02638cam a2200277 4500001000800000005001700008008004100025020001500066035001200081040001800093100002600111245004600137300001000183500002000193500008900213500002800302502006200330520091500392520069501307533007402002650001702076710004002093773005002133856015202183984002502335377323520070613151042.0060808s2003 miu|||||bm||||||| ||eng d a0542252341 a3773235 aUnMbengcUnM1 aDevelin, Michael Lee.10aTopics in discrete geometryh[microform]. a80 p. a(UnM)AAI3183802 aSource: Dissertation Abstracts International, Volume: 66-07, Section: B, page: 3741. aChair: Bernd Sturmfels. aThesis (Ph.D.)--University of California, Berkeley, 2003. aTaking the polytope as our central object of investigation, we delve into three disparate areas of mathematics where this object of rich combinational and geometric structure arises naturally. We first enter the field of linear programming and optimization, presenting a combinatorial investigation of so-called LP-orientations on the graphs of polytopes. These orientations result from directing the edges of graphs by means of some realization in Rn and some linear functional on that space. Holt and Klee [9] constructed a set of conditions satisfied by such orientations, which allowed them to prove the strict monotone Hirsch conjecture in dimension 4; we demonstrate that these conditions are highly insufficient for n-cubes, and present an extension of the conditions in question for crosspolytopes. Next, we construct Cayley compactifications of abelian groups presented by distinguished generators. aThese compactifications are derived from function algebras produced by Cayley graphs; their topological structure in the case of the integer lattice Zn is intimately connected to the polytope P Delta (proper subset of) Rn polar to the convex hull of the generating set. Cayley compactifications also exhibit a strong connection to commutative algebra. The third section is dedicated to the hull complex, a polyhedral complex which produces a canonical free resolution of a monomial k[ x1,..., x n]-module in the Laurent polynomial ring k[ x+/-11 ,..., x+/-1n ]. We resolve a conjecture of Bayer and Sturmfels by showing that the faces of the hull complex have at most n! vertices. aMicrofiche.bAnn Arbor, Mich. :cUniversity Microfilms International. 4aMathematics.2 aUniversity of California, Berkeley.0 tDissertation Abstracts Internationalg66-07B. uhttp://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3183802 aANLcmc 252 31-83802