Search Results (1 - 6 of 6 Results)

Sort By  
Sort Dir
 
Results per page  

Miller, Jason AOkounkov Bodies of Borel Orbit Closures in Wonderful Group Compactifications
Doctor of Philosophy, The Ohio State University, 2014, Mathematics
This thesis draws a connection between two areas of algebraic geometry, spherical varieties and Okounkov bodies, in order to study the structure of Borel orbit closures in wonderful group compactifications. Spherical varieties are a natural generalization of many classes of varieties equipped with group actions such as flag varieties, symmetric varieties, and toric varieties. The theory of Okounkov bodies is a fascinating recent development generalizing the polytopes that appear in toric geometry to any projective algebraic variety. Let X be a projective spherical G-variety equipped with a very ample G-line bundle L. Choosing a reduced decomposition of the longest element of the Weyl group determines a valuation vN on the ring of sections, R(X,L). One can then use Okounkov theory to encode information about the G-orbits of a spherical variety in terms of the associated Newton polytope. Each G-orbit closure of X determines a face of the Newton polytope. This correspondence allows one to use the combinatorial methods of convex geometry to answer questions about the G-orbit closures of the spherical variety X. However for nontoric spherical varieties, the G-orbit structure is too coarse-grained. A great deal of information about the spherical variety, such as the intersection theory, is determined by the structure of the Borel orbits. In this thesis we consider wonderful group compactifications. We prove that one can extend the correspondence between G-orbits and faces to the Borel orbits for this class of varieties. Given any Borel orbit closure of a wonderful group compactification, we show that the Okounkov construction will yield a finite union of faces of the Newton polytope. This correspondence can be shown to enjoy many of the same nice properties as in the case of G-orbits: the dimension of the space of global sections of L is given by the number of lattice points in the union of faces, and the degree of any Borel orbit closure is the sum of the normalized volumes of the associated faces.

Committee:

Gary Kennedy, Ph.D. (Advisor); Roy Joshua, Ph.D. (Committee Member); James Cogdell, Ph.D. (Committee Member)

Subjects:

Mathematics

Keywords:

spherical varieties; Okounkov ; wonderful group compactifications; Borel orbits; toric varieties; flag varieties; Newton-Okounkov; polytopes; string polytope; moment polytope; algebraic geometry; Schubert varieties; standard monomial theory; crystal basis

Hoehner, Steven DouglasThe Surface Area Deviation of the Euclidean Ball and a Polytope
Doctor of Philosophy, Case Western Reserve University, 2016, Mathematics
While there is extensive literature on approximation of convex bodies by inscribed or circumscribed polytopes, much less is known in the case of generally positioned polytopes. In this thesis we give upper and lower bounds for the approximation of the Euclidean unit ball by arbitrarily positioned polytopes with a fixed number of vertices or facets in the surface area deviation. The main results of this paper are joint work with Dr. Carsten Schuett and Dr. Elisabeth Werner. In the introduction, we collect some definitions and background material from analysis and convex geometry that will be used throughout the paper. The main results of this dissertation are stated in Chapter 1. In Chapter 2, we collect several results related to our main results. Our focus there will be on results related to best and random approximation with respect to the volume, surface area, and mean width deviations. In Chapter 3 we state several auxiliary lemmas needed for the proofs of our main results. The proofs of our main results are given in Chapters 4 and 5.

Committee:

Elisabeth Werner (Advisor); Carsten Schuett (Committee Member); Woyczynski Wojbor (Committee Member); Mathur Harsh (Committee Member)

Subjects:

Mathematics

Keywords:

convexity; convex body; polytope; approximation; surface area; surface area deviation; Euclidean ball; convex geometry

Knapp, GregMinkowski's Linear Forms Theorem in Elementary Function Arithmetic
Master of Sciences, Case Western Reserve University, 2017, Mathematics
A classical question in formal logic is "how much mathematics do we need to know in order to prove a given theorem?" Of particular interest is Harvey Friedman's grand conjecture: that every known mathematical theorem involving only finitary mathematical objects can be proven from Elementary Function Arithmetic (EFA)—a fragment of Peano Arithmetic. If Friedman's conjecture is correct, this would imply that Fermat's Last Theorem is derivable from the axioms of EFA. The vast task of proving Fermat's Last Theorem from the axioms of EFA seems to require certain theorems on convex polytopes and an important corollary: Minkowski's Linear Forms Theorem. I show that certain theorems of convex geometry—e.g. representation of polytopes both in vertex form and as the intersection of half-spaces, monotonicity of volume, the existence of a separating plane between disjoint polytopes, and Minkowski's Linear Forms Theorem—can be both interpreted and derived in EFA, assuming the well-definedness of volume of a convex polytope along with one other technical lemma.

Committee:

Colin McLarty (Advisor); Mark Meckes (Committee Member); David Singer (Committee Member); Elisabeth Werner (Committee Member)

Subjects:

Logic; Mathematics

Keywords:

minkowskis linear forms theorem; elementary function arithmetic; first-order arithmetic; convex geometry; polytope; volume; triangulation

Reisdorf, Stephen R.Cellohedra
Master of Science, University of Akron, 2012, Mathematics
The associahedron has been generalized to a great variety of combinatorial structures. In each example the convex polytope is found whose face poset is the same as a certain poset structure on the combinatorial structures. Here we find polytopes whose face poset models the containment order of certain order ideals of the face poset of a cell complex. This is progress in a program which asks which posets in general have their ideals modeled by convex polytopes.

Committee:

Stefan Forcey, Dr. (Advisor); James P. Cossey, Dr. (Committee Member); Jeffrey Riedl, Dr. (Committee Member)

Subjects:

Mathematics

Keywords:

associahedra; associahedron; graph associahedra; graph associahedron; pseudograph associahedron; cell complex; geometric combinatorics; polytope; polytopes

Cimren, EmrahValid Inequalities for The 0-1 Mixed Knapsack Polytope with Upper Bounds
Doctor of Philosophy, The Ohio State University, 2010, Industrial and Systems Engineering

The polyhedral structure of the convex hull of the 0-1 mixed knapsack polytope defined by a knapsack inequality with continuous and binary variables with upper bounds is investigated. This polytope arises as subpolytope of more general mixed integer problems such as network flow problems, facility location problems, and lotsizing problems.

Two sets of valid inequalities for the polytope are developed. We derive the first set of valid inequalities by adding a new subset of variables to the flow cover inequalities. We show that, under some conditions, this set is facet defining. Computational results show the effectiveness of these inequalities.

The second set of valid inequality is generated by sequence independent lifting of the flow cover inequalities. We show that computing exact lifting coefficients is NP-hard. As a result, an approximate lifting procedure is developed. We give computational results that show the effectiveness of the valid inequalities and the lifting procedure.

Committee:

Marc Posner (Committee Chair); Nicholas Hall (Committee Member); Simge Kucukyavuz (Committee Member)

Subjects:

Industrial Engineering

Keywords:

valid inequalities;polyhedral theory;mixed integer programming;knapsack polytope;sequence independent lifting

Berry, Lisa TredwayPainted Trees and Pterahedra
Master of Science, University of Akron, 2013, Mathematics
Associahedra can be realized by taking the convex hull of coordinates derived from binary trees. Similarly, permutahedra can be found using leveled trees. In this paper we will introduce a new type of painted tree, (T ◦ Y)n where n is the number of interior nodes. We create these painted trees by composing binary trees on leveled trees. We define a coordinate system on these trees and take the convex hull of these points. We explore the resulting polytope and prove, using a bijection to tubings, that for n ≤ 4 the poset of the painted face trees with n+1 leaves is isomorphic to the face poset of an n-dimensional polytope, specifically KF1,n, the graph-associahedron for a fan graph, F1,n.

Committee:

Stefan Forcey, Dr. (Advisor); W. Stuart Clary, Dr. (Committee Member); Hung Nguyen, Dr. (Committee Member)

Subjects:

Mathematics

Keywords:

painted trees; polytope; pterahedra; graph-associahedra; tubings