Browsing by Subject "Surfaces, Algebraic"
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Item Error-correcting codes on low néron-severi rank surfaces(2006) Zarzar, Marcos Augusto; Voloch, José FelipeIn this work we construct and estimate the parameters of error-correcting codes on algebraic surfaces whose N´eron-Severi group has low rank. If the rank of the N´eron-Severi group of a surface is 1, the intersection of this surface with an irreducible surface of lower degree will be an irreducible curve, and this allows the construction of ”good” codes and we can have a good estimate for its parameters. Surfaces with rank 1 and many points are not easy to find, but we are able to find some surfaces with low rank that gave us ”good” codes too. We also present an efficient decoding algorithm for such codes. It is based on the realization of the code as an LDPC code, and it was inpired by the Luby-Mitzenmacher algorithm.Item Essential surfaces in hyperbolic three-manifolds(2002-05) Leininger, Christopher Jay; Reid, Alan W.This dissertation is concerned with existence and behavior problems for essential surfaces in hyperbolic three-manifolds. In Chapter 2, we provide new examples of finite volume hyperbolic manifolds which are not fibered, but are virtually fibered. In Chapter 3, we prove that there are Dehn fillings arbitrarily close to infinity in the hyperbolic Dehn surgery space of the figure eight knot complement in which some closed totally geodesic surface compresses.Item Geodesics in the complex of curves of a surface(2002) Leasure, Jason Paige; Gordon, Cameron, 1945-The curve complex of a closed surface S of genus g ≥ 2, C(S), is the complex whose vertices are isotopy classes of simple closed curves on S, and ([x0], . . . , [xn]) is a simplex of C if and only if there are disjoint representatives xi and xj for all i, j. The curve complex of the torus is similar, with ([x0], . . . , [xn]) a simplex if and only if there are representatives xi and xj which meet in a single point for each i = j. We use the path metric on C(S). This dissertation introduces several tools for studying geodesics in the curve complexes of closed orientable surfaces. In the simplest case, when S is a torus, we prove a structure theorem for C(S) and deduce some results about its global geometry. For the higher genus cases, we introduce two methods for approximating distances. The first yields an elementary proof of the known result [10], [6] that the curve complex has infinite diameter, and a constructive means for estimating distance. The second bounds certain intersection numbers and results in an algorithm to compute distance precisely. All results are expressly constructive and elementary.