We consider some aspects of the global geometry of cellular complexes. Motivated by techniques in graph theory, we develop combinatorial versions of isoperimetric and Poincare inequalities, and use them to derive various geometric and topological estimates. This has a progression of three major topics: 1. We define isoperimetric inequalities for normed chain complexes. In the graph case, these quantities boil down to various notions of graph expansion. We also develop some randomized algorithms which provide (in expectation) solutions to these isoperimetric problems. 2. We use these isoperimetric inequalities to derive topological and geometric estimates for certain models of random simplicial complexes. These models are generalizations of the well-known models of random graphs. 3. Using these random complexes as examples, we show that there are simplicial complexes which cannot be embedded into Euclidean space while faithfully preserving the areas of minimal surfaces.

We consider some aspects of the global geometry of cellular complexes. Motivated by techniques in graph theory, we develop combinatorial versions of isoperimetric and Poincare inequalities, and use them to derive various geometric and topological estimates. This has a progression of three major topics: 1. We define isoperimetric inequalities for normed chain complexes. In the graph case, these quantities boil down to various notions of graph expansion. We also develop some randomized algorithms which provide (in expectation) solutions to these isoperimetric problems. 2. We use these isoperimetric inequalities to derive topological and geometric estimates for certain models of random simplicial complexes. These models are generalizations of the well-known models of random graphs. 3. Using these random complexes as examples, we show that there are simplicial complexes which cannot be embedded into Euclidean space while faithfully preserving the areas of minimal surfaces.

No description available.

No description available.

In this thesis the Brill-Noether loci W of rank 2 stable vector bundles of canonical determinant over an algebraic curve are studied. We analyse three conjectures on the nonexistence, dimension and smoothness of W, collectively known as the Brill- Noether conditions. The local structure of W is described by a symmetric Petri map; assuming that W ≠ Ø, the injectivity of this map ensures the dimension and smoothness conditions we are aiming for. The nonemptiness of W is shown by constructing the appropriate bundles from extensions of line bundles. In a similar vein the nonexistence conjecture is addressed by showing that certain bundles are extensions of line bundles that are prohibited on the curve. Finally, subject to an assumption, the Petri map is shown to be injective for genus ≤10; which allows us to prove that the Brill-Noether conditions hold for genus ≤10, improving on the genus ≤7 results of Bertram-Feinberg [4] .

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No description available.

Thesis (M.F.A.)--University of Hawaii at Manoa, 2004. / Includes bibliographical references. / vi, 23 leaves, bound col. plates 29 cm

Thesis (Ph.D.)--Tufts University, 2005. / Submitted to the Dept. of Computer Science. Adviser: Diane L. Souvaine. Includes bibliographical references (leaves 154-169). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

Thesis (Ph. D.)--University of Adelaide, 1988. / Includes bibliographical references (leaves 211-217).