 28). We also investigate another combinatorial question related to curvature. We prove a combinatorial isoperimetric inequality by finding an exact answer for the largest possible number of interior vertices in a triangulated n-gon satisfying the property that every interior vertex has degree at least six.
dc.format.extent 74 p.
dc.format.mimetype application/pdf
dc.language.iso eng
dc.subjectMathematics
dc.title Discrete Morse theory and the geometry of nonpositively curved simplicial complexes
dc.type.genre Thesis
dc.type.material Text
thesis.degree.department Mathematics
thesis.degree.discipline Natural Sciences
thesis.degree.grantor Rice University
thesis.degree.level Doctoral
thesis.degree.name Doctor of Philosophy
dc.identifier.citation Crowley, Katherine Dutton. "Discrete Morse theory and the geometry of nonpositively curved simplicial complexes." (2001) Diss., Rice University. http://hdl.handle.net/1911/17951.


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