Forman, Robin2009-06-042009-06-042001Crowley, Katherine Dutton. "Discrete Morse theory and the geometry of nonpositively curved simplicial complexes." (2001) Diss., Rice University. <a href="https://hdl.handle.net/1911/17951">https://hdl.handle.net/1911/17951</a>.https://hdl.handle.net/1911/17951Understanding 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.74 p.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.MathematicsThesisTHESIS MATH. 2001 CROWLEY